EP3706119A1 - Spatialised audio encoding with interpolation and quantifying of rotations - Google Patents

Abstract

The invention relates to the compression coding of sound signals forming a succession in time of frames of samples, in each of N channels in ambisonic representation of order greater than 0, the method comprising: - forming, from the channels for a current frame, a covariance matrix between channels and find (S3) eigenvectors of the covariance matrix to obtain an eigenvector matrix, - test (S5) the eigenvector matrix to verify that it represents a rotation in a space of dimension N and correct (S6) if not the matrix of eigenvectors until a rotation matrix is obtained, for the current frame, and - apply said rotation matrix (S7) to the signals of the N channels before encoding by channels separated from said signals.

Description

The present invention relates to the encoding / decoding of spatialized sound data, in particular in a surround sound context (hereinafter also referred to as “ambisonic”).

The coders / decoders (hereinafter called “codecs”) which are currently used in mobile telephony are mono (a single signal channel for reproduction on a single loudspeaker). The 3GPP EVS (for “Enhanced Voice Services”) codec makes it possible to offer “Super-HD” quality (also called “High Definition +” or HD + voice) with an audio band in super-wide band (SWB for “super- wideband "in English) for signals sampled at 32 or 48 kHz or full band (FB for" Fullband ") for signals sampled at 48 kHz; the audio bandwidth is 14.4 to 16 kHz in SWB mode (9.6 to 128 kbit / s) and 20 kHz in FB mode (16.4 to 128 kbit / s).

The next quality development in conversational services offered by operators should be immersive services, using terminals such as smartphones, for example, equipped with several microphones or spatialized audio conferencing or tele-presence type videoconferencing equipment. , or even “live” content sharing tools, with spatialized 3D sound rendering, which is far more immersive than a simple 2D stereo rendering. With the more and more widespread use of listening on mobile phones with an audio headset and the appearance of advanced audio equipment (accessories such as a 3D microphone, voice assistants with acoustic antennas, virtual reality headsets, etc.) and specific tools (for example the production of 360 ° video content) the capture and rendering of spatialized sound scenes are now common enough to offer an immersive communication experience.

• Format multicanal (channel-based en anglais) de type stéréo, 5.1 où chaque canal vient alimenter un haut-parleur (par exemple L et R en stéréo, ou L, R, Ls, Rs et C en 5.1)
• Format objet (object-based en anglais) où des objets sonores sont décrits comme un signal audio (en général mono) associé à des métadonnées décrivant les attributs de cet objet (position dans l'espace, largeur spatiale de la source, etc.), et
• Format ambisonique (scene-based en anglais) qui décrit le champ sonore en un point donné, en général capté par un microphone sphérique ou synthétisé dans le domaine des harmoniques sphériques.

As such, the future 3GPP standard "IVAS" (for "Immersive Voice And Audio Services") proposes the extension of the EVS codec to immersive by accepting as input format of the codec at least the spatialized sound formats listed below. below (and their combinations):

• Multichannel format (channel-based in English) of stereo type, 5.1 where each channel feeds a speaker (for example L and R in stereo, or L, R, Ls, Rs and C in 5.1)
• Object-based format where sound objects are described as an audio signal (generally mono) associated with metadata describing the attributes of this object (position in space, spatial width of the source, etc.) , and
• Ambisonic format (scene-based in English) which describes the sound field at a given point, generally picked up by a spherical microphone or synthesized in the domain of spherical harmonics.

Hereinafter, we are typically interested in the coding of a sound in ambisonic format, by way of example of an embodiment (at least certain aspects presented in connection with the invention below can also be applied to other formats. than ambisonics).

Ambisonics is a recording method ("encoding" in the acoustic sense) of spatialized sound and a reproduction system ("decoding" in the acoustic sense). An ambisonic microphone (at order 1) comprises at least four capsules (typically of the cardoid or sub-cardoid type) arranged on a spherical grid, for example the vertices of a regular tetrahedron. The audio channels associated with these capsules are called “A-format”. This format is converted into a “B-format”, in which the sound field is broken down into four components (spherical harmonics) denoted W, X, Y, Z, which correspond to four coincident virtual microphones. The W component corresponds to an omnidirectional capture of the sound field while the X, Y and Z components, which are more directive, are comparable to pressure gradients oriented along the three dimensions of space. An ambisonic system is a flexible system in the sense that recording and playback are separate and decoupled. It allows decoding (in the acoustic sense) on any speaker configuration (for example, binaural, 5.1-type surround sound or 7.1.4-type periphery (with elevation)). Of course, the ambisonic approach can be generalized to more than four channels in B-format and this generalized representation is commonly called “HOA” (for “Higher-Order Ambisonics”). Breaking down the sound into more spherical harmonics improves the spatial accuracy of reproduction when rendering on loudspeakers.

An ambisonic signal at order N comprises (N + 1) ² components and, at order 1 (if N = 1), we find the four components of the original ambisonic which is commonly called FOA (for First-Order Ambisonics). There is also a so-called "planar" variant of ambisonics which breaks down the sound defined in a plane which is generally the horizontal plane. In this case, the number of components is 2N + 1 channels. The first order ambisonics (4 channels: W, X, Y, Z) and the first order planar ambisonics (3 channels: W, X, Y) are hereinafter referred to as “ambisonics” indiscriminately to facilitate reading, the treatments presented being applicable regardless of planar type or not. If, however, in some passages it is necessary to make a distinction, the terms "first-order ambisonics" and "first-order planar ambisonics" are used. Note that we can derive from the 1st order B-format a stereo signal (2 channels) corresponding to coincident stereo pickups of the Blumlein Crossed Pair (X + Y and XY) or Mid-Side type (by combining W and X for Mid and taking Y as Side).

In the following, a B-format signal with a predetermined order is called “ambisonic sound”. In variations, the ambisonic sound can be defined in another format such as A-format or pre-combined channels by fixed matrixing (keeping the number of channels or reducing it to a 3 or 2 channel case), as will be seen. further.

The signals to be processed by the encoder / decoder are presented as successions of blocks of sound samples called “frames” or “sub-frames” below.

• Vecteur : u (minuscule, gras)
• Matrice : A (majuscule, gras)

In addition, hereafter, mathematical notations follow the following convention:

• Vector: u ( lowercase, bold )
• Matrix: A ( uppercase, bold )

The simplest approach to encoding a stereo or ambisonic signal is to use a mono encoder and apply it in parallel to all channels with possibly a different bit allocation depending on the channels. This approach is called here “multi-mono” (even if in practice we can generalize the approach to multi-stereo or a use of several parallel instances of the same core codec).

Such an achievement is presented at the figure 1 . The input signal is divided into channels (mono) by block 100. These channels are individually encoded by blocks 120 through 122 according to a predetermined allocation. Their binary train is multiplexed (block 130) and after transmission and / or storage it is demultiplexed (block 140) to apply a decoding of each of the channels (blocks 150 to 152) which are recombined (block 160).
The associated quality varies according to the mono coding used, and it is generally satisfactory only at very high speed, for example with a data rate of at least 48 kbit / s per mono channel for an EVS coding. Thus at order 1 we obtain a minimum bit rate of 4x48 = 192 kbit / s.

The solutions currently proposed for more sophisticated codecs, for ambisonic spatialization in particular, are not satisfactory, in particular in terms of complexity, delay and efficient use of the bit rate, to ensure efficient decorrelation between ambisonic channels.

For example, the MPEG-H codec for ambisonic sounds uses an add-overlap operation which adds delay and complexity, as well as linear interpolation on direction vectors which is suboptimal and introduces defects. A basic problem with this codec is that it implements a decomposition into predominant components and ambience because the predominant components are supposed to be perceptually distinct from ambience, but this decomposition is not completely specified. The MPEG-H encoder suffers from the problem of non-correspondence between the directions of the principal components from one frame to another: the order of the components (signals) can be swapped just like the associated directions. This is the reason why the MPEG-H codec uses a “matching” and overlap-add technique in order to solve this problem.

Furthermore, it would be possible to use frequency coding approaches (in the FFT or MDCT domain) rather than a temporal coding as in the MPEG-H codec, but signal processing in the frequency domain (sub-bands) makes it necessary to transmit data by sub-band to a decoder, thus increasing the bit rate necessary for this transmission.

• former, à partir des canaux pour une trame courante, une matrice de covariance entre canaux et rechercher des vecteurs propres de la matrice de covariance pour obtenir une matrice de vecteurs propres,
• tester la matrice de vecteurs propres pour vérifier qu'elle représente une rotation dans un espace de dimension N et corriger sinon la matrice de vecteurs propres jusqu'à obtenir une matrice de rotation, pour la trame courante, et
• appliquer ladite matrice de rotation aux signaux des N canaux avant un encodage par canaux séparés desdits signaux.

The present invention improves this situation.
To this end, it proposes a method of coding in compression of sound signals forming a succession in time of frames of samples, in each of N channels in ambisonic representation of order greater than 0, the method comprising:

• form, from the channels for a current frame, a covariance matrix between channels and search for eigenvectors of the covariance matrix to obtain a matrix of eigenvectors,
• test the eigenvector matrix to verify that it represents a rotation in a space of dimension N and otherwise correct the eigenvector matrix until obtaining a rotation matrix, for the current frame, and
• applying said rotation matrix to the signals of the N channels before encoding by separate channels of said signals.

Thus, the present invention makes it possible to improve a decorrelation between the N channels to be encoded separately subsequently. This separate encoding is hereinafter also referred to as “multi-mono encoding”.

• coder des paramètres tirés de la matrice de rotation en vue d'une transmission via un réseau. Ces paramètres peuvent être typiquement des valeurs de quaternion et/ou d'angle de rotation et/ou d'angle d'Euler comme on le verra plus loin, ou encore simplement des éléments de cette matrice par exemple.

In one embodiment, the method may further include:

• encoding parameters taken from the rotation matrix for transmission over a network. These parameters can typically be quaternion and / or rotation angle and / or Euler angle values as will be seen below, or even simply elements of this matrix for example.

• comparer la matrice de vecteurs propres obtenue pour la trame courante à une matrice de rotation obtenue pour une trame précédant la trame courante, et
• permuter des colonnes de la matrice de vecteurs propres de la trame courante pour assurer une cohérence avec la matrice de rotation de la trame précédente.

Une telle réalisation permet de conserver une homogénéité globale et d'éviter notamment des clics audibles d'une trame à l'autre, pendant la restitution sonore.In one embodiment, the method may further include:

• compare the eigenvector matrix obtained for the current frame with a rotation matrix obtained for a frame preceding the current frame, and
• permute columns of the eigenvector matrix of the current frame to ensure consistency with the rotation matrix of the previous frame.

Such an embodiment makes it possible to maintain overall homogeneity and in particular to avoid audible clicks from one frame to another, during sound reproduction.

• vérifier, pour chaque vecteur propre de la trame courante, une cohérence de direction avec un vecteur-colonne de position correspondante de la matrice de rotation de la trame précédente, et
• en cas d'incohérence, inverser le signe des éléments de ce vecteur propre dans la matrice de vecteurs propres de la trame courante.

However, certain transformations implemented to obtain the eigenvectors from the covariance matrix (such as the “PCA / KLT” seen later) are likely to reverse the direction of some of the eigenvectors and it is then necessary to check at the same time a coherence of axis, then of direction on this axis, of each eigenvector of the matrix of the current frame. To this end, in one embodiment, the aforementioned permutation of the columns already making it possible to ensure a coherence of the axes of the vectors, the method further comprises:

• check, for each eigenvector of the current frame, a coherence of direction with a column vector of corresponding position of the rotation matrix of the previous frame, and
• in the event of inconsistency, invert the sign of the elements of this eigenvector in the matrix of eigenvectors of the current frame.

Typically, a permutation between columns of the eigenvector matrix inverting the sign of a determinant of the eigenvector matrix and the determinant of a rotation matrix being equal to 1,
we can estimate the determinant of the eigenvector matrix, and if the latter is equal to -1, we can then invert the signs of the elements of a chosen column of the eigenvector matrix, so that the determinant is equal to 1 and thereby form a rotation matrix.

• une estimation d'écart entre la matrice de rotation obtenue pour la trame courante et une matrice de rotation obtenue pour une trame précédant la trame courante,
• en fonction de l'écart estimé, déterminer si au moins une interpolation est à opérer entre la matrice de rotation de la trame courante et la matrice de rotation de la trame précédente.

Une telle interpolation permet alors de lisser (« moyenner progressivement ») les matrices de rotation appliquées respectivement à la trame précédente et la trame courante et atténuer ainsi un effet de clic audible d'une trame à l'autre à la restitution.In one embodiment, the method may further comprise:

• an estimate of the difference between the rotation matrix obtained for the current frame and a rotation matrix obtained for a frame preceding the current frame,
• as a function of the estimated difference, determining whether at least one interpolation is to be made between the rotation matrix of the current frame and the rotation matrix of the previous frame.

Such an interpolation then makes it possible to smooth (“progressively average”) the rotation matrices applied respectively to the previous frame and the current frame and thus to attenuate an audible click effect from one frame to another on playback.

• en fonction de l'écart estimé, il est déterminé un nombre d'interpolations à opérer entre la matrice de rotation de la trame courante et la matrice de rotation de la trame précédente,
• la trame courante est découpée en un nombre de sous-trames correspondant au nombre d'interpolations à opérer, et
• on peut coder au moins ce nombre d'interpolations en vue d'une transmission via le réseau précité.

In such a realization:

• as a function of the estimated difference, a number of interpolations to be made between the rotation matrix of the current frame and the rotation matrix of the previous frame is determined,
• the current frame is divided into a number of sub-frames corresponding to the number of interpolations to be operated, and
• at least this number of interpolations can be coded with a view to transmission via the aforementioned network.

In one embodiment, the ambisonic representation is of order 1 and the number N of channels is four, and the rotation matrix of the current frame is represented by two quaternions.

In this embodiment and in the case of an interpolation, each interpolation for a current sub-frame is a linear spherical interpolation (or “SLERP”), carried out as a function of the interpolation of the sub-frame preceding the sub-frame. current frame and from the quaternions of the previous subframe.

$$Q_{R,\mathit{interp}}\left(\alpha \right)=Q_{R,t-1}\frac{{\sin \left(1-\alpha \right)\mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}+Q_{R,t}\frac{{\sin \alpha \mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}$$

Où:

• Q _L,t-1 est l'un des quaternions de la sous-trame précédente t-1,
• Q _R,t-1 est l'autre des quaternions de la sous-trame précédente t-1,
• Q _L,t est l'un des quaternions de la sous-trame courante t,
• Q_R,t est l'autre des quaternions de la sous-trame courante t,
• Ω _L = Arccos (Q _L,t-1 · Q _L,t ) ; Ω _R = Arccos (Q _R,t-1 · Q_R,t )
• et α correspond à un facteur d'interpolation.

For example, linear spherical interpolation of the current subframe can be conducted to obtain quaternions of the current subframe as follows: $$Q_{L,\mathit{interp}}\left(\alpha \right)=Q_{L,t-1}\frac{{\sin \left(1-\alpha \right)\mathrm{\Omega }}_L}{{\mathrm{sin\Omega }}_L}+Q_{L,t}\frac{{\sin \alpha \mathrm{\Omega }}_L}{{\mathrm{sin\Omega }}_L}$$

$$Q_{R,\mathit{interp}}\left(\alpha \right)=Q_{R,t-1}\frac{{\sin \left(1-\alpha \right)\mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}+Q_{R,t}\frac{{\sin \alpha \mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}$$

Or:

• Q _{L, t -1} is one of the quaternions of the previous subframe t-1,
• Q _{R , t -1} is the other of the quaternions of the previous subframe t-1,
• Q _{L , t} is one of the quaternions of the current subframe t,
• Q _{R, t} is the other of the quaternions of the current sub-frame t,
• Ω _L = Arccos (Q _{L , t -1} · Q _{L , t} ); Ω _R = Arccos ( Q _{R, t} -1Q _{R, t} )
• and α corresponds to an interpolation factor.

In one embodiment, the search for the eigenvectors is performed by principal component analysis (or "PCA") or by Karhunen Loeve transform (or "KLT"), in the time domain.
Of course, other embodiments can be considered (decomposition into singular values, or others).

• pour chaque canal ambisonique, une estimation d'énergie acoustique courante dans le canal,
• la sélection dans une mémoire d'un score prédéterminé, de qualité, fonction de ce canal ambisonique et d'un débit courant dans le réseau,
• l'estimation d'une pondération à opérer pour l'allocation de bits à ce canal, par multiplication du score sélectionné à l'énergie estimée.

Cette réalisation permet alors de gérer une allocation de bits optimale à attribuer pour chaque canal à coder. Elle est avantageuse en tant que telle et pourrait éventuellement faire l'objet d'une protection séparée.In one embodiment, the method comprises a preliminary step of forecasting the budget for allocation of bits per ambisonic channel, comprising:

• for each ambisonic channel, an estimate of current acoustic energy in the channel,
• the selection in a memory of a predetermined score, of quality, function of this ambisonic channel and of a current flow in the network,
• estimating a weighting to be operated for the allocation of bits to this channel, by multiplying the selected score at the estimated energy.

This embodiment then makes it possible to manage an optimal allocation of bits to be allocated for each channel to be coded. It is advantageous as such and could possibly be the subject of separate protection.

• recevoir, pour une trame courante, en plus des signaux des N canaux de cette trame courante, des paramètres d'une matrice de rotation,
• construire une matrice de rotation inverse à partir desdits paramètres,
• appliquer ladite matrice de rotation inverse à des signaux issus des N canaux reçus, avant un décodage par canaux séparés desdits signaux.

Une telle réalisation permet d'améliorer aussi au décodage une décorrélation entre les N canaux.The present invention also relates to a method of decoding sound signals forming a succession in time of frames of samples, in each of N channels in ambisonic representation of order greater than 0, the method comprising:

• receive, for a current frame, in addition to the signals of the N channels of this current frame, parameters of a rotation matrix,
• construct an inverse rotation matrix from said parameters,
• applying said inverse rotation matrix to signals from the N received channels, before separate channel decoding of said signals.

Such an embodiment also makes it possible to improve on decoding a decorrelation between the N channels.

The present invention is also aimed at a coding device comprising a processing circuit for implementing the coding method presented above.

It also relates to a decoding device comprising a processing circuit for implementing the above decoding method.

It also relates to a computer program comprising instructions for implementing the above method, when these instructions are executed by a processor of a processing circuit.
It also relates to a non-transient memory medium storing the instructions of such a computer program.

• la figure 1 illustre un codage multi-mono (état de l'art),
• la figure 2 illustre une succession d'étapes principales d'un exemple procédé au sens de l'invention,
• la figure 3 présente la structure générale d'un exemple de codeur selon l'invention,
• la figure 4 présente détaille l'analyse et la transformation PCA/KLT réalisée par le bloc 310 du codeur de la figure 3,
• la figure 5 présente un exemple de décodeur selon l'invention,
• la figure 6 présente le décodage et la synthèse PCA/KLT inverse de la figure 4, au décodage,
• la figure 7 illustre des exemples de réalisation structurelle d'un codeur et d'un décodeur au sens de l'invention.

Other advantages and characteristics and characteristics of the invention will become apparent on reading the exemplary embodiments presented in the detailed description below, and on examining the appended drawings in which:

• the figure 1 illustrates multi-mono coding (state of the art),
• the figure 2 illustrates a succession of main steps of an example process within the meaning of the invention,
• the figure 3 presents the general structure of an example of an encoder according to the invention,
• the figure 4 presents details the analysis and the PCA / KLT transformation carried out by the block 310 of the encoder of the figure 3 ,
• the figure 5 presents an example of a decoder according to the invention,
• the figure 6 presents the reverse PCA / KLT decoding and synthesis of the figure 4 , when decoding,
• the figure 7 illustrates structural embodiments of a coder and a decoder within the meaning of the invention.

• un matriçage adaptatif en temporel (en particulier avec une transformation adaptative obtenue par PCA/KLT (« PCA » désignant une analyse en composante principale et « KLT » désignant une transformée de Karhunen Loeve),
• suivi préférentiellement par un codage multi-mono.

Le matriçage adaptatif permet une décomposition en canaux plus efficaces qu'un matriçage fixe. Le matriçage selon l'invention permet avantageusement de décorréler les canaux avant codage multi-mono, de sorte que le bruit de codage introduit par le codage de chacun des canaux déforme globalement le moins possible l'image spatiale lorsque les canaux sont recombinés pour reconstruire un signal ambisonique au décodage.
De plus, l'invention permet d'assurer une adaptation douce des paramètres de matriçage afin d'éviter des artéfacts de type « clics » en bordure de trame ou des fluctuations trop rapides d'image spatiale, ou encore des artéfacts de codage dus à des variations trop fortes (par exemple liées à des permutations intempestives de sources sonores entre canaux) dans les différents canaux individuels issus du matriçage qui sont ensuite codés par des instances différentes d'un codec mono. Il est présenté ci-après un codage multi-mono avec allocation préférentiellement variable des bits entre canaux (après matriçage adaptatif), mais dans des variantes plusieurs instances d'un codec coeur stéréo ou autre peuvent être utilisées.The invention aims to allow an optimized coding by:

• a temporal adaptive matrixing (in particular with an adaptive transformation obtained by PCA / KLT (“PCA” designating a principal component analysis and “KLT” designating a Karhunen Loeve transform),
• preferably followed by multi-mono coding.

Adaptive matrixing allows more efficient channelization than fixed matrixing. The matrixing according to the invention advantageously makes it possible to decorrelate the channels before multi-mono coding, so that the coding noise introduced by the coding of each of the channels globally distorts the spatial image as little as possible when the channels are recombined to reconstruct a ambisonic signal on decoding.
In addition, the invention makes it possible to ensure a gentle adaptation of the mastering parameters in order to avoid "click" type artifacts at the edge of the frame or too rapid fluctuations in the spatial image, or even coding artifacts due to too strong variations (for example linked to untimely permutations of sound sources between channels) in the various individual channels resulting from the mastering which are then coded by different instances of a mono codec. Multi-mono coding is presented below with preferentially variable allocation of the bits between channels (after adaptive matrixing), but in variants several instances of a stereo or other core codec can be used.

In order to facilitate understanding of the invention, certain explanatory concepts relating to rotations in dimension n, decompositions of the PCA / KLT or SVD type (“SVD” denoting a decomposition into singular values) are recalled below.

Rotations and "quaternions"

The signals are represented by successive blocks of sound samples, these blocks being called “sub-frames” below.
The invention uses a representation of the rotations in dimension n with parameters suitable for a quantization per frame and especially an efficient interpolation per sub-frame. The representations of rotations used in dimension 2, 3 and 4 are defined below.

• L'amplitude du vecteur est préservée
• Le produit vectoriel de vecteurs définissant un repère orthonormé avant rotation est préservé après rotation (il n'y a pas de réflexion).

Une matrice M de taille n x n est une matrice de rotation si et seulement si M^T.M=I_n où I_n désigne la matrice identité de taille n x n (c'est-à-dire que M est une matrice unitaire, M^T désignant la transposée de M) et son déterminant vaut +1.A rotation (around the origin) is a transformation of space in dimension n which changes a vector into another vector, such as:

• The amplitude of the vector is preserved
• The cross product of vectors defining an orthonormal coordinate system before rotation is preserved after rotation (there is no reflection).

A matrix M of size n x n is a rotation matrix if and only if M ^T .M = I _n where I _n denotes the identity matrix of size n x n (i.e. M is a unit matrix , M ^T denoting the transpose of M ) and its determinant is +1.

Several representations are used in the invention which are equivalent to the representation by rotation matrix:
In two dimensions (in a 2D plane) (n = 2): We use as representation the angle of rotation as follows.

Given the angle of rotation θ we deduce the rotation matrix: $${\mathit{<figure-callout\; id="2"\; label="M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">M</figure-callout>}}_2\left(\theta \right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

Given a rotation matrix, we can calculate the angle θ by observing that the trace of the matrix is 2cos θ . It will be noted that it is also possible to estimate θ directly from a covariance matrix before applying a decomposition into main components (PCA) and decomposition into eigenvalues (EVD) presented later.

The interpolation between two rotations of respective angles θ ₁ and θ ₂ can be done by linear interpolation between θ ₁ and θ ₂ , taking into account the constraint of the shortest path on the unit circle between these two angles.

In three-dimensional space (3D) ( n = 3): Euler angles and quaternions are used as representation. In variants, it is also possible to use a representation by axis-angle which is not recalled here.

$${\mathit{M}}_{3,y}\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right)$$

$${\mathit{M}}_{3,z}\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)$$

A rotation matrix of size 3x3 can be decomposed into a product of 3 elementary rotations of angle θ along the x, y, or z axes. $${\mathit{<figure-callout\; id="3"\; label="M"\; filenames="imgf0002.png"\; state="\{\{state\}\}">M</figure-callout>}}_{3,x}\left(\theta \right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right)$$

$${\mathit{<figure-callout\; id="3"\; label="M"\; filenames="imgf0002.png"\; state="\{\{state\}\}">M</figure-callout>}}_{3,y}\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right)$$

$${\mathit{<figure-callout\; id="3"\; label="M"\; filenames="imgf0002.png"\; state="\{\{state\}\}">M</figure-callout>}}_{3,z}\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)$$

According to the combinations of axes, the angles are said to be Eulerian or Cardanic.

Another representation of 3D rotations however is given by quaternions. Quaternions are a generalization of complex number representations with four components in the form of a number q = a + bi + cj + dk where i ² = j ² = k ² = ijk = -1.

Les quaternions unitaires (de norme 1) représentent les rotations - cependant cette représentation n'est pas unique ; ainsi, si q représente une rotation, -q représente la même rotation.The real part a is called a scalar and the three imaginary parts ( b, c , d ) form a 3D vector. The norm of a quaternion is $\left|\mathit{q}\right|=\sqrt{{\mathrm{at}}^2+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">b</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="vs"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">vs</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">d</figure-callout>}}^2}\mathrm{.}$

Unit quaternions (of norm 1) represent rotations - however this representation is not unique; thus, if q represents a rotation, - q represents the same rotation.

Given a unitary quaternion q = a + bi + cj + dk (with a ² + b ² + c ² + d ² = 1), the associated rotation matrix is: $${\mathit{<figure-callout\; id="3"\; label="M"\; filenames="imgf0002.png"\; state="\{\{state\}\}">M</figure-callout>}}_{3,\mathit{quat}}\left(\mathit{q}\right)=\left(\begin{array}{ccc}{\mathrm{at}}^2-b^2-{\mathrm{vs}}^2-{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">d</figure-callout>}}^2& 2\left(\mathit{bc}-\mathit{<figure-callout\; id="2"\; label="ad"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">ad</figure-callout>}\right)& 2\left(\mathit{ac}+\mathit{comic}\right)\\ 2\left(\mathit{ad}+\mathit{bc}\right)& {\mathrm{at}}^2-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">b</figure-callout>}}^2+{\mathrm{vs}}^2-{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">d</figure-callout>}}^2& 2\left(\mathit{CD}-\mathit{<figure-callout\; id="2"\; label="ab"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">ab</figure-callout>}\right)\\ 2\left(\mathit{comic}-\mathit{<figure-callout\; id="2"\; label="ac"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">ac</figure-callout>}\right)& 2\left(\mathit{ab}+\mathit{CD}\right)& {\mathrm{at}}^2-b^2-{\mathrm{<figure-callout\; id="2"\; label="vs"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">vs</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">d</figure-callout>}}^2\end{array}\right)$$

où 0 ≤ α ≤ 1 est le facteur d'interpolation pour aller de q ₁ à q ₂ et Ω est l'angle entre les deux quaternions: $$\mathrm{\Omega }=\arccos \left({\mathit{q}}_1\mathrm{.}{\mathit{q}}_2\right)$$

où q _1· q ₂ désigne le produit scalaire entre deux quaternions (identique au produit scalaire entre deux vecteurs de dimension 4).Euler angles do not allow correct interpolation of 3D rotations; to do this, we use quaternions or the axis-angle representation instead. The SLERP interpolation method (for "spherical linear interpolation") consists of interpolating according to the formula: $$\mathit{slerp}\left({\mathit{q}}_1,{\mathit{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}}_2,\alpha \right)=\frac{\sin \left(1-\alpha \right)\mathrm{<figure-callout\; id="1"\; label="\Omega \; sin\Omega \; q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\Omega \; </figure-callout>}}{\mathrm{sin\Omega }}{\mathit{q}}_{}{\mathit{}}_1+\frac{\sin \alpha \mathrm{<figure-callout\; id="2"\; label="\Omega \; sin\Omega \; q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">\Omega \; </figure-callout>}}{\mathrm{sin\Omega }}{\mathit{q}}_{}{\mathit{}}_2$$

where 0 ≤ α ≤ 1 is the interpolation factor to go from q ₁ to q ₂ and Ω is the angle between the two quaternions: $$\mathrm{\Omega }=\mathrm{<figure-callout\; id="1"\; label="arccos\; q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">arccos\; </figure-callout>}\left({\mathit{q}}_{}\right)\left({\mathit{}}_1\mathrm{.}{\mathit{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}}_2\right)$$

where q _{1 ·} q ₂ denotes the dot product between two quaternions (identical to the dot product between two vectors of dimension 4).

This amounts to interpolating by following a large circle on a 4D sphere with a constant angular speed as a function of α . It should be ensured that the shortest path is used for the interpolant by changing the sign of one of the quaternions when q _{1 ·} q ₂ <0. Note that other quaternion interpolation methods can be used (normalized linear interpolation or nlerp, splines, ...).
Note that it is also possible to interpolate 3D rotations through the axis-angle representation; in this case the angle is interpolated as in the 2D case and the axis can be interpolated for example by the SLERP method (in 3D) by ensuring that the shortest path is taken on a 3D unit sphere and by taking into account because the representation given by the r axis and the angle θ is equivalent to that given by the opposite direction axis - r and the angle 2π - θ .

associées à des quaternions q ₁ = a + bi + cj + dk et q ₂ = w + xi + yj + zk donne une matrice de rotation de taille 4x4.
Il est possible de retrouver le double quaternion associé (q ₁, q ₂) et des matrices de quaternion et antiquaternion associées telles que : $${\mathit{Q}}_1=\left(\begin{array}{cccc}a& b& c& d\\ -b& a& -d& c\\ -c& d& a& -b\\ -d& -c& b& a\end{array}\right)$$

et $${\mathit{Q}}_2^{★}=\left(\begin{array}{cccc}w& -x& -y& -z\\ x& w& -z& y\\ y& z& w& -x\\ z& -y& x& w\end{array}\right)$$

In dimension 4 ( n = 4), a rotation can be parameterized by 6 angles ( n ( n -1) / 2) and we show that the multiplication of two matrices of size 4x4 called quaternion ( Q ₁ ) and antiquaternion $\left({\mathit{Q}}_{}^{}\right)$$\left({\mathit{}}_2^{★}\right)$

associated with quaternions q ₁ = a + bi + cj + dk and q ₂ = w + xi + yj + zk gives a rotation matrix of size 4x4.
It is possible to find the associated double quaternion ( q ₁ , q ₂ ) and associated quaternion and antiquaternion matrices such as: $${\mathit{<figure-callout\; id="1"\; label="Q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">Q</figure-callout>}}_1=\left(\begin{array}{cccc}\mathrm{at}& b& \mathrm{vs}& d\\ -b& \mathrm{at}& -d& \mathrm{vs}\\ -\mathrm{vs}& d& \mathrm{at}& -b\\ -d& -\mathrm{vs}& b& \mathrm{at}\end{array}\right)$$

and $${\mathit{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2^{★}=\left(\begin{array}{cccc}w& -x& -y& -z\\ x& w& -z& y\\ y& z& w& -x\\ z& -y& x& w\end{array}\right)$$

et il est possible de vérifier que cette matrice vérifie les propriétés d'une matrice de rotation (matrice unitaire et déterminant égal à 1).Their product gives a 4x4 size matrix: $${\mathit{M}}_{4,\mathit{quat}}\left(q_1,{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}}_2\right)={\mathit{Q}}_1{\mathit{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2^{★}$$

and it is possible to check that this matrix satisfies the properties of a rotation matrix (unit matrix and determinant equal to 1).

par exemple avec la méthode dite « factorisation de Cayley ». Cela implique de calculer une matrice intermédiaire appelée « transformée tétragonale » (ou matrice associée) et d'en déduire les quaternions à une indétermination près sur le signe des deux quaternions (qui peut être levée par une contrainte supplémentaire de « plus court chemin » évoquée plus loin).Conversely, given a 4x4 rotation matrix, one can factorize this matrix into a product of matrices in the form ${\mathit{Q}}_1{\mathit{<figure-callout\; id="2"\; label="Q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2^{★},$

for example with the method known as “Cayley factorization”. This involves calculating an intermediate matrix called a "tetragonal transform" (or associated matrix) and deducing the quaternions from it up to an indeterminacy on the sign of the two quaternions (which can be removed by an additional constraint of "shortest path" mentioned further).

Singular value decomposition (or "SVD ")

où U est une matrice unitaire ( U ^T U = I _m ) de taille m x m, ∑ est une matrice diagonale rectangulaire de taille m x n à coefficients réels et positifs σ_i ≥ 0 (i = 1 ... p où p = min (m, n)), V est une matrice unitaire ( V ^T V = I _n ) de taille n x n et V^T est la transposée de V. Les coefficients σ_i dans la diagonale de ∑ sont les valeurs singulières de la matrice A. Par convention, elles sont en général listées par ordre décroissant, et dans ce cas la matrice diagonale ∑ associée à A est unique.The singular value decomposition (SVD in English) consists in factoring a real matrix A of size mxn in the form: $$\mathit{AT}=\mathit{U\Sigma }{\mathit{V}}^{\mathit{T}}$$

where U is a unit matrix ( U ^T U = I _m ) of size mxm, ∑ is a rectangular diagonal matrix of size mxn with real and positive coefficients σ _i ≥ 0 (i = 1 ... p where p = min (m , n)), V is a unit matrix ( V ^T V = I _n ) of size nxn and V ^T is the transpose of V. The coefficients σ _i in the diagonal of ∑ are the singular values of the matrix A. By convention, they are generally listed in decreasing order, and in this case the diagonal matrix ∑ associated with A is unique.

où U_r = [u ₁, u ₂,..., u _r] sont les vecteurs singuliers à gauche (ou vecteurs de sortie) de A, ∑ _r = diag(σ₁, ..., σ_r) and V _r = [v ₁, v ₂, ..., v _r] sont les vecteurs singuliers à droite (ou vecteurs d'entrée) de A. Cette formulation matricielle peut être aussi ré-écrite comme: $$\mathit{A}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{r}}}{\mathrm{\sigma }}_{\mathrm{i}}{\mathbf{u}}_{\mathrm{i}}{{\mathbf{v}}_{\mathrm{i}}}^{\mathrm{T}}$$

The rank r of A is given by the number of non-zero coefficients σ _i . We can therefore rewrite the decomposition in singular values as: $$\mathit{AT}=\left[{\mathbf{U}}_{\mathrm{r}}{\tilde{\mathbf{U}}}_{\mathrm{r}}\right]\left[\begin{array}{cc}{\sum }_{\mathrm{r}}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{\mathrm{r}}^{\mathrm{T}}\\ {\tilde{\mathbf{V}}}_{\mathrm{r}}^{\mathrm{T}}\end{array}\right]$$

where U _r = [ u ₁ , u ₂ , ..., u _r ] are the left singular vectors (or output vectors) of A, ∑ _r = diag (σ ₁ , ..., σ _r ) and V _r = [ v ₁ , v ₂ , ..., v _r ] are the right singular vectors (or input vectors) of A. This matrix formulation can also be rewritten as: $$\mathit{AT}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{r}}}{\mathrm{\sigma }}_{\mathrm{i}}{\mathbf{u}}_{\mathrm{i}}{{\mathbf{v}}_{\mathrm{i}}}^{\mathrm{T}}$$

If the sum is limited to an index i <r, a “filtered” matrix is obtained which represents only the “preponderant” information.
We can also write: $${\mathit{AT}\mathbf{v}}_i={\mathrm{\sigma }}_{\mathrm{i}}{\mathbf{u}}_{\mathrm{i}}$$

Which shows that the matrix A transforms v _i into σ _i u _i .

$$\mathit{A}{\mathit{A}}^T=\mathit{U}\left(\mathbf{\Sigma }{\mathbf{\Sigma }}^{\mathrm{T}}\right){\mathit{U}}^{\mathrm{T}}$$

The SVD of A has a relation with the eigenvalue decomposition of A ^T A and AA ^T because: $${\mathit{AT}}^T\mathit{AT}=\mathit{V}\left({\mathbf{\Sigma }}^{\mathrm{T}}\mathbf{\Sigma }\right){\mathit{V}}^{\mathrm{T}}$$

$$\mathit{AT}{\mathit{AT}}^T=\mathit{U}\left(\mathbf{\Sigma }{\mathbf{\Sigma }}^{\mathrm{T}}\right){\mathit{U}}^{\mathrm{T}}$$

Les colonnes de U sont les vecteurs propres de A A ^T, tandis que les colonnes de V sont les vecteurs propres de A ^T A. The eigenvalues of ∑ ^T ∑ and ∑ ∑ ^T are ${\sigma }_1^2,...,{\mathrm{<figure-callout\; id="2"\; label="\sigma \; r"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">\sigma \; </figure-callout>}}_r^{}\mathrm{.}$

The columns of U are the eigenvectors of AA ^T , while the columns of V are the eigenvectors of A ^T A.

The SVD can be interpreted in a geometric way: the image of a sphere in dimension n by the matrix A is in dimension m a hyper-ellipse having main axes in the directions u ₁ , u ₂ , ..., u _m and of length σ ₁ , ..., σ _m .

Karhunen Loeve Transform (or "KLT" for "Karhunen Loeve Transform")

où V est la matrice de vecteurs propres (avec la convention que les vecteurs propres sont des vecteurs colonne) obtenue par décomposition en valeurs propres de R _xx $${\mathit{R}}_{\mathbf{xx}}=\mathit{V}\mathrm{\Lambda }{\mathit{V}}^{\mathrm{T}}$$

où Λ = diag(λ₁, ..., λ_n) est une matrice diagonale dont les coefficients sont les valeurs propres. La matrice V = [v ₁, v ₂,..., v _n] contient les vecteurs propres (colonnes) de R _xx, tels que $${\mathit{R}}_{\mathbf{xx}}{\mathbf{v}}_{\mathrm{i}}={\mathrm{\lambda }}_{\mathrm{n}}{\mathbf{v}}_{\mathrm{i}}$$

The Karhunen Loeve (KLT) transformation of a random vector x centered at 0 and a covariance matrix R _xx = E [ xx ^T ] is defined by: $$\mathbf{y}={\mathit{V}}^{\mathrm{T}}\mathbf{x}$$

where V is the matrix of eigenvectors (with the convention that eigenvectors are column vectors) obtained by decomposition into eigenvalues of R _xx $${\mathit{R}}_{\mathbf{xx}}=\mathit{V}\mathrm{\Lambda }{\mathit{V}}^{\mathrm{T}}$$

where Λ = diag (λ ₁ , ..., λ _n ) is a diagonal matrix whose coefficients are the eigenvalues. The matrix V = [ v ₁ , v ₂ , ..., v _n ] contains the eigenvectors (columns) of R _xx , such that $${\mathit{R}}_{\mathbf{xx}}{\mathbf{v}}_{\mathrm{i}}={\mathrm{\lambda }}_{\mathrm{not}}{\mathbf{v}}_{\mathrm{i}}$$

We can see the KLT as a change of basis, because the product V ^T x expresses the vector x in the basis given by the eigenvectors.
The inverse transformation is given by: $$\mathbf{x}=\mathit{V}\mathbf{y}$$

The KLT makes it possible to decorrelate the components of x ; the variances of the transformed vector y are the eigenvalues of R _xx .

Principal component analysis (or "PCA" for "principal component analysis ")

Principal Component Analysis (PCA) is a dimensionality reduction technique that produces orthogonal variables and maximizes the variance of the variables after projection (or equivalently minimizes the reconstruction error).

en supposant que ces vecteurs sont centrés : $${\mathit{m}}_{\mathbf{x}}=\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}}{\mathit{x}}_i=0$$

The PCA presented below, although also based on an eigenvalue decomposition like the KLT, is such that the estimated covariance matrix R̂ _xx is calculated from N observed vectors x _i , i = 1 ... N of dimension n: $${\hat{\mathit{R}}}_{\mathbf{xx}}=\frac{1}{\mathrm{NOT}-1}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{NOT}}}{\mathit{x}}_i{{\mathit{x}}_i}^T$$

assuming that these vectors are centered: $${\mathit{m}}_{\mathbf{x}}=\frac{1}{\mathrm{NOT}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{NOT}}}{\mathit{x}}_i=0$$

The eigenvalue decomposition of R̂ _xx in the form R̂ _xx = V Λ V ^T makes it possible to calculate the principal components: y _n = V ^T x _n .
The PCA is a transformation by the matrix V ^T which projects the data into a new basis to maximize the variance of the variables after projection.
Note that the PCA can also be obtained from an SVD of the signal x _i put in the form of a matrix X of size nx N. In this case, we can write: $$\mathit{X}=\mathit{U}\mathbf{D}{\mathit{V}}^{\mathrm{T}}$$

It is checked that XX DD ^T = U ^T U ^T which corresponds to a diagonalization XX ^T. Thus the projection vectors of the PCA correspond to the column vectors of U and the projection gives as result U ^T X = D V ^T.

It will also be noted that PCA is generally seen as a dimensionality reduction technique, to "compress" a large-dimensional dataset into a set. comprising few principal components. In the invention, the PCA advantageously makes it possible to decorrelate the multidimensional input signal but one avoids eliminating channels (therefore reducing the number of channels) in order to avoid introducing artefacts. A minimum encoding rate is thus forced to avoid "truncating" the spatial image, except in specific variants where eigenvalues are so low that a zero rate can be authorized (for example to better encode ambisonic sounds created artificially. with a single synthetically spatialized source).

• permuter à nouveau deux vecteurs propres (par exemple associés à des canaux de faible énergie, donc peu représentatifs), ou
• préférentiellement d'inverser le signe de tous les éléments d'une colonne (par exemple associée à un canal de faible énergie) à l'étape S6.

On obtient alors une matrice de vecteurs propres pour la trame courante t correspondant effectivement à une rotation à l'étape S7.
On peut alors coder sur un nombre de bits alloués à cet effet des paramètres de cette matrice (comme par exemple la valeur d'angle, d'un axe et d'un angle, ou de quaternion(s) de cette matrice) à l'étape S8. Dans une autre réalisation optionnelle mais avantageuse, dans le cas où il est constaté à l'étape S9 un écart significatif (supérieur à un seuil par exemple) entre la matrice de rotation estimée pour la trame courante t et la matrice de rotation de la trame précédente t-1, on peut déterminer un nombre variables de sous-trames d'interpolation : autrement on fixe ce nombre de sous-trames à une valeur pré-déterminée. L'étape S10 consiste à :

• découper la trame courante en sous-trames, et
• interpoler des matrices à appliquer aux sous-trames successives depuis la matrice de la trame précédente t-1 jusqu'à la matrice de la trame courante t, afin de lisser dans le temps la différence entre les deux matrices.

A l'étape S11, on applique les matrices de rotation interpolées à une matrice n X (L/K) représentant chacune des K sous-trames des signaux des canaux ambisoniques de l'étape S1 (ou optionnellement S2) pour décorréler autant que possible ces signaux avant l'encodage multi-mono de l'étape S14. Il est rappelé en effet qu'il est souhaité dé-corréler autant que possible ces signaux avant cette transformation multi-mono, selon une approche générale. Une allocation binaire aux canaux séparés est faite à l'étape S12 et codée à l'étape S13.
A l'étape S14, avant d'opérer le multiplexage de l'étape S15 et finir ainsi le procédé de codage en compression, on peut décider d'un nombre de bits à allouer par canal en fonction de la représentativité de ce canal et du débit disponible sur le réseau RES (figure 7). Dans une forme de réalisation, on estime l'énergie dans chaque canal pour une trame courante et on multiplie cette énergie par un score prédéfini pour ce canal et pour un débit donné (ce score étant par exemple une note MOS explicitée plus loin en référence à la figure 3). On pondère ainsi le nombre de bits à allouer pour chaque canal. Une telle réalisation est avantageuse en tant que tel et peut éventuellement faire l'objet d'une protection séparée en contexte ambisonique.We now refer to the figure 2 in order to describe the general principles of the steps which are implemented in a method within the meaning of the invention, for a current frame t.
Step S1 consists in obtaining the respective signals of the ambisonic channels (here four channels W, Y, Z, X in the example described using a channel order according to the ACN convention for Ambisonics Channel Number), for each frame t. These signals can be put in the form of an n x L matrix (for n ambisonic channels (here 4) and L samples per frame).
In the following step S2, the signals of these channels can optionally be pre-processed, for example by a high-pass filter as described below with reference to figure 3 .
In the next step S3, we apply to these signals a PCA principal component analysis or in an equivalent way a Karhunen Loeve KLT transform, to obtain eigenvalues and an eigenvector matrix from a covariance matrix of the n canals. In variants of the invention an SVD could be used.
In step S4, this matrix of eigenvectors, obtained for the current frame t, undergoes signed permutations so that it is as aligned as possible with the matrix of the same nature of the previous frame t -1. In principle, we make sure that the axis of the column vectors in the matrix of eigenvectors corresponds as much as possible to the axis of the column vectors at the same place in the matrix of the previous frame and if not, we swap the positions eigenvectors of the matrix of the current frame t which do not correspond. Then, one also makes sure that the directions of the eigenvectors from one matrix to another also coincide. In other words, one is interested initially only with the lines which carry the eigenvectors (just the direction, without the direction) and one seeks for each line the line closest in the matrix of the frame previous t-1. For this, vectors are permuted in the matrix of the current frame. Then in a second step, we try to do match the orientation of the vectors (direction). For that, one reverses the sign of the eigenvectors which would not have the good sense. Such an embodiment makes it possible to ensure maximum consistency between the two matrices and thus to avoid audible clicks between two frames during sound reproduction.

In step S5, it is also ensured that the matrix of eigenvectors of the current frame t, thus corrected by signed permutations, indeed represents the application of a rotation (of an angle for n = 2 channels, of three Euler angles, of an axis and of an angle or of a quaternion for n = 3 corresponding to the ambisonic representation of order 1 planar W, Y, Z, and of two quaternions for n = 4 in ambisonic representation of order 1 of type W, Y, Z, X).
To ensure that it is indeed a rotation, the determinant of the eigenvector matrix of the current frame t, corrected by permutations, must be positive and equal to (or, in practice, close to) + 1 in step S6. If it is equal to (or close to) -1, then it is advisable to:

• permute again two eigenvectors (for example associated with low energy channels, therefore not very representative), or
• preferably to invert the sign of all the elements of a column (for example associated with a low energy channel) in step S6.

We then obtain a matrix of eigenvectors for the current frame t corresponding effectively to a rotation in step S7.
Parameters of this matrix (such as for example the angle value, of an axis and of an angle, or of quaternion (s) of this matrix) can then be coded on a number of bits allocated for this purpose) at l 'step S8. In another optional but advantageous embodiment, in the case where a significant difference (greater than a threshold for example) is observed in step S9 between the rotation matrix estimated for the current frame t and the rotation matrix of the frame previous t -1, it is possible to determine a variable number of interpolation sub-frames: otherwise, this number of sub-frames is fixed at a predetermined value. Step S10 consists of:

• split the current frame into sub-frames, and
• interpolate matrices to be applied to successive subframes from the matrix of the preceding frame t -1 to the matrix of the current frame t, in order to smooth the difference between the two matrices over time.

In step S11, the interpolated rotation matrices are applied to a matrix n X ( L / K ) representing each of the K sub-frames of the signals of the ambisonic channels of step S1 (or optionally S2) in order to decorrelate as much as possible these signals before the multi-mono encoding of step S14. It is recalled in fact that it is desired to de-correlate as much as possible these signals before this multi-mono transformation, according to a general approach. A bit allocation to the separate channels is made in step S12 and encoded in step S13.
In step S14, before carrying out the multiplexing of step S15 and thus ending the compression coding method, it is possible to decide on a number of bits to be allocated per channel as a function of the representativeness of this channel and of the speed available on the RES network ( figure 7 ). In one embodiment, the energy in each channel is estimated for a current frame and this energy is multiplied by a predefined score for this channel and for a given bit rate (this score being for example an MOS score explained later with reference to the figure 3 ). The number of bits to be allocated for each channel is thus weighted. Such an embodiment is advantageous as such and may optionally be the subject of separate protection in an ambisonic context.

We illustrated on the figure 7 a DCOD encoding device and a DDEC decoding device, within the meaning of the invention, these devices being dual to each other (in the sense of “reversible”) and connected to each other by a RES communication network.

• une mémoire MEM1 pour stocker des données d'instructions d'un programme informatique au sens de l'invention (ces instructions pouvant être réparties entre le codeur DCOD et le décodeur DDEC) ;
• une interface INT1 de réception de signaux ambisoniques répartis sur différents canaux (par exemple quatre canaux W, Y, Z, X à l'ordre 1) en vue de leur codage en compression au sens de l'invention ;
• un processeur PROC1 pour recevoir ces signaux et les traiter en exécutant les instructions de programme informatique que stocke la mémoire MEM1, en vue de leur codage ; et
• une interface de communication COM 1 pour transmettre les signaux codés via le réseau.

The DCOD coding device comprises a processing circuit typically including:

• a memory MEM1 for storing instruction data of a computer program within the meaning of the invention (these instructions being able to be distributed between the DCOD encoder and the DDEC decoder);
• an interface INT1 for receiving ambisonic signals distributed over different channels (for example four channels W, Y, Z, X at order 1) with a view to their coding in compression within the meaning of the invention;
• a processor PROC1 for receiving these signals and processing them by executing the computer program instructions stored in the memory MEM1, with a view to their coding; and
• a COM 1 communication interface for transmitting the coded signals via the network.

• une mémoire MEM2 pour stocker des données d'instructions d'un programme informatique au sens de l'invention (ces instructions pouvant être réparties entre le codeur DCOD et le décodeur DDEC comme indiqué précédemment) ;
• une interface COM2 pour recevoir du réseau RES les signaux codés en vue de leur décodage en compression au sens de l'invention ;
• un processeur PROC2 pour traiter ces signaux en exécutant les instructions de programme informatique que stocke la mémoire MEM2, en vue de leur décodage ; et
• une interface de sortie INT2 pour délivrer les signaux décodés sous forme de canaux ambisoniques W', Y', Z', X', par exemple en vue de leur restitution.

The DDEC decoding device has its own processing circuit, typically including:

• a memory MEM2 for storing instruction data of a computer program within the meaning of the invention (these instructions being able to be distributed between the DCOD encoder and the DDEC decoder as indicated above);
• an interface COM2 for receiving from the network RES the encoded signals with a view to their compression decoding within the meaning of the invention;
• a processor PROC2 for processing these signals by executing the computer program instructions stored in the memory MEM2, with a view to their decoding; and
• an output interface INT2 for delivering the decoded signals in the form of ambisonic channels W ', Y', Z ', X', for example with a view to their reproduction.

Of course, this figure 7 illustrates an example of a structural embodiment of a codec (coder or decoder) within the meaning of the invention. The figures 3 to 6 commented on below describe in detail rather functional implementations of these codecs.

We now refer to the figure 3 to describe an encoder device within the meaning of the invention.

The coder's strategy is to decorrelate the channels of the ambisonic signal as much as possible and to encode them with a core codec. This strategy makes it possible to limit the artefacts in the decoded ambisonic signal. More particularly, one seeks to apply an optimized decorrelation of the input channels before multi-mono coding here. In addition, an interpolation of which the computation cost for the encoder and the decoder is limited because it is carried out in a specific domain (angle in 2D, quaternion in 3D, double quaternion in 4D) makes it possible to interpolate the covariance matrices calculated for the PCA / KLT analysis rather than repeating a decomposition into eigenvalues and eigenvectors several times per frame.

Nevertheless, before approaching the core coding carried out within the meaning of the invention, a few functions of the coder which are advantageous, such as in particular the optimization of the budget of bits allocated to coding as a function of perceptual criteria, are presented here below.

In the embodiment described here of the encoder, the latter can typically be an extension of the standardized 3GPP EVS (for “Enhanced Voiced Services”) encoder. Advantageously, the EVS coding rates can be used without then modifying the structure of the EVS binary train. Thus, the multi-mono coding (block 340 of the figure 3 described later) operates here with a possible allocation to each transformed channel, restricted to the following bit rates for super-wide audio band coding: 9.6; 13.2; 16.4; 24.4; 32; 48; 64; 96 and 128 kbit / s.
Of course, it is possible to add additional bit rates (to have a finer allocation granularity) by modifying the EVS codec. It is also possible to use a codec other than the EVS type, for example the OPUS® codec.

In general, it is retained that the finer the coding granularity, the more bits must be reserved to represent the possible bit rate combinations. A compromise between fineness of allocation and additional information describing the binary allocation must be made. This allocation is optimized here by block 320 of the figure 3 , which is described later. This is an advantageous characteristic as such and independent of the decomposition into eigenvectors with a view to establishing a rotation matrix within the meaning of the invention. As such, the bit allocation performed by block 320 can be the subject of separate protection.

With reference to the figure 3 , the block 300 receives an input signal Y in the current frame of index t. The index is not indicated here so as not to weigh down the ratings. This is a matrix of size nx L. In an embodiment adapted to an ambisonic context of order 1, we have n = 4 channels W, Y, Z, X (thus defined according to the order ACN) which can be standardized according to the SN3D convention. In a variant, the order of the channels can be alternately, for example W, X, Y, Z (following the FuMA convention) and the normalization can be different (N3D or FuMa). Thus the channels W, Y, Z, X correspond to the successive lines: y _{1, l} , y _{2, l} , y _{3, l} , y _{4, l} , which will be denoted in the form of one-dimensional signals y _i ( l ), l = 1, ..., L. This is therefore a succession of samples from 1 to L occupying frame t.

It is assumed that the signal (in each channel) is sampled at 48 kHz, without loss of generality. The frame length is fixed at 20 ms, ie L = 960 successive samples, without loss of generality. In variants, it is possible for example to use a frame length of L = 640 samples for sampling at 32 kHz.

The PCA / KLT analysis and the PCA / KLT transformation which are described later are performed in the time domain. It is thus understood that one remains here in the time domain without necessarily having to operate a sub-band or more generally frequency transform.

At each frame, the block 300 of the encoder applies a pre-processing (optional) to obtain the pre-processed input signal denoted Y. This may be a high-pass filtering (with a cutoff frequency typically at 20 Hz) of each new one. 20 ms frame of the input signal channels. This operation removes the DC component likely to bias estimating the covariance matrix so that at the output of block 300 the signal can be considered to have zero mean. The transfer function is denoted H _pre ( z ), so we have for each channel: X _i ( z ) = H _pre ( z ) Y _i ( z ) . If the block 300 is not implemented, we have X = Y. A low pass filter of block 340 can also be implemented to perform multi-mono encoding but when block 300 is implemented the high pass filtering in preprocessing of the mono encoding which can be used in block 340 is preferably disabled to avoid repeating the same preprocessing and thus reduce overall complexity.

en appliquant ce filtre à chacun des n canaux du signal d'entrée dont les coefficients peuvent être tels que présentés dans le tableau ci-dessous : 8 kHz 16 kHz 32 kHz 48 kHz b ₀ 0.988954248067140 0.994461788958.195 0.997227049904470 0.998150511190452 b ₁ -1.977908496134280 -1.988923577916390 -1.994454099808940 -1.996301022380904 b ₂ 0.988954248067140 0.994461788958195 0.997227049904470 0.998150511190452 a ₁ 1.977786483776764 1.988892905899653 1.994446410541927 1.996297601769122 a ₂ -0.978030508491796 -0.988954249933127 -0.994461789075954 -0.996304442992686 The transfer function noted H _pre ( z ) above can be of the type: $$H_{\mathit{pre}}\left(z\right)=\frac{b_0+b_1{z}^{-1}+b_2{z}^{-2}}{1-{\mathrm{at}}_1{z}^{-1}-{\mathrm{at}}_2{z}^{-2}}$$

by applying this filter to each of the n channels of the input signal, the coefficients of which can be as presented in the table below: 8 kHz 16 kHz 32 kHz 48 kHz b ₀ 0.988954248067140 0.994461788958.195 0.997227049904470 0.998150511190452 b ₁ -1.977908496134280 -1.988923577916390 -1.994454099808940 -1.996301022380904 b ₂ 0.988954248067140 0.994461788958195 0.997227049904470 0.998150511190452 a ₁ 1.977786483776764 1.988892905899653 1.994446410541927 1.996297601769122 a ₂ -0.978030508491796 -0.988954249933127 -0.994461789075954 -0.996304442992686

Alternatively, another type of filter can be used, for example a 6th order Butterworth filter with a frequency of 50 Hz.
In variants, the pre-processing could include a fixed die-stamping step which could keep the same number of channels or reduce the number of channels. An example of matrixing applied to the four channels of an ambisonic signal in B-format is given below: $$M_{B\to \mathrm{AT}}=\left[\begin{array}{cccc}1/2& \frac{1}{\sqrt{6}}& 0& \frac{1}{\sqrt{12}}\\ 1/2& \frac{-1}{\sqrt{6}}& 0& \frac{1}{\sqrt{12}}\\ 1/2& 0& \frac{1}{\sqrt{6}}& \frac{-1}{\sqrt{12}}\\ 1/2& 0& \frac{-1}{\sqrt{6}}& \frac{-1}{\sqrt{12}}\end{array}\right]$$

It will be noted that in this case this preprocessing will have to be reversed on decoding by applying a matrixing by M _{A → B} = M _{B → A} ^-1 of the decoded signal to find the channels in the original format.

The following block 310 estimates at each frame t a transformation matrix obtained by determining the eigenvectors by PCA / KLT and checking that the transformation matrix formed by these eigenvectors indeed characterizes a rotation. Details of the block operation 310 are given below with reference to figure 4 . This transformation matrix performs a matrixing of the channels to de-correlate them making it possible to apply an independent coding of the multi-mono type by the block 340. As detailed below, the block 310 transmits to the multiplexer quantization indices representing the matrix of transformation and optionally information encoding the number of interpolations of the transformation matrix, per sub-frame of the current frame t, as also detailed further below.

où B_overhead correspond au budget de bits pour l'information supplémentaire codée par trame (allocation binaire + données de rotation) comme décrit plus loin. Par exemple, B_overhead peut être de l'ordre de B_overhead = 55 bits par trame de 20 ms (soit 2,75 kbit/s) pour le cas d'un codage ambisonique à quatre canaux ; cela comprend 51 bits pour coder la matrice de rotation et 4 bits (comme décrit ci-desous) pour coder l'allocation des bits pour le codage des canaux séparés. Pour un débit global de 4x13.2 = 52,8 kbits/s, cela laisse donc un budget de B_multimono =50.05 kbit/s.Block 320 determines the optimal rate allocation for each channel (after PCA / KLT transformation) as a function of a given B-bit budget. This block seeks a distribution of the bit rate between channels by calculating a score for each possible combination of bit rates; the optimal allocation is found by looking for the combination maximizing this score.
Several criteria can be used to define a score for each combination.
For example, the number of possible rates for mono encoding of a channel may be limited to the nine discrete rates of the EVS codec having super-wide audio band: 9.6; 13.2; 16.4; 24.4; 32; 48; 64; 96 and 128 kbit / s. However if the codec according to the invention operates at a given bit rate associated with a budget of B bits in the current frame of index t, in general only a subset of these listed bit rates can be used. For example if the codec bit rate is fixed at 4x13.2 = 52.8 kbits / s to represent four channels and if each channel receives a minimum budget of 9.6 kbit / s to guarantee a super-wide band for each of the channels , the possible combinations of bit rates for coding separate channels must respect the constraint that the bit rate used remains lower than the available bit rate which corresponds to: $$B_{\mathit{multimono}}=B-B_{\mathit{overhead}},$$

where B _overhead corresponds to the bit budget for the additional information encoded per frame (binary allocation + rotation data) as described later. For example, B _overhead may be of the order of B _overhead = 55 bits per 20 ms frame (ie 2.75 kbit / s) for the case of four-channel ambisonic coding; this includes 51 bits for encoding the rotation matrix and 4 bits (as described below) for encoding the bit allocation for the encoding of the separate channels. For an overall speed of 4x13.2 = 52.8 kbits / s, this therefore leaves a budget of B _multimono = 50.05 kbit / s.

• Singleton (9.6, 9.6, 9.6, 9.6) - total = 38.4
• Permutations de (13.2, 9.6, 9.6, 9.6) - total = 42 kbit/s
• Permutations de (13.2, 13.2, 9.6, 9.6) - total = 45.6 kbit/s
• Permutations de (13.2, 13.2, 13.2, 9.6) - total = 49.2 kbit/s
• Permutations de (16.4, 9.6, 9.6, 9.6) - total = 45.2 kbit/s
• Permutations de (16.4, 13.2, 9.6, 9.6) - total = 48.8 kbit/s

On observe que certaines combinaisons respectant la limite de budget maximal ont un débit très inférieur aux autres, et finalement seules deux combinaisons pertinentes peuvent être retenues :

• Permutations de (13.2, 13.2, 13.2, 9.6) - 4 cas et débit non utilisé de 50.5 - 49.2 = 1.3 kbit/s
• et Permutations de (16.4, 13.2, 9.6, 9.6) - 12 cas et débit non utilisé de 50.5 - 48.8 = 1.7 kbit/s

Cela permet d'illustrer que seize combinaisons sont intéressantes en particulier et peuvent être codées sur 4 bits (16 valeurs). Par ailleurs un certain nombre de bits restent potentiellement inutilisés en fonction de l'allocation choisie.In terms of bit rates per channel, this gives the following permutations of bit rates per channel:

• Singleton (9.6, 9.6, 9.6, 9.6) - total = 38.4
• Permutations of (13.2, 9.6, 9.6, 9.6) - total = 42 kbit / s
• Permutations of (13.2, 13.2, 9.6, 9.6) - total = 45.6 kbit / s
• Permutations of (13.2, 13.2, 13.2, 9.6) - total = 49.2 kbit / s
• Permutations of (16.4, 9.6, 9.6, 9.6) - total = 45.2 kbit / s
• Permutations of (16.4, 13.2, 9.6, 9.6) - total = 48.8 kbit / s

We observe that some combinations respecting the maximum budget limit have a much lower flow rate than others, and finally only two relevant combinations can be retained:

• Permutations of (13.2, 13.2, 13.2, 9.6) - 4 cases and unused rate of 50.5 - 49.2 = 1.3 kbit / s
• and Permutations of (16.4, 13.2, 9.6, 9.6) - 12 cases and unused rate of 50.5 - 48.8 = 1.7 kbit / s

This makes it possible to illustrate that sixteen combinations are of particular interest and can be coded on 4 bits (16 values). Furthermore, a certain number of bits remain potentially unused depending on the allocation chosen.

• l'énergie de chaque canal, et
• une note moyenne qui peut être pré-mémorisée et issue de tests subjectifs ou objectifs, laquelle note, notée MOS (pour « Mean Opinion Score », s'agissant d'une note moyenne sur un panel de testeurs), est associée au débit alloué.

Ce score peut alors être défini par l'équation $$S\left(b_{t,1},\mathrm{...},b_{t,n}\right)={\displaystyle \sum _{i=1}^n}Q\left(v_{t,i}\right){\cdot E}_i$$

où E_i est l'énergie dans la trame courante (d'indice t) du signal s(l), l = ···. L - 1 sur le canal i, avec : $$E_i={\displaystyle \sum _{l=0}^{L-1}}{s}^2\left(l\right)$$

It is observed that the coding of adaptive matrixing based on PCA / KLT processing and allowing flexible bit allocation can result in unused bits and, for some channels, a lower bit rate (for example 9.6 kbit / s) than rate equally distributed among each of the channels (for example 13.2 kbit / s per channel).
To improve this situation, the block 320 can then evaluate all the possible (relevant) combinations of bit rates for the 4 channels resulting from the PCA / KLT transformation (at the output of the block 310) and attribute a score to them. This score is calculated based on:

• the energy of each channel, and
• an average score which can be pre-memorized and resulting from subjective or objective tests, which score, denoted MOS (for "Mean Opinion Score", being an average score on a panel of testers), is associated with the allocated throughput .

This score can then be defined by the equation $$S\left(b_{t,1},\mathrm{...},b_{t,\mathrm{not}}\right)={\displaystyle \sum _{i=1}^{\mathrm{not}}}Q\left(v_{t,i}\right){\cdot E}_i$$

where E _i is the energy in the current frame (of index t ) of signal s ( l ), l = ···. L - 1 on channel i, with: $$E_i={\displaystyle \sum _{l=0}^{L-1}}{s}^2\left(l\right)$$

The optimal allocation can be such that: $$b_{t,1}^{\mathit{opt}},\mathrm{..},b_{t,1}^{\mathit{opt}}=\arg \underset{b_{t,1},\mathrm{..},b_{t,\mathrm{not}}|\sum _{i=1}^{\mathrm{not}}{b}_{t,i}\le B}{\max }S\left(b_{t,1},\mathrm{..},b_{t,\mathrm{not}}\right)$$

As a variant, the factor E _i can be fixed at the value taken by the eigenvalue associated with the channel i resulting from the decomposition into eigenvalues of the signal at the input of block 310 and after possible signed permutation.

The MOS Q ( b _i ) score is preferably the subjective quality score of the codec used for multi-mono coding in block 340 for a budget b _i (in numbers of bits) per frame of 20 ms corresponding to a bit rate R _i = 50 b _i (in bits / sec). The subjective (average) MOS scores of an EVS standardized encoder given by: κ _i 0 1 2 3 4 5 6 7 8 b _i 192 264 328 488 640 960 1280 1920 2560 R _i 9600 13200 16400 24400 32000 48000 64000 96000 128000 Q ( b _i ) 3.62 3.79 4.25 4.60 4.53 4.82 4.83 4.85 4.87

Alternatively, other MOS score values for each of the listed bit rates can be derived from other tests (subjective or objective) predicting the quality of the codec. It is also possible to adapt the MOS notes used in the current frame according to a classification of the type of signal (for example a speech signal without background noise, or speech with ambient noise, or music or mixed content), by reusing classification methods implemented by the EVS codec and by applying them to the W channel of the ambisonic input signal before performing the binary allocation. The MOS score can also correspond to an average score resulting from different types of methodologies and rating scales: MOS (absolute) from 1 to 5, DMOS (from 1 to 5), MUSHRA (from 0 to 100).

In a variant where the EVS coder is replaced by another codec, the list of bit rates b _i and the notes Q ( b _i ) can be replaced as a function of this other codec. It is also possible to add additional coding rates to the EVS encoder and therefore complete the list of rates and MOS notes, or even modify the EVS encoder and potentially the associated MOS notes.

Alternatively again, the allocation between the channels is refined by weighting the energy by a power α where α takes a value between 0 and 1. By varying the value of α , we can thus control the influence of the energy in the allocation: the closer α is to 1, the more important the energy is in the score, and therefore the more unequal the allocation is between the channels. Conversely, the closer α is to 0, the less important the energy is and the more the allocation is evenly distributed between the channels. The score is therefore expressed in the form: $$S\left(b_{t,1},\mathrm{..},b_{t,\mathrm{not}}\right)={\displaystyle \sum _{i=1}^{\mathrm{not}}}Q\mathrm{.}\left(b_{t,i}\right)E_i^{\alpha }$$

où β_i a pour valeur une constante pré-déterminée (par exemple 0.1) quand b_t,i =b _t-1,i et β_i = 0 quand b_t,i ≠ b _t-1,i .
Cette pondération supplémentaire permet de limiter les fluctuations trop fréquentes de débit entre les canaux. Avec cette pondération, seuls les changements significatifs d'énergie entrainent un changement de débit. On peut en outre varier la valeur de la constante pour régler une stabilité de l'allocation.Alternatively again, to make the allocation more stable, a second weighting can be added to the score function to penalize inter-frame rate changes. A penalty is added to the score if the rate combination is not the same in frame t as in frame t - 1. The score is then expressed in the form: $$S\left(b_{t,1},\mathrm{..},b_{t,\mathrm{not}}\right)={\displaystyle \sum _{i=1}^{\mathrm{not}}}Q\left(b_{t,i}\right)\mathrm{.}{E}_i^{\alpha }\mathrm{.}\left(1+{\beta }_i\right)$$

where β _i has for value a predetermined constant (for example 0.1) when b _{t, i} = b _{t -1, i} and β _i = 0 when b _{t, i} ≠ b _{t -1, i} .
This additional weighting makes it possible to limit the too frequent fluctuations in flow rate between the channels. With this weighting, only significant changes in energy cause a change in flow. It is also possible to vary the value of the constant to adjust a stability of the allocation.

Cependant on peut préférer énumérer (au départ, hors ligne) les différentes combinaisons de débits pertinentes pour le budget de bits donné et utiliser le débit minimal pour représenter ces combinaisons. L'indice peut alors être représenté par un codage de type « code de la permutation » + « offset de la combinaison » ; par exemple dans l'exemple où on code sur un indice de 4 bits les 16 combinaisons de débit comprenant 4 permutations de (13.2, 13.2, 13.2, 9.6) et 12 permutations de (16.4, 13.2, 9.6, 9.6), on pourra utiliser les indices 0-3 pour coder les 4 premières permutations possibles (avec un offset à 0 et un code allant de 0 à 3) et les indices 4-15 pour coder les 12 autres permutations possibles (avec un offset à 4 et un code de 0 à 11).With reference again to the figure 3 , once the bit rate has been calculated for each frame, this bit rate is coded by block 330, for example exhaustively for all the bit rate combinations. In the case of 9 bit rates and 4 channels, the necessary bit rate is ┌ log ₂ (9 ⁴ ) ┐ = 13 bits, where ┌.┐ corresponds to the rounding to the upper integer. The combination of the 4 bit rates can be coded in the form of the index: $\sum _{i=1}^{\mathrm{not}}{9}^i{\kappa }_i\mathrm{.}$

However, one may prefer to enumerate (initially, offline) the different combinations of rates relevant to the given bit budget and use the minimum rate to represent these combinations. The index can then be represented by a coding of the type "permutation code" + "offset of the combination"; for example in the example where we code on a 4-bit index the 16 bit rate combinations comprising 4 permutations of (13.2, 13.2, 13.2, 9.6) and 12 permutations of (16.4, 13.2, 9.6, 9.6), we can use indices 0-3 to code the first 4 possible permutations (with an offset of 0 and a code ranging from 0 to 3) and the indices 4-15 to code the 12 other possible permutations (with an offset of 4 and a code of 0 to 11).

bits. Dans des variantes, le budget de bits restant peut être redistribué au codage des canaux transformés afin d'utiliser tout le budget disponible et si le codage multimono est basé sur une technologie de type EVS, on peut alors modifier l'algorithme de codage 3GPP EVS spécifié pour introduire des débits supplémentaires. Dans ce cas, il est également possible d'intégrer ces débits supplémentaires dans la table définissant la correspondance entre b_i et Q(b_i ).
On peut en outre réserver un bit pour pouvoir commuter entre deux modes de codage :

• Codage selon l'invention avec codage de la matrice de rotation, et
• Codage selon l'invention avec une matrice de rotation restreinte à la matrice identité (donc non transmise) ce qui revient à un codage multi-mono direct si la matrice de rotation de la trame précédente était aussi une matrice identité (par exemple quand le signal ambisonique comprend des sources sonores très diffuses ou de multiples sources étalées spatialement autour de certaines directions privilégiées, auquel cas les canaux ambisoniques sont moins corrélés que pour des sons mélangeant des sources plus ponctuelles et isolées).

Le choix entre ces deux modes implique d'utiliser un bit dans le train pour indiquer si la trame courante utilise un matrice de rotation restreinte à la matrice identité sans transmission de paramètres de rotation (bit=0) ou si une matrice de rotation est codée (bit=1). Quand bit=0, on pourra dans des variantes utiliser une allocation des bits fixes aux canaux séparés et ne pas transmettre d'allocation binaire.With reference again to the figure 3 , the multiplexing block 350 takes as input the n matrixed channels coming from the block 310 and the bit rates allocated to each channel coming from the block 320 to then separately code the different channels with a core codec which corresponds to the EVS codec for example. If the core codec used allows stereo or multi-channel coding, the multi-mono approach can be replaced by multi-stereo or multi-channel coding. Once the channels are coded, the associated bit stream is sent to the multiplexer (block 350).
In frames where part of the overall budget is not fully utilized, the multiplexer (block 350) can add zero stuffing bits to reach the bit budget allocated to the current frame, i.e. $B-\sum _{i=1}^{\mathrm{not}}{b}_{t,i}^{\mathit{opt}}$

bits. In variants, the remaining bit budget can be redistributed to the coding of the transformed channels in order to use all the available budget and if the multimono coding is based on an EVS type technology, then the 3GPP EVS coding algorithm can be modified. specified to introduce additional flow rates. In this case, it is also possible to integrate these additional rates in the table defining the correspondence between b _i and Q ( b _i ).
A bit can also be reserved in order to be able to switch between two coding modes:

• Coding according to the invention with coding of the rotation matrix, and
• Coding according to the invention with a rotation matrix restricted to the identity matrix (therefore not transmitted), which amounts to direct multi-mono coding if the rotation matrix of the previous frame was also an identity matrix (for example when the signal ambisonics includes very diffuse sound sources or multiple sources spatially spread around certain preferred directions, in which case the ambisonic channels are less correlated than for sounds mixing more point and isolated sources).

The choice between these two modes implies using a bit in the stream to indicate whether the current frame uses a rotation matrix restricted to the identity matrix without transmission of rotation parameters (bit = 0) or if a rotation matrix is encoded. (bit = 1). When bit = 0, it is possible in variants to use an allocation of fixed bits to the separate channels and not to transmit a binary allocation.

We now refer to the figure 4 to describe in detail the block 310 applying the analysis and the PCA / KLT transformation. In this block, the encoder calculates the covariance matrix from the ambisonic (preprocessed) channels in block 400: $$\mathit{VS}=\frac{1}{L-1}{\mathit{XX}}^T$$

Alternatively, this matrix can be replaced by the correlation matrix, where the channels are pre-normalized by their respective standard deviation, or generally weights reflecting a relative importance can be applied to each of the channels; moreover, the normalization term 1 / ( L - 1) can be omitted or replaced by another value (for example 1 / L ). The values C _ij correspond to the variance between x _i and x _j .

The encoder then performs in block 410 an eigenvalue decomposition (EVD for “Eigenvalue Decomposition”), by calculating the eigenvalues and the eigenvectors of the matrix C. The eigenvectors are noted here V _t to indicate the frame index t because the eigenvectors V _{t -1} obtained in the previous frame of index t - 1 are preferably stored and used subsequently. The eigenvalues are denoted λ ₁ , λ ₂ , ..., λ _n .

et les vecteurs propres V_t sont donnés par les n vecteurs (colonne) singuliers à gauche U. Alternatively, a singular value decomposition (SVD) of the X preprocessed channels can be used. We thus obtain the singular vectors (left U and right V ) and the singular values σ _i . In this case we can consider that the eigenvalues λ _i are ${\mathrm{\lambda }}_{\mathrm{i}}={\sigma }_i^2$

and the eigenvectors V _t are given by the n vectors (column) singular on the left U.

The encoder then applies in block 420 a first signed permutation of the columns of the transformation matrix for the frame t (the columns of which are the eigenvectors) in order to avoid too much disparity with the transformation matrix of the previous frame t - 1, which would cause click problems at the border with the previous frame.
Thus, once a blank of the transformation matrix is obtained for frame t, block 430 takes n estimated eigenvectors V _t = v _{t , 0} , ..., v _{t , n} of the current frame of index t and n eigenvectors V _{t -1} stored from the previous frame of index t - 1, and applies a signed permutation to the estimated vectors V _t so that they are as close as possible to V _{t -1} . Thus the eigenvectors of frame t are permuted so that the associated basis are as close as possible to the basis of frame t - 1. This has the effect of improving the continuity of the transformed signal frames (once the transformation matrix is applied to the channels).

Another constraint is that the transformation matrix must correspond to a rotation. This constraint makes it possible to guarantee that the encoder can convert the transformation matrix into generalized Euler angles (block 430) in order to quantize them (block 440) with a predetermined bit budget as seen previously. For this purpose, the determinant of this matrix must be positive (equal to +1 typically).

• La première étape (S4 sur la figure 2 précédemment présentée) fait correspondre les vecteurs les plus proches entre deux trames en se souciant uniquement de l'axe et non de la direction (du sens) de l'axe. Ce problème peut être formulé comme un problème combinatoire d'affectation de tâches, où l'objectif est de trouver la configuration qui minimise un coût. Le coût peut être défini ici comme la trace de la valeur absolue de l'inter-corrélation entre les matrices de vecteurs propres des trames t et t - 1. $${\mathit{C}}_{\mathit{t}}=\mathit{tr}\left(\mathit{abs}\left(\mathit{corr}\left({\mathit{V}}_{\mathit{t}},{\mathit{V}}_{\mathit{t}-\mathbf{1}}\right)\right)\right)$$
  
  où tr(.) désigne la trace d'une matrice, abs(.) revient à appliquer l'opération de valeur absolue à tous les coefficients d'une matrice et corr(V1,V2) donne la matrice de corrélation entre les vecteurs V1 et V2.
  Dans un mode de réalisation la méthode « hongroise » (ou « algorithme hongrois ») sert à déterminer l'assignation optimale qui donne une permutation des vecteurs propres de la trame t;
• La seconde étape (S6 sur la figure 2) consiste à déterminer la direction/sens de chaque vecteur propre permuté. Le bloc 420 calcule l'inter-corrélation entre les vecteurs propres permutés Ṽ_t de la trame t et le vecteur propre de la trame t - 1 $${\mathbf{\Gamma }}_{\mathit{t}}=\mathit{corr}\left({\tilde{\mathit{V}}}_{\mathit{t}},{\mathit{V}}_{\mathit{t}-\mathbf{1}}\right)$$
  

Preferably, the optimal signed permutation is obtained in two steps:

• The first step (S4 on the figure 2 previously presented) matches the closest vectors between two frames, looking only at the axis and not the direction (sense) of the axis. This problem can be formulated as a combinatorial task assignment problem, where the objective is to find the configuration which minimizes a cost. The cost can be defined here as the trace of the absolute value of the inter-correlation between the eigenvector matrices of frames t and t - 1. $${\mathit{VS}}_{\mathit{t}}=\mathit{tr}\left(\mathit{abs}\left(\mathit{corr}\left({\mathit{V}}_{\mathit{t}},{\mathit{V}}_{\mathit{t}-\mathbf{1}}\right)\right)\right)$$
  
  where tr ( . ) denotes the trace of a matrix, abs ( . ) amounts to applying the operation of absolute value to all the coefficients of a matrix and corr (V1, V2) gives the correlation matrix between the vectors V1 and V2.
  In one embodiment, the “Hungarian” method (or “Hungarian algorithm”) is used to determine the optimal assignment which gives a permutation of the eigenvectors of the frame t;
• The second step (S6 on the figure 2 ) consists in determining the direction / sense of each permuted eigenvector. The block 420 calculates the inter-correlation between the permuted eigenvectors Ṽ _t of the frame t and the eigenvector of the frame t - 1 $${\mathbf{\Gamma }}_{\mathit{t}}=\mathit{corr}\left({\tilde{\mathit{V}}}_{\mathit{t}},{\mathit{V}}_{\mathit{t}-\mathbf{1}}\right)$$
  

If a value on the diagonal of the inter-correlation matrix Γ _t is negative, that denotes a change of sign between the directions of eigenvectors. A sign inversion is then operated on the eigenvector corresponding to Ṽ _t .
At the end of the two steps, the frame transformation matrix t is designated by V _t so that at the following frame the stored matrix becomes V _{t -1} .

ou $V_t{V}_{t-1}^{-1}$

qui est convertie en 3D ou 4D et en convertissant cette matrice de passage respectivement en un quaternion unitaire ou deux quaternions unitaires. La recherche devient alors une recherche du plus proche voisin avec un dictionnaire représentant l'ensemble des permutations signées possibles. Par exemple dans le cas 4D les douze permutations paires possibles (sur 24 permutations totales) de 4 valeurs sont associées aux doubles quaternions unitaires suivants écrits comme des vecteurs 4D:

• (1, 0, 0, 0) et (1, 0, 0, 0)
• (0, 0, 0, 1) et (0, 0, -1, 0)
• (0, 1, 0, 0) et (0, 0, 0, -1)
• (0, 0, 1, 0) et (0, -1, 0, 0)]
• (0.5, -0.5, -0.5, -0.5) et (0.5, 0.5, 0.5, 0.5)
• (0.5, 0.5, 0.5, 0.5) et (0.5, -0.5, -0.5, -0.5)
• (0.5, -0.5, 0.5, -0.5) et (0.5, -0.5, 0.5, 0.5)
• (0.5, -0.5, 0.5, 0.5) et (0.5, -0.5, -0.5, 0.5)
• (0.5, 0.5, -0.5, 0.5) et (0.5, 0.5, -0.5, -0.5)
• (0.5, -0.5, -0.5, 0.5) et (0.5, 0.5, -0.5, 0.5)
• (0.5, 0.5, -0.5, -0.5) et (0.5, 0.5, 0.5, -0.5)
• (0.5, 0.5, 0.5, -0.5) et (0.5, -0.5, 0.5, -0.5)

La recherche de la permutation (paire) optimale peut se faire en utilisant la liste ci-dessus comme un dictionnaire de double quaternion prédéfini et en effectuant une recherche du plus proche voisin par rapport au double quaternion associé à la matrice de passage. Un avantage de cette méthode est de réutiliser les paramètres de rotation de type quaternion et double quaternion.In a variant, the search for the optimal signed permutation can be done by calculating the passage matrix $V_{t-1}^{-1}{V}_t$

or $V_t{V}_{t-1}^{-1}$

which is converted to 3D or 4D and converting this pass matrix respectively into a unit quaternion or two unit quaternions. The search then becomes a search for the nearest neighbor with a dictionary representing the set of possible signed permutations. For example in the 4D case the twelve possible even permutations (out of 24 total permutations) of 4 values are associated with the following double unit quaternions written as 4D vectors:

• (1, 0, 0, 0) and (1, 0, 0, 0)
• (0, 0, 0, 1) and (0, 0, -1, 0)
• (0, 1, 0, 0) and (0, 0, 0, -1)
• (0, 0, 1, 0) and (0, -1, 0, 0)]
• (0.5, -0.5, -0.5, -0.5) and (0.5, 0.5, 0.5, 0.5)
• (0.5, 0.5, 0.5, 0.5) and (0.5, -0.5, -0.5, -0.5)
• (0.5, -0.5, 0.5, -0.5) and (0.5, -0.5, 0.5, 0.5)
• (0.5, -0.5, 0.5, 0.5) and (0.5, -0.5, -0.5, 0.5)
• (0.5, 0.5, -0.5, 0.5) and (0.5, 0.5, -0.5, -0.5)
• (0.5, -0.5, -0.5, 0.5) and (0.5, 0.5, -0.5, 0.5)
• (0.5, 0.5, -0.5, -0.5) and (0.5, 0.5, 0.5, -0.5)
• (0.5, 0.5, 0.5, -0.5) and (0.5, -0.5, 0.5, -0.5)

The search for the optimal permutation (pair) can be done by using the above list as a pre-defined double quaternion dictionary and performing a closest neighbor search against the double quaternion associated with the passage matrix. An advantage of this method is to reuse the quaternion and double quaternion type rotation parameters.

The operation which is implemented in the following block 460 assumes that the transformation matrix after signed permutation is indeed a rotation matrix; the transformation matrix is necessarily unitary, but its determinant must also be equal to 1 $$\det \left({\mathit{V}}_{\mathit{t}}\right)=1$$

Now, the transformation matrix resulting from blocks 410 and 420 (after EVD and signed permutations) is an orthogonal (unitary) matrix which can have a determinant at -1 or 1, that is to say a reflection or rotation matrix.

If the transformation matrix is a reflection matrix (if its determinant is equal to -1), it can be modified into a rotation matrix by inverting an eigenvector (for example the eigenvector associated with the lowest value) or by inverting two columns (eigenvectors).
Certain methods of decomposition into eigenvalues (for example by Givens rotation) or of decomposition into singular values can lead to transformation matrices which are intrinsically matrices of rotation (with a determinant at +1); in this case, the step of verifying that the determinant is +1 will be optional.

Block 430 converts the rotation matrix into parameters. In the preferred embodiment, an angular representation is used for the quantification (6 generalized Euler angles for the 4D case, 3 Euler angles for the 3D case, and one 2D angle). For the ambisonic case (four channels) we obtain six Euler angles generalized according to the method described in the article “Generalization of Euler Angles to N-Dimensional Orthogonal Matrices” by David K. Hoffman, Richard C. Raffenetti, and Klaus Ruedenberg , published in Journal of Mathematical Physics 13, 528 (1972); for the case of planar ambisonics (three channels), three Euler angles are obtained and for the stereo case, an angle of rotation is obtained according to the methods well known in the state of the art. The values of the angles are quantized in block 440 with a predetermined budget of bits. In the preferred embodiment, a scalar quantization is used and the quantization step is for example identical for each angle. For example in the case of 4 channels, we code 6 Euler angles generalized with 3x (8 + 9) = 51 bits (3 angles defined on an interval of [-π / 2, π / 2] coded on 8 bits with a step of π / 256 and the 3 other angles defined on an interval of [-π, π] coded on 9 bits with one with a step of π / 256). The quantization indices of the transformation matrix are sent to the multiplexer (block 350). Moreover, the block 440 will be able to convert the quantized parameters into a quantized rotation matrix V̂ _t , if the parameters used for the quantization do not correspond to the parameters used for the interpolation.

• Le bloc 430 peut effectuer une conversion des matrices de rotations en un double quaternion unitaire (cas de 4 canaux), en quaternion unitaire (cas de 3 canaux) et en un angle (cas de 2 canaux).

Alternatively, blocks 430 and 440 can be replaced as follows:

• Block 430 can convert the rotation matrices into a unitary double quaternion (case of 4 channels), unitary quaternion (case of 3 channels) and into an angle (case of 2 channels).

• Calcul de la matrice associée A[i,j] avec : $$\mathrm{A}\left[\mathrm{0},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{1},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{2},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{3},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{0},\mathrm{1}\right]=\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{1},\mathrm{1}\right]=-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{2},\mathrm{1}\right]=-\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{3},\mathrm{1}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{0},\mathrm{2}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{1},\mathrm{2}\right]=\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[\mathrm{2},\mathrm{2}\right]=-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[3,\mathrm{2}\right]=-\mathrm{R}\left[1,\mathrm{0}\right]-\mathrm{R}\left[0,\mathrm{1}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]-\mathrm{R}\left[2,\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[0,3\right]=\mathrm{R}\left[3,\mathrm{0}\right]-\mathrm{R}\left[2,\mathrm{1}\right]+\mathrm{R}\left[1,\mathrm{2}\right]-\mathrm{R}\left[0,\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[1,3\right]=-\mathrm{R}\left[2,\mathrm{0}\right]-\mathrm{R}\left[3,\mathrm{1}\right]-\mathrm{R}\left[0,\mathrm{2}\right]-\mathrm{R}\left[1,\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[2,3\right]=\mathrm{R}\left[1,\mathrm{0}\right]+\mathrm{R}\left[0,\mathrm{1}\right]-\mathrm{R}\left[3,\mathrm{2}\right]-\mathrm{R}\left[2,\mathrm{3}\right]$$
  
  $$\mathrm{A}\left[3,3\right]=-\mathrm{R}\left[0,\mathrm{0}\right]+\mathrm{R}\left[1,\mathrm{1}\right]+\mathrm{R}\left[2,\mathrm{2}\right]-\mathrm{R}\left[3,\mathrm{3}\right]$$
  
  $$\mathrm{A}=\mathrm{A}/\mathrm{4}$$
  
• Calcul des 2 quaternions à partir de la matrice associée
  • A2 = square(A) # carré de coefficients
  • q1 = sqrt(A2.sum(axis=1)) # somme sur le lignes
  • q2= sqrt(A2.sum(axis=0)) # somme sur les colonnes

Détermination des signes

• Pour k=0..3 : Si sign(A[i,k])<0, Alors q2[k] = -q2[k]
• Pour k=0..3 : Si sign(A[k,j])!=sign(q1[k]*q2[j]), Alors q1[k] = -q1[k]

La conversion en quaternion pour le cas 3D peut être réalisée comme suit pour une matrice R[i,j] i,j=0...2 de taille 3x3:
Calcul de la matrice associée simplifiée : $$\mathrm{q}\left[\mathrm{0}\right]={\left(\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{q}\left[\mathrm{1}\right]={\left(\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{q}\left[\mathrm{2}\right]={\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]\right)}^{\wedge }{\left(\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{q}\left[\mathrm{3}\right]={\left(\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}$$

Pour i=0..3 : q[i] = sqrt(q[i])/4This conversion into a double quaternion for the 4D case could be carried out for a rotation matrix whose coefficients are denoted R [i, j], i, j = 0 ... 3, by the following pseudo-code:

• Calculation of the associated matrix A [i, j] with: $$\mathrm{AT}\left[\mathrm{0},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{1},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{2},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{3},\mathrm{0}\right]=\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{0},\mathrm{1}\right]=\mathrm{R}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{1},\mathrm{1}\right]=-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{2},\mathrm{1}\right]=-\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{3},\mathrm{1}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{0},\mathrm{2}\right]=\mathrm{R}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{1},\mathrm{2}\right]=\mathrm{R}\left[\mathrm{3},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{1},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[\mathrm{2},\mathrm{2}\right]=-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{R}\left[\mathrm{3},\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[3,\mathrm{2}\right]=-\mathrm{R}\left[1,\mathrm{0}\right]-\mathrm{R}\left[0,\mathrm{1}\right]-\mathrm{R}\left[\mathrm{3},\mathrm{2}\right]-\mathrm{R}\left[2,\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[0,3\right]=\mathrm{R}\left[3,\mathrm{0}\right]-\mathrm{R}\left[2,\mathrm{1}\right]+\mathrm{R}\left[1,\mathrm{2}\right]-\mathrm{R}\left[0,\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[1,3\right]=-\mathrm{R}\left[2,\mathrm{0}\right]-\mathrm{R}\left[3,\mathrm{1}\right]-\mathrm{R}\left[0,\mathrm{2}\right]-\mathrm{R}\left[1,\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[2,3\right]=\mathrm{R}\left[1,\mathrm{0}\right]+\mathrm{R}\left[0,\mathrm{1}\right]-\mathrm{R}\left[3,\mathrm{2}\right]-\mathrm{R}\left[2,\mathrm{3}\right]$$
  
  $$\mathrm{AT}\left[3,3\right]=-\mathrm{R}\left[0,\mathrm{0}\right]+\mathrm{R}\left[1,\mathrm{1}\right]+\mathrm{R}\left[2,\mathrm{2}\right]-\mathrm{R}\left[3,\mathrm{3}\right]$$
  
  $$\mathrm{AT}=\mathrm{AT}/\mathrm{4}$$
  
• Calculation of the 2 quaternions from the associated matrix
  • A2 = square (A) # square of coefficients
  • q1 = sqrt (A2.sum (axis = 1)) # sum on the lines
  • q2 = sqrt (A2.sum (axis = 0)) # sum over columns

Determination of signs

• For k = 0..3: If sign (A [i, k]) <0, then q2 [k] = -q2 [k]
• For k = 0..3: If sign (A [k, j])! = Sign (q1 [k] * q2 [j]), then q1 [k] = -q1 [k]

The quaternion conversion for the 3D case can be carried out as follows for a matrix R [i, j] i, j = 0 ... 2 of size 3x3:
Calculation of the simplified associated matrix: $$\mathrm{q}\left[\mathrm{0}\right]={\left(\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]+\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{1}\right]={\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{2}\right]={\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]\right)}^{\wedge }{\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{1}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{2}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{1}\right]+\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}$$

$$\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{3}\right]={\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]+\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{1}\right]+\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)}^{\wedge }\mathrm{2}+{\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{2}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{0}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{1}\right]+\mathrm{1}\right)}^{\wedge }\mathrm{2}$$

For i = 0..3: q [i] = sqrt (q [i]) / 4

• Si $$\left(\mathrm{R}\right)$$$$\left(\mathrm{}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{1}\right]=-\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{1}\right]$$
  
• Si $$\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{2}\right]=-\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{2}\right]$$
  
• Si $$\left(\mathrm{R}\right)$$$$\left(\mathrm{}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{3}\right]=-\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{3}\right]$$
  

Calculation of the quaternion q

• Yes $$\left(\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{1}\right]-\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{2}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{1}\right]=-\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{1}\right]$$
  
• Yes $$\left(\mathrm{R}\left[\mathrm{0},\mathrm{2}\right]-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{2},\mathrm{0}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{2}\right]=-\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{2}\right]$$
  
• Yes $$\left(\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">R</figure-callout>}\left[\mathrm{1},\mathrm{0}\right]-\mathrm{R}\left[\mathrm{0},\mathrm{1}\right]\right)<\mathrm{0},\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{3}\right]=-\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="imgf0002.png"\; state="\{\{state\}\}">q</figure-callout>}\left[\mathrm{3}\right]$$
  

• Le bloc 440 peut réaliser une quantification dans le domaine indiqué :
  • * Cas de 4 canaux : la paire de quaternions unitaires q ₁ et q ₂ est quantifiée par un dictionnaire de quantification sphérique en dimension 4 ; par convention on quantifie q ₁ avec un dictionnaire hémisphérique (car q ₁ et -q ₁ correspondent à une même rotation 3D) et q ₂ est quantifié avec un dictionnaire sphérique. Des exemples de dictionnaires peuvent être donnés par des points prédéfinis à partir de polyèdres de dimension 4 ; dans des variantes on pourra quantifier une double représentation axe-angle associés qui serait équivalente au double quaternion ;
  • * Cas de 3 canaux : le quaternion unitaire est quantifié par un dictionnaire de quantification sphérique en dimension 4 - des exemples de dictionnaires peuvent être donnés par des points prédéfinis à partir de polyèdres de dimension 4 ;
  • * Cas de 2 canaux : l'angle est quantifié par quantification scalaire uniforme.

The calculation of the angle for the case of a 2x2 matrix is done according to the methods of the state of the art already known.
In variants, the unit quaternions q1, q2 (4D case) and q (3D case) could be converted into axis-angle representations known from the state of the art.

• Block 440 can perform quantization in the indicated domain:
  • * Case of 4 channels: the pair of unitary quaternions q ₁ and q ₂ is quantized by a spherical quantization dictionary in dimension 4; by convention, q ₁ is quantified with a hemispherical dictionary (because q ₁ and - q ₁ correspond to the same 3D rotation) and q ₂ is quantized with a spherical dictionary. Examples of dictionaries can be given by points predefined from polyhedra of dimension 4; in variants, it is possible to quantify a double associated axis-angle representation which would be equivalent to the double quaternion;
  • * Case of 3 channels: the unitary quaternion is quantified by a spherical quantization dictionary in dimension 4 - examples of dictionaries can be given by predefined points from polyhedra of dimension 4;
  • * Case of 2 channels: the angle is quantized by uniform scalar quantization.

We now describe the block 460 of interpolation of the rotation matrices between two successive frames. It makes it possible to smooth the discontinuities of the channels after the application of these matrices. Typically, if two sets of angles or quaternions are too different from a previous frame t-1 to the next t , audible clicks are to be feared if a smooth transition has not been made between these two frames in sub-frames between these two frames. A passage interpolation is then carried out between the rotation matrix calculated for the frame t -1 and the rotation matrix calculated for the frame t. The encoder interpolates in block 460 the (quantized) representation of the rotation between the current frame and the previous frame to avoid excessively rapid fluctuations of the different channels after transformation. The number of interpolations can be fixed (equal to a predetermined value) or adaptive. Each frame is then divided into sub-frames as a function of the number of interpolations determined in the block 450. Thus, if an adaptive interpolation is used, the block 450 can code on a chosen number of bits the number of interpolations to be performed, and therefore the number of sub-frames to be provided, in the case where this number is determined adaptively; in the case of a fixed interpolation, no information is to be coded.

Next, block 460 converts the rotation matrices to a specific domain representing a rotation matrix. The frame is divided into sub-frames, and in the chosen domain the interpolation is performed for each sub-frame.
For an ambisonic input signal of order 1 (with 4 channels W, X, Y, Z), in block 460, the encoder reconstructs from the 6 quantized Euler angles a quantized 4D rotation matrix and that- ci is then converted to two unit quaternions for interpolation purposes. In a variant where the input of the encoder is a planar ambisonic signal (3 channels W, X, Y), in block 460 the encoder reconstructs from the 3 quantized Euler angles a quantized 3D rotation matrix and the latter is then converted to a unitary quaternion for interpolation purposes. In an alternative where the input of the encoder is a stereo signal, the encoder uses in block 460 the representation of the 2D rotation quantized with a rotation angle.
In the embodiment with 4 channels, for the interpolation of the rotation matrix between frame t and frame t - 1, the rotation matrix calculated for frame t is factored into 2 quaternions (a double quaternion) thanks to the Cayley factorization and we use the double quaternion memorized for the previous frame t -1 and noted ( Q _{L, t -1} , Q _{R, t -1} ).

où α correspond au facteur d'interpolation (α=1/K, 2/K, ... 1) et Ω _L = arccos(Q _L,t-1 · Q _L,t )For each sub-frame, the quaternions two by two are interpolated in each sub-frame.
For the left quaternion ( Q _{L , t} ), the block determines the shortest path between the two possible ( Q _{L , t} or -Q _{L , t} ). Depending on the case, the sign of the quaternion of the current frame is reversed. Then the interpolation is calculated for the left quaternion with linear spherical interpolation (SLERP): $$Q_{L,\mathit{interp}}\left(\alpha \right)=Q_{L,t-1}\frac{{\sin \left(1-\alpha \right)\Omega }_L}{{\mathrm{sin\Omega }}_L}+Q_{L,t}\frac{{\sin \alpha \mathrm{\Omega }}_L}{{\mathrm{sin\Omega }}_L}$$

where α corresponds to the interpolation factor ( α = 1 / K, 2 / K, ... 1) and Ω _L = arccos ( Q _{L, t -1} · Q _{L , t} )

où α correspond au facteur d'interpolation (α=1/K, 2/K, ... 1) et Ω _R = arccos(Q _R,t-1 · Q _R,t )For the right quaternion ( Q _{R , t} ), if there has been an inversion for the left quaternion then we must respect the parity and force the sign of the right quaternion. This sign constraint is hereinafter referred to as the “joint shortest path constraint”. Then the interpolation is calculated similarly to the left quaternion: $$Q_{R,\mathit{interp}}\left(\alpha \right)=Q_{R,t-1}\frac{{\sin \left(1-\alpha \right)\mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}+Q_{R,t}\frac{\sin \alpha \mathrm{\Omega }}{{\mathrm{sin\Omega }}_R}$$

where α corresponds to the interpolation factor ( α = 1 / K, 2 / K, ... 1) and Ω _R = arccos ( Q _{R, t -1} · Q _{R , t} )

• Cas 4D : pour un double quaternion
  • Comme décrit précédemment on calcule les matrices de quaternion et antiquaternion et on calcule le produit matriciel.
• Cas 3D : pour quaternion q=(w,x,y,z) on obtient la matrice M[i,j], i,j=0...2, de taille 3x3 $$\mathrm{xy}=\mathrm{2}*\mathrm{x}*\mathrm{y}$$
  
  $$\mathrm{xz}=\mathrm{2}*\mathrm{x}*\mathrm{z}$$
  
  $$\mathrm{yz}=\mathrm{2}*\mathrm{y}*\mathrm{z}$$
  
  $$\mathrm{wx}=\mathrm{2}*\mathrm{w}*\mathrm{x}$$
  
  $$\mathrm{wy}=\mathrm{2}*\mathrm{w}*\mathrm{y}$$
  
  $$\mathrm{wz}=\mathrm{2}*\mathrm{w}*\mathrm{z}$$
  
  $$\mathrm{xx}=\mathrm{2}*\mathrm{x}*\mathrm{x}$$
  
  $$\mathrm{yy}=\mathrm{2}*\mathrm{y}*\mathrm{y}$$
  
  $$\mathrm{zz}=\mathrm{2}*\mathrm{z}*\mathrm{z}$$
  
  $$\mathrm{M}\left[\mathrm{0}\right]\left[\mathrm{0}\right]=\mathrm{1}-\left(\mathrm{yy}+\mathrm{zz}\right)$$
  
  $$\mathrm{M}\left[0\right]\left[1\right]=\left(\mathrm{xy}-\mathrm{wz}\right)$$
  
  $$\mathrm{M}\left[0\right]\left[2\right]=\left(\mathrm{xz}+\mathrm{wy}\right)$$
  
  $$\mathrm{M}\left[1\right]\left[0\right]=\left(\mathrm{xy}+\mathrm{wz}\right)$$
  
  $$\mathrm{M}-\left[1\right]\left[1\right]=1-\left(\mathrm{xx}+\mathrm{<figure-callout\; id="1"\; label="zz\; M"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">zz</figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[1\right]\left[2\right]=\left(\mathrm{yz}-\mathrm{<figure-callout\; id="2"\; label="wx\; M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">wx</figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[2\right]\left[0\right]=\left(\mathrm{xz}-\mathrm{wy}\right)$$
  
  $$\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">M</figure-callout>}\left[2\right]\left[1\right]=\left(\mathrm{yz}+\mathrm{<figure-callout\; id="2"\; label="wx\; M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">wx</figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[2\right]\left[2\right]=1-\left(\mathrm{xx}+\mathrm{yy}\right);$$
  

Once the interpolation has been calculated for the two quaternions, the rotation matrix of 4x4 dimension is calculated (respectively 3x3 for planar ambisonics or 2x2 for the stereo case).
This conversion into a rotation matrix can be carried out according to the following pseudo-codes:

• 4D case: for a double quaternion
  • As previously described, the quaternion and antiquaternion matrices are calculated and the matrix product is calculated.
• 3D case: for quaternion q = (w, x, y, z) we obtain the matrix M [i, j], i, j = 0 ... 2, of size 3x3 $$\mathrm{xy}=\mathrm{2}*\mathrm{x}*\mathrm{y}$$
  
  $$\mathrm{xz}=\mathrm{2}*\mathrm{x}*\mathrm{z}$$
  
  $$\mathrm{yz}=\mathrm{2}*\mathrm{y}*\mathrm{z}$$
  
  $$\mathrm{wx}=\mathrm{2}*\mathrm{w}*\mathrm{x}$$
  
  $$\mathrm{wy}=\mathrm{2}*\mathrm{w}*\mathrm{y}$$
  
  $$\mathrm{wz}=\mathrm{2}*\mathrm{w}*\mathrm{z}$$
  
  $$\mathrm{xx}=\mathrm{2}*\mathrm{x}*\mathrm{x}$$
  
  $$\mathrm{yy}=\mathrm{2}*\mathrm{y}*\mathrm{y}$$
  
  $$\mathrm{zz}=\mathrm{2}*\mathrm{z}*\mathrm{z}$$
  
  $$\mathrm{M}\left[\mathrm{0}\right]\left[\mathrm{0}\right]=\mathrm{1}-\left(\mathrm{yy}+\mathrm{zz}\right)$$
  
  $$\mathrm{M}\left[0\right]\left[1\right]=\left(\mathrm{xy}-\mathrm{wz}\right)$$
  
  $$\mathrm{M}\left[0\right]\left[2\right]=\left(\mathrm{xz}+\mathrm{wy}\right)$$
  
  $$\mathrm{M}\left[1\right]\left[0\right]=\left(\mathrm{xy}+\mathrm{wz}\right)$$
  
  $$\mathrm{M}-\left[1\right]\left[1\right]=1-\left(\mathrm{xx}+\mathrm{<figure-callout\; id="1"\; label="zz\; M"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">zz\; </figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[1\right]\left[2\right]=\left(\mathrm{yz}-\mathrm{<figure-callout\; id="2"\; label="wx\; M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">wx\; </figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[2\right]\left[0\right]=\left(\mathrm{xz}-\mathrm{wy}\right)$$
  
  $$\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">M</figure-callout>}\left[2\right]\left[1\right]=\left(\mathrm{yz}+\mathrm{<figure-callout\; id="2"\; label="wx\; M"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">wx\; </figure-callout>}\right)$$
  
  $$\mathrm{M}$$$$\mathrm{}\left[2\right]\left[2\right]=1-\left(\mathrm{xx}+\mathrm{yy}\right);$$
  

(ou leurs transposées) calculées par sous-trame dans le bloc 460 d'interpolation sont ensuite utilisées dans le bloc 470 de transformation qui produit n canaux transformés par application des matrices de rotation ainsi trouvées, aux canaux ambisoniques qui ont été prétraités par le bloc 300.Finally, the matrices ${\mathit{V}}_{\mathit{t}}^{\mathit{interp}}\left(\alpha \right)$

(or their transposed) computed by subframe in the interpolation block 460 are then used in the transformation block 470 which produces n channels transformed by applying the rotation matrices thus found, to the ambisonic channels which have been preprocessed by the block 300.

We return below to the number K of subframes to be determined in block 450 for the case where this number is adaptive. The final difference between the current frame and the previous frame or directly from the angular difference of the parameters describing the rotation matrix is measured. In the latter case, an attempt is made to ensure that the angular variation between successive sub-frames is not perceptible. The realization of an adaptive number of sub-frames is above all advantageous for reducing the average complexity of the codec, but if it is chosen to reduce the complexity, it may be preferable to use interpolation with a fixed number of sub-frames.

où I_n est la matrice identité, V_t les vecteurs propres à la trame d'indice t, et ∥M∥ est une norme de la matrice M qui correspond ici à la somme des valeurs absolues de tous les coefficients. D'autres normes matricielles peuvent être utilisées (par exemple la norme de Frobenius).The final difference between the corrected rotation matrix of the frame t and the rotation matrix of the frame t - 1 gives a measure of the importance of the difference in matrixing of the channels between the two frames. The greater this difference, the greater the number of subframes for the interpolation made in block 460. We use the sum of the absolute value of the inter-correlation matrix between the transformation matrix of the current frame and the previous frame, as follows, to measure this difference: $${\delta }_t\Vert {\mathit{I}}_{\mathit{not}}-\mathit{corr}\left({\mathit{V}}_{\mathit{t}},{\mathit{V}}_{\mathit{t}-\mathbf{1}}\right)\Vert$$

where I _n is the identity matrix, V _t the vectors specific to the frame of index t, and ∥ M ∥ is a norm of the matrix M which corresponds here to the sum of the absolute values of all the coefficients. Other matrix standards can be used (for example the Frobenius standard).

• Seuils : {4.0, 5.0, 6.0, 7.0}
• Nombre K des sous-trames pour interpolation : {10, 48, 96, 192}
  Ainsi seuls deux bits peuvent suffire à coder les quatre valeurs possibles donnant le nombre de subdivisions (sous-trames).
  Le nombre K d'interpolations déterminé par le bloc 450 est ensuite envoyé au module d'interpolation 460 et dans le cas adaptatif le nombre de sous-trames est codé sous la forme d'un indice binaire qui est envoyé au multiplexeur (bloc 350).

If the two matrices are identical then this difference is equal to 0. The more the matrices are dissimilar, the more the value of the difference δ _t is high. Predetermined thresholds can be applied to δ _t , with each threshold is associated a predefined number of interpolations, for example according to the following decision logic:

• Thresholds: {4.0, 5.0, 6.0, 7.0}
• Number K of subframes for interpolation: {10, 48, 96, 192}
  Thus only two bits can suffice to encode the four possible values giving the number of subdivisions (sub-frames).
  The number K of interpolations determined by the block 450 is then sent to the interpolation module 460 and in the adaptive case the number of subframes is encoded in the form of a binary index which is sent to the multiplexer (block 350) .

The realization of the interpolation makes it possible to apply in fine an optimization of the decorrelation of the input channels before multi-mono coding. In fact, the rotation matrices calculated respectively for a previous frame t -1 and a current frame t can be very different because of this search for decorrelation, but the interpolation nevertheless makes it possible to smooth this difference.
The interpolation used requires a limited calculation cost for the encoder and the decoder since it is carried out in a specific domain (angle in 2D, quaternion in 3D, double quaternion in 4D). This approach is more advantageous than interpolating covariance matrices calculated for the PCA / KLT analysis and repeating an eigenvalue decomposition type EVD (for “EigenValue Decomposition”) several times per frame.

où X (α) correspond aux sous-blocs de taille n x (L/K) pour α=1/K, 2/K, ... 1. Le signal contenu dans ces canaux est ensuite envoyé au bloc 340 pour l'encodage multi-monos.Block 470 then performs the matrixing of the ambisonic channels by subframe using the transformation matrices calculated in block 460. This matrixing amounts to calculating by subframe. ${\mathit{V}}_{\mathit{t}}^{\mathit{interp}}{\left(\alpha \right)}^T\mathit{X}\left(\alpha \right),$

where X ( α ) corresponds to the sub-blocks of size nx ( L / K ) for α = 1 / K, 2 / K, ... 1. The signal contained in these channels is then sent to block 340 for encoding multi-monos.

We now refer to the figure 5 to describe a decoder in an exemplary embodiment of the invention.
After demultiplexing of the binary train for the current frame t by block 500, the allocation information is decoded (block 510) which makes it possible to de-multiplex and decode (block 520) the binary train (s) (s) received for each of the n transformed channels.
Block 520 calls for multiple instances executed separately from core decoding. The core decoding can be of the EVS type optionally modified to improve its performance. Using a multi-mono approach, each channel is decoded separately. If the Previously used encoding is stereo or multi-channel encoding, the multi-mono approach can be replaced by multi-stereo or multi-channel for decoding. The channels thus decoded are sent to block 530 which decodes the rotation matrix for the current frame and optionally the number K of subframes to be used for the interpolation (if the interpolation is adaptive). For each matrix, the interpolation block 460 splits the frame into sub-frames whose number K can be read in the stream encoded by block 610 ( figure 6 ) and interpolates the rotation matrices, the aim being to find - in the absence of transmission errors - the same matrices as in block 460 of the encoder in order to be able to reverse the transformation which was previously done in block 470.

où X̂ (α) correspond aux sous-blocs successifs de taille n x (L/K) pour α=1/K, 2/K, ... 1.
Le bloc 530 effectue globalement le décodage et de la synthèse PCA/KLT inverse qui a été effectué par le bloc 310 de la figure 3. Les indices de quantification des paramètres de quantification de la rotation dans la trame courante sont décodés dans le bloc 600. Une quantification scalaire peut être utilisée et le pas de quantification est identique pour chaque angle. Dans le cas adaptatif le nombre de sous-trames d'interpolation est décodé (bloc 610) pour retrouver le nombre K de sous-trames parmi l'ensemble {10, 48, 96, 192} ; dans des variantes où la longueur de trames L est différente, cet ensemble de valeurs pourra être adapté. L'interpolation du décodeur est identique à celle effectuée à l'encodeur (bloc 460).Block 530 performs the matrixing inverting that of block 470 to reconstruct a decoded signal, as detailed below with reference to figure 6 . This matrixing amounts to calculating by sub-frame ${\mathit{V}}_{\mathit{t}}^{\mathit{interp}}\left(\alpha \right)\hat{\mathit{X}}\left(\alpha \right),$

where X̂ ( α ) corresponds to the successive sub-blocks of size nx ( L / K ) for α = 1 / K, 2 / K, ... 1.
Block 530 globally performs the decoding and reverse PCA / KLT synthesis that was performed by block 310 of the figure 3 . The quantization indices of the rotation quantization parameters in the current frame are decoded in block 600. Scalar quantization can be used and the quantization step is identical for each angle. In the adaptive case, the number of interpolation sub-frames is decoded (block 610) to find the number K of sub-frames among the set {10, 48, 96, 192}; in variants where the length of frames L is different, this set of values may be adapted. The interpolation of the decoder is identical to that performed at the encoder (block 460).

Block 620 performs the reverse matrixing of the ambisonic channels per subframe using the inverses (transposed in practice) of the transformation matrices calculated in block 460.

Thus, the invention uses a completely different approach than the add / overlap MPEG-H codec based on a specific representation of transformation matrices which are restricted to matrices of rotation from one frame to another, in the temporal domain, allowing in particular an interpolation of the transformation matrices, with a mapping which ensures a coherence in direction (including by taking into account the direction by the sign).

The general approach of the invention is a coding of ambisonic sounds in the time domain by PCA with in particular PCA transformation matrices forced to be rotation matrices and interpolated by sub-frames in an optimized manner (in particular in the field of quaternions / double quaternions) to improve the quality. The interpolation step is either fixed or adaptive as a function of a criterion of difference between an inter-correlation matrix and a reference matrix (identity) or between matrices to be interpolated. The quantification of the rotation matrices can be implemented in the domain of generalized Euler angles. However, it may be preferentially chosen to quantify the matrices of dimension 3 and 4 in the domain of quaternions and double quaternions (respectively), which makes it possible to remain in the same domain for the quantization and the interpolation.
In addition, eigenvector alignment is used to avoid the problems of clicks and channel inversion from frame to frame.

Of course, the present invention is not limited to the embodiments described above by way of example and extends to other variants.

Thus, the foregoing description has dealt with the cases of four channels.
However, in variants, it is also possible to encode a number of channels greater than four. The implementation remains identical (in terms of functional blocks) to the case n = 4, but the interpolation by double quaternion is replaced by the general method below.
The transformation matrices at frames t - 1 and t are denoted V _{t -1} and V _t . The interpolation can be performed with a factor α between V _{t -1} and V _t such that: $${\mathit{V}}_{\mathit{t}}^{\mathit{interp}}\left(\alpha \right)={\mathit{V}}_{\mathit{t}-\mathbf{1}}{\left({\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\right)}^{\alpha }$$

peut se calculer directement par décomposition en valeurs propres de ${\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\mathrm{.}$

En effet, si ${\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}={\mathit{QLQ}}^{\mathit{T}},$

on a: ${\left({\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\right)}^{\alpha }={\mathit{QL}}^{\alpha }{\mathit{Q}}^{\mathit{T}}\mathrm{.}$

On notera également que cette variante pourrait aussi remplacer l'interpolation par double quaternion unitaire (cas 4D), quaternion unitaire (cas 3D) ou angle, cependant elle serait moins avantageuse car elle nécessiterait une étape de diagonalisation supplémentaire et des calculs de puissance, alors que le mode de réalisation décrit précédemment est plus efficace pour ces cas de 2, 3 ou 4 canaux.The term ${\left({\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\right)}^{\alpha }$

can be calculated directly by decomposition in eigenvalues of ${\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\mathrm{.}$

Indeed, if ${\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}={\mathit{QLQ}}^{\mathit{T}},$

we have: ${\left({\mathit{V}}_{\mathit{t}-\mathbf{1}}^{\mathit{T}}{\mathit{V}}_{\mathit{t}}\right)}^{\alpha }={\mathit{QL}}^{\alpha }{\mathit{Q}}^{\mathit{T}}\mathrm{.}$

Note also that this variant could also replace the interpolation by double unitary quaternion (4D case), unitary quaternion (3D case) or angle, however it would be less advantageous because it would require an additional diagonalization step and power calculations, then that the embodiment described above is more effective for these cases of 2, 3 or 4 channels.

Claims (15)

Method for the compression coding of sound signals forming a succession in time of frames (t-1, t) of samples, in each of N channels in ambisonic representation of order greater than 0, the method comprising: - form, from the channels for a current frame (t), a covariance matrix between channels and search for eigenvectors of the covariance matrix to obtain a matrix of eigenvectors, - test the eigenvector matrix to check that it represents a rotation in a space of dimension N and otherwise correct the eigenvector matrix until obtaining a rotation matrix, for the current frame (t), and applying said rotation matrix to the signals of the N channels before encoding by separate channels of said signals. A method according to claim 1, further comprising: - compare the eigenvector matrix obtained for the current frame (t) with a rotation matrix obtained for a frame (t-1) preceding the current frame (t), and - permute columns of the eigenvector matrix of the current frame (t) to ensure consistency with the rotation matrix of the previous frame (t-1). A method according to claim 2, wherein said permutation of the columns ensures a consistency of axes of the vectors, and the method further comprises: - check, for each eigenvector of the current frame (t), a coherence of direction with a column vector of corresponding position of the rotation matrix of the previous frame (t-1), and - in the event of inconsistency, invert the sign of the elements of this eigenvector in the matrix of eigenvectors of the current frame (t). Method according to one of the preceding claims, further comprising: - an estimate of the difference between the rotation matrix obtained for the current frame (t) and a rotation matrix obtained for a frame (t-1) preceding the current frame, - depending on the estimated difference, determine whether at least one interpolation is to be made between the rotation matrix of the current frame (t) and the rotation matrix of the previous frame (t-1). A method according to claim 4, wherein: - as a function of the estimated difference, a number of interpolations to be performed between the rotation matrix of the current frame (t) and the rotation matrix of the previous frame (t-1) is determined, - the current frame is divided into a number of sub-frames corresponding to the number of interpolations to be operated, and - at least this number of interpolations is coded with a view to transmission via a network. Method according to one of the preceding claims, in which a permutation between columns of the eigenvector matrix inverting the sign of a determinant of the eigenvector matrix and the determinant of a rotation matrix being equal to 1,
if the determinant of the eigenvector matrix is equal to -1, the signs of the elements of a chosen column of the eigenvector matrix are reversed, so that the determinant is equal to 1 and thus form a rotation matrix. Method according to one of the preceding claims, in which the ambisonic representation is of order 1 and the number N of channels is four, and in which the rotation matrix of the current frame is represented by two quaternions. A method according to claim 7, taken in combination with claim 6, wherein each interpolation for a current subframe is linear spherical interpolation (SLERP), carried out as a function of the interpolation of the subframe preceding the subframe. current and from the quaternions of the previous subframe. A method according to claim 8, wherein the linear spherical interpolation of the current subframe is carried out to obtain the quaternions of the current subframe as follows: $$Q_{L,\mathit{interp}}\left(\alpha \right)=Q_{L,t-1}\frac{{\sin \left(1-\alpha \right)\Omega }_L}{{\mathrm{sin\Omega }}_L}+Q_{L,t}\frac{{\sin \alpha \mathrm{\Omega }}_L}{{\mathrm{sin\Omega }}_L}$$

$$Q_{R,\mathit{interp}}\left(\alpha \right)=Q_{R,t-1}\frac{{\sin \left(1-\alpha \right)\mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}+Q_{R,t}\frac{{\sin \alpha \mathrm{\Omega }}_R}{{\mathrm{sin\Omega }}_R}$$

Or: Q _{L, t -1} is one of the quaternions of the previous subframe t-1, Q _{R, t -1} is the other of the quaternions of the previous subframe t-1, Q _{L, t} is one of the quaternions of the current subframe t, Q _{R, t} is the other of the quaternions of the current sub-frame t, $${\mathrm{\Omega }}_L=\mathrm{Arccos}\left(Q_{L,t-1}\cdot Q_{L,t}\right);{\mathrm{\Omega }}_R=\mathrm{Arccos}\left(Q_{R,t-1}\cdot Q_{R,t}\right)$$

and α corresponds to an interpolation factor. Method according to one of the preceding claims, in which the search for the eigenvectors is carried out by principal component analysis (PCA), or by Karhunen Loeve transform (KLT), in the time domain. Method according to one of the preceding claims, in which a preliminary step of forecasting the budget for allocating bits per ambisonic channel is implemented and comprises: - for each ambisonic channel, an estimate of current acoustic energy in the channel, - the selection in a memory of a predetermined score, of quality (MOS), depending on this ambisonic channel and a current flow in the network, the estimation of a weighting to be operated for the allocation of bits to this channel, by multiplying the score selected at the estimated energy. Method for decoding sound signals forming a succession in time of frames (t-1, t) of samples, in each of N channels in ambisonic representation of order greater than 0, the method comprising: - receive, for a current frame (t), in addition to the signals of the N channels of this current frame, parameters of a rotation matrix, - construct an inverse rotation matrix from said parameters, - Applying said reverse rotation matrix to signals from the N received channels, before separate channel decoding of said signals. Coding device comprising a processing circuit for implementing the method according to one of claims 1 to 11. Decoding device comprising a processing circuit for implementing the method according to claim 12. Computer program comprising instructions for implementing the method according to one of claims 1 to 12, when said instructions are executed by a processor of a processing circuit.

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