FR3134198A1 - Method and device for decoding a digital hologram, method and device for coding a digital hologram and associated computer program - Google Patents

Abstract

A method for decoding a digital hologram (Hk+1) comprises the following steps: - obtaining (E30) a reference digital hologram (Hk) represented by a representation in a spatio-frequency domain D; - decoding (E40) of a residue (ek+1) from data received (D3, D4); - prediction (E42) of a digital hologram predicted by transformation of the reference digital hologram (Hk); - obtaining (E44) the decoded digital hologram (Hk+1) by combining the residue (ek+1) and the predicted hologram. The prediction step (E42) comprises the application of a function χ from D to D such that the subset of elements of the product space D x D of form (e, χ(e)) is the sub -set defined by the elements of the form (x, y, , , , , ηx,ηy) for (x, y, ηx, ηy) varying in ℝ4, where φ is the scalar product of a first vector and a vector resulting from the application of the transformation to a second vector. Figure for abstract: figure 7

Description

Method and device for decoding a digital hologram, method and device for coding a digital hologram and associated computer program Technical field of the invention

The present invention relates to the technical field of digital holography.

It relates more particularly to a method and a device for decoding a digital hologram, as well as a method and a device for coding a digital hologram and an associated computer program.

State of the art

A digital hologram is the recording in digital form of a section, at the level of a reference plane, of a light field propagating in a three-dimensional space, with a view to being able to subsequently restore this light field to a user.

The physical properties of light fields have led to digital holograms being represented by means of coefficients respectively associated with elements of a spatio-frequency domain defined by two spatial dimensions (which correspond to the reference plane) and by the two associated frequency dimensions.

In the context of coding a sequence of such digital holograms, the patent application published as WO 2021/004 797 proposes to use residues defined respectively by difference between first coefficients associated with elements of the spatio-frequency domain to represent a first digital hologram and second coefficients representing a second digital hologram and associated with respective images of these elements by a transformation.

Presentation of the invention

The invention proposes a method for decoding a digital hologram propagating in a three-dimensional propagation space from received data, comprising the following steps:

- obtaining a reference digital hologram defined in a reference plane and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;

- decoding of a residue from the data received;

- prediction of a digital hologram predicted by transformation of the reference digital hologram;

- obtaining the decoded digital hologram by combining the residue and the predicted hologram,

characterized in that the step of predicting the predicted digital hologram comprises, for the determination of the predicted digital hologram at the level of a plane parallel to the reference plane, the application of a function χ having the set of departure the spatio-frequency domain D and for arrival set the spatio-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e)) is the sub -set defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:

- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and

- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables.

The use of the χ function makes it possible to precisely take into account the effect of the transformation on the representation of the digital hologram in the spatio-frequency domain D.

The representation of the reference digital hologram may include, for example, a set of first coefficients respectively associated with elements of the spatio-frequency domain D.

The predicted digital hologram can be represented by a set of second coefficients respectively associated with elements of the spatio-frequency domain D.

The step of predicting the predicted digital hologram can then comprise, for the determination of a second coefficient associated with a given element of the spatio-frequency domain D, the application of the function χ to said given element in order to obtain a other element of the spatio-frequency domain D and the use of the first coefficient associated with this other element.

In other words, to determine the second coefficient associated with a given element of the spatio-frequency domain D (in the representation of the predicted digital hologram), the first associated coefficient (in the representation of the reference digital hologram) is used. like this element given by the function χ.

By considering the set of first coefficients as a distribution over the spatio-frequency domain D, the prediction step therefore includes a composition of this distribution and the function χ.

The prediction step may further comprise a sub-step of obtaining the second coefficient associated with the given element by multiplication by a spatio-frequency distribution function with real values on the spatio-frequency domain D and/or variable in function only according to the two frequency dimensions.

In practice, said representation can result from a decomposition of the reference digital hologram into Gabor frames; said Gabor frames can in this case correspond respectively to said first coefficients.

The decoding method may comprise, for each of a plurality of objects in the three-dimensional scene represented by the digital reference hologram, the application of a function χ_mfrom the spatio-frequency domain D towards itself such that the subset of the elements of the space produces D x D of form (e, χ_m(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ_mis a non-quadratic generating function with four variables defined as the scalar product of said first vector and a vector resulting from the application to said second vector of a transformation associated with the object concerned.

In this case, several objects of the three-dimensional scene are processed separately using a transformation relating to this object (determined for example by analysis of the movement of this object) and the function χ _m associated with this transformation.

It can also be envisaged that the step of predicting the predicted digital hologram includes (for the determination of a coefficient participating in the representation of the predicted digital hologram) a step of determining a corrective term as a function of the acquisition wavelength of the digital hologram.

The coefficient associated with the element (x ₀ , y ₀ , η ₀ , ξ ₀ ) participating in the representation of the predicted digital hologram can be determined for example by using the formula:

W _k (χ(x ₀ , y ₀ , η ₀ , ξ ₀ ) ) + (x ₀ , y ₀ , η ₀ , ξ ₀ )

where W _k (ε) is the first coefficient associated with an element ε of the spatio-frequency domain D, is the acquisition wavelength of the digital hologram and is a complex function defined on the spatio-frequency domain D.

According to another possibility, the coefficient associated with the element (x ₀ , y ₀ , η ₀ , ξ ₀ ) participating in the representation of the predicted digital hologram can be determined using the formula:

[W _k (χ(x ₀ , y ₀ , η ₀ , ξ ₀ )) + (x ₀ , y ₀ , η ₀ , ξ ₀ )].Δ(x ₀ , y ₀ , η ₀ , ξ ₀ )

where W _k (ε) is the first coefficient associated with an element ε of the spatio-frequency domain D, is the acquisition wavelength of the digital hologram, is a complex function defined on the spatio-frequency domain D and Δ is the spatio-frequency distribution function.

It can also be provided that, the reference digital hologram or the residue being represented by a set of coefficients respectively associated with elements of the spatio-frequency domain D each corresponding to a multiplet formed by a first spatial parameter, a second spatial parameter , a first frequency parameter and a second frequency parameter, the method comprises the following steps, carried out for at least one pair of respective values of the first frequency parameter and the second frequency parameter:

- decoding of data indicative of positions associated respectively with non-zero coefficients among a set of positions associated in a predefined manner with the coefficients associated respectively with the different bytes comprising said pair of values of the first frequency parameter and the second frequency parameter;

- decoding of the coefficients associated respectively with the positions indicated by said indicative data.

According to a first possibility of realization, at least one indicative data among said indicative data represents a difference between a position associated with a non-zero coefficient and another position associated with another non-zero coefficient.

According to a second possibility of realization, said indicative data designate at least one position associated with a non-zero coefficient without designating at least one other position associated with another non-zero coefficient and indicated by indicative data decoded for another pair of values respective of the first frequency parameter and the second frequency parameter.

Furthermore, the coefficients associated with the different bytes defined by said pair of values of the first frequency parameter and the second frequency parameter can be ordered in a predefined manner in an ordered sequence of coefficients; the data indicative of a position associated with a non-zero coefficient can then indicate the position of this non-zero coefficient in said ordered sequence of coefficients.

In this case, we can predict that at the coefficient decoding step, the non-zero coefficients are decoded in the order defined by the ordered sequence.

The invention also proposes a method for coding a digital hologram propagating in a three-dimensional propagation space, comprising the following steps:

- prediction of a digital hologram predicted by transformation of a reference digital hologram defined in a reference plane and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;

- determination of a residue between said digital hologram and the predicted digital hologram,

- coding of the determined residue;

characterized in that the step of predicting the predicted digital hologram comprises, for the determination of the predicted digital hologram at the level of a plane parallel to the reference plane, the application of a function χ having the set of departure the spatio-frequency domain D and for arrival set the spatio-frequency domain D, and such that the subset of elements of the product space D x D of form (e, χ(e)) is the sub -set defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:

- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and

- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables.

The invention also proposes a device for decoding a digital hologram propagating in a three-dimensional propagation space from received data, comprising:

- a first decoding unit designed to obtain a reference digital hologram defined in a reference plane and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;

- a second decoding unit designed to decode a residue from the received data;

- a prediction unit designed to predict a digital hologram predicted by transforming the reference digital hologram;

- a combination unit designed to obtain the decoded digital hologram by combining the residue and the predicted hologram,

characterized in that the prediction unit is designed to apply, for the determination of the digital hologram predicted at the level of a plane parallel to the reference plane, a function χ having as starting set the spatio-frequency domain D and for arrival set the spatio-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e)) is the subset defined by the elements of the shape (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:

- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and

- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables.

The invention further proposes a device for coding a digital hologram propagating in a three-dimensional propagation space, comprising the following steps:

- a prediction unit designed to predict a digital hologram predicted by transformation of a reference digital hologram defined in a reference plane and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;

- a unit for determining a residue between said digital hologram and the predicted digital hologram,

- a coding unit designed to code the determined residue;

characterized in that the prediction unit is designed to apply, for the determination of the digital hologram predicted at the level of a plane parallel to the reference plane, a function χ having as starting set the spatio-frequency domain D and for arrival set the spatio-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e)) is the subset defined by the elements of the shape (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:

- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and

- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables

We also propose a decoding device comprising a processor configured to implement a decoding method as defined above.

The invention also proposes a computer program comprising instructions designed to implement one of the methods as defined above when these instructions are executed by a processor.

The methods proposed above can thus be implemented by this processor, for example when this processor equips a computer or a digital holography device as mentioned above and described below.

The invention finally proposes an information medium (for example non-transitory) readable by a computer, on which such a computer program is recorded.

Of course, the different characteristics, variants and embodiments of the invention can be associated with each other in various combinations as long as they are not incompatible or exclusive of each other.

Detailed description of the invention

In addition, various other characteristics of the invention emerge from the appended description made with reference to the drawings which illustrate non-limiting forms of embodiment of the invention and where:

schematically represents a digital hologram;

represents an example of a digital holography device;

represents an example of a method implemented by a processing module of the digital holography device of the ;

represents a method of decoding a digital hologram from data received in a data stream;

represents an example of a coding device according to the invention;

represents an example of a decoding device according to the invention;

represents a method of exchanging data between the coding device of the and the decoding device of the ;

represents a variant embodiment of the process of ;

represents a method for partitioning the spatio-frequency domain D;

represents a method for coding coefficients associated respectively with elements of the spatio-frequency domain D; And

represents a decoding method corresponding to the coding method of the .

There schematically represents a digital hologram defined by means of a function ψ₀ in a reference plane P_refof equation (z=0) in a propagation space E where the three-dimensional coordinates of a point are defined in an orthonormal coordinate system (O,x,y,z). In other words, the axes (Ox) and (Oy) (or more simply the x and y axes) are included in the reference plane P_refand/or define the reference plane P_ref. The direction of propagation (before application of the transformation mentioned below) thus corresponds here to the axis (Oz); in this situation, the direction of propagation (z) is perpendicular (or orthogonal) to the reference plane P_ref.

The wave field (here representing the monochromatic light field of wavelength λ) associated with this hologram at a point in the propagation space E with coordinates r=(x,y,z) is then written (before application of the transformation mentioned below):

where the Fourier transform of the function ψ ₀ is defined (in the reference plane P _ref ) by

λ is the wavelength of the (monochromatic) light concerned, η = (η _x , η _y , η _z ) with η _z = SQRT(1 - η _x ² - η _y ² ), SQRT() being the square root operator, <r,η> is the scalar product of r and η, and D _λ is the domain defined by {η _x ² + η _y ² < 1}.

The wave field defined by the digital hologram can thus be seen as a sum (in theory continuous or infinite) of plane waves of wavelength λ.

In order to take into account the particularities of holograms, we use representations of these holograms in the form of distribution in a 4-dimensional spatio-frequency domain D (or phase space), the first 2 dimensions of which correspond to the spatial coordinates in the plane P of the hologram and the 2 other dimensions correspond to the spatial frequencies in this same plane. Such a distribution includes for example the light energy values respectively associated with the spatial coordinates and the spatial frequencies of the space-frequency domain D.

According to a first example, we can apply a windowed Fourier transform (in English: “ Windowed Fourier Transform ”) to the wave field ψ; the distribution W _ψ ^Φ is then written:

where Φ is a so-called window function, ψ* is the (complex) conjugate of ψ, X represents the spatial coordinates (x, y) and Ξ the spatial frequencies (ξ _x , ξ _y ).

According to a second example, we can use a Wigner-Ville distribution W _ψ obtained by applying a Wigner-Ville transform to the wave field ψ:

with as previously X representing the spatial coordinates (x, y) and Ξ the spatial frequencies (ξ _x , ξ _y ).

It is therefore a particular windowed Fourier transform, in which the window function used is the function ψ.

According to a third example, we can use a decomposition into Gabor frames (in English: “ Gabor frames ”), which corresponds to a discretization of the windowed Fourier transform.

To do this, we define a mesh Λ of the spatio-frequency domain D as follows:

where A is an invertible matrix of dimensions 4x4.

The decomposition into Gabor frames can then be written:

where Φ ^D is the dual window of the function Φ, π(ζ) is the translation in the spatio-frequency domain D of vector ζ and with the following modulation/translation operator:

where plan).

We study here the field ψ _T obtained when we apply a transformation T of the three-dimensional space E to the digital hologram.

According to Kyoji Matsushima, Hagen Schimmel & Frank Wyrowski, “ Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves ”, in J. Opt. Soc. Am. A 20, 1755-1762 (2003), the wave field (or light field) modified by the transformation T is written as follows (at the level of a plane parallel to the reference plane P _ref and with coordinate z ):

This expression implies in particular the scalar product of the coordinate vector (xyz) ^t of the three-dimensional propagation space E and the vector Tη resulting from the application of the transformation T to the wave vector (η _x , η _y , η _z ) of predefined standard C and whose first two coordinates are the frequency variables (η _x , η _y ) respectively associated with the spatial coordinates (x,y) in the reference plane P _ref of the digital hologram. The predefined norm C depends on the representation used and is here worth 1 (normalized representation). Alternatively, the predefined standard could be equal to the wavelength λ.

In writing :

,

the expression of the function ψ _z ^T given above can be seen as a Fourier integral operator used in semi-classical analysis in the field of quantum mechanics, where the function φ defined above is the generating function (in English " generating function ") associated with this operator.

By focusing on a particular plane P _z parallel to the reference plane, with coordinate z in the direction perpendicular to the reference plane P _ref , we can write the generating function as a function with four variables (i.e. a function with values on ℝ ⁴ ) as follows:

The generating function φ _z is therefore the scalar product:

- a first vector (of three-dimensional coordinates), whose first two coordinates correspond to the first two variables of the four variables and whose third coordinate corresponds to the z coordinate of the plane P _z concerned, and

- a vector resulting from the application of the transformation T to a second vector, whose first two coordinates correspond to the last two variables of the four variables and, here, whose norm C is a predefined constant (this norm C being equivalent here 1 due to the normalization already mentioned; alternatively, in other representations, this norm C could for example be equal to the wavelength λ).

The explanations are given below for a plane P _z with given z coordinate (plane parallel to the reference plane P _ref ), but are valid for any plane parallel to the reference plane P _ref ( ie for any z). For the sake of simplification, we note in the following φ the generating function φ _z associated with this plane P _z .

In the context defined above, if we note χ the symplectomorphism of the spatio-frequency domain D towards itself which corresponds to the transformation T, the subset (or graph) Γ of ℝ ⁴ x ℝ ⁴ comprising the elements of the form (e, χ(e)), for e varying in ℝ ⁴ , can also be defined as the set of elements of the form:

for (x,y,η _u , η _v ) varying in ℝ ⁴ .

We can refer to the work “ Semi-classical Analysis ”, by Victor Guillemin and Shlomo Sternberg, International Press, 2013, ISBN 1571462767, 9781571462763 for more explanations on this subject.

We recall that a symplectomorphism is a diffeomorphism which preserves the symplectic form, that is to say whose Jacobian matrix M satisfies: J=M ^t JM,

with :

Noting

we can give the expression of the variables η _u and η _v as a function of e = (x,y,ξ _x ,ξ _y ):

The ^5th and ^6th coordinates (u, v) of the elements of the subset Γ can therefore be written:

We thus obtain a parametric expression of the subset Γ as a function of its first four coordinates (x,y,ξ _x ,ξ _y ), which makes it possible to obtain the following expression of the symplectomorphism χ (remembering that the sub- set Γ corresponds to the elements of the form (e, χ(e)) for e varying in ℝ ⁴ ):

Considering now that the transformation T studied is a rigid transformation of the propagation space E, this transformation T is therefore the combination of a translation of vector (t _x ,t _y ,t _z ) and a rotation R which can be decomposed into 3 rotations using Euler angles: R = R ^z _α R ^x _θ R ^z _β ,

where R ^z _α and R ^z _β are rotations around the z axis with respective angles α and β, and R ^x _θ is a rotation around the x axis with angle θ.

We can thus define symplectomorphisms χ _t , χ _α , χ _θ , χ _β respectively associated with the translation and these 3 rotations R ^z _α , R ^x _θ , R ^z _β , and whose composition is the symplectomorphism χ associated with the transformation T:

χ = χ_βo χ_θo χ_α o χ_t, where o is the composition operator.

The symplectomorphisms χ _t , χ _α , χ _β associated with the translation and rotations R ^z _α , R ^x _θ , R ^z _β around the Z axis are simply written:

χ _t (x,y,ξ _x ,ξ _y ) = (x+t _x ,y+t _y ,ξ _x ,ξ _y )

χ _α (x,y,ξ _x ,ξ _y ) = (x.cos α–y.sin α,x.sin α+y.cos α, ξ _x .cos α–ξ _y .sin α, ξ _x . sin α+ξ _y .cos α)

χ _β (x,y,ξ _x ,ξ _y ) = (x.cos β –y.sin β,x.sin β +y.cos β, ξ _x .cos β –ξ _y .sin β, ξ _x . sin β +ξ _y .cos β)

We note that the third component t _z of the translation is taken into account in the propagation, that is to say by considering the plane P _z+tz .

We propose to use the developments presented above to determine the symplectomorphism χ _θ associated with the rotation R ^x _θ of angle θ around the x axis.

In this case, the generating function φ is written:

either :

Using the fact that:

(see above the definition of the subset Γ) and replacing η _z by its expression as a function of η _x and η _y given above, we obtain:

_ξx = _ηx

ξ _y = (η _y .cos θ) – SQRT(1 – η _x ² – η _y ² ).sin θ

By taking the square of the two terms of this equation, we obtain a quadratic equation whose only solution having a physical meaning is:

η _y = (ξ _y .cos θ) + SQRT(1 – ξ _x ² – ξ _y ² ).sin θ

Using the definition of the subset Γ again, we can finalize the expression of the symplectomorphism χ _θ as follows:

χ _θ (x,y,ξ _x ,ξ _y ) = (U,V,ξ _x , (ξ _y .cos θ) + SQRT(1 – ξ _x ² – ξ _y ² ).sin θ)

with

Note that the variables η _x and η _y in these latter expressions are expressed as a function of ξ _x and ξ _y a few lines above.

We are now interested in the consequences of the transformation T of the three-dimensional propagation space E on the representation W _ψ ^Φ , W _ψ of the digital hologram (also called distribution).

The representations W _ψ ^Φ , W _ψ of the hologram in the spatio-frequency domain D can be seen as the symbol P of a pseudo-differential operator Op(P). Continuing the parallel with the theories of quantum mechanics, we can refer to the article " How Wigner Functions Transform Under Symplectic Maps ", by Alex J. Dragt and Salman Habib, in Proceedings of the Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics (Monterey, CA) 1998.

In the first example given above (windowed Fourier transform), the symbol differential operator W _ψ ^Φ is

Op(W _ψ ^Φ ) = | Φ><ψ |

where we use the usual notation in quantum mechanics introduced in the article " A new notation for quantum mechanics ", by PAM Dirac, in Mathematical Proceedings of the Cambridge Philosophical Society, July 1939, p. 416-418.

In the second example (Wigner-Ville distribution), the symbol differential operator W _ψ is:

Op(W _ψ ) = | ψ><ψ|.

Concerning the third example (Gabor frames), we notice that the decomposition uses functions of the same type as that used in the first example (windowed Fourier transforms), within a sum over the elements of the mesh, and the developments below therefore apply because they apply to the different terms of the sum.

Continuing the parallel with the theories of semi-classical analysis (we can for example refer to the article " Uniform semiclassical estimates for the propagation of quantum observables " by A. Bouzouina and D. Robert in Duke Mathematical Journal, 111 (2) 223-252, February ¹ , 2002), the effects of the transformation T on the hologram can then be rendered by applying the propagation operator Op(χ) (associated with the symplectomorphism χ) to the pseudo-operator differential Op(P) so that the symbol P' of the propagated differential operator Op(χ)Op(P)[Op(χ)] ^-1 forms a representation of the hologram resulting from the transformation T.

According to Egorov's theorem (see for example the article by A. Bouzouina and D. Robert cited above), the symbol P' of the propagated differential operator Op(χ)Op(P)[Op(χ)] ^-1 can be written:

where χ is the symplectomorphism associated with the transformation T as explained above, λ is the wavelength of the light concerned as already mentioned, and where the functions f _n are defined as a sum of integrals of differential operators bi- differentials; for example, in the Weyl quantification used, the function f ₁ is zero and the functions f _n can be defined as follows for n > 1:

where H _t and χ _t are respectively the Hamiltonian and the Hamiltonian flow at time t (see below), i is here as previously the imaginary number such that i²=-1, and where the extended Poisson bracket is defined as following :

where g and h are scalar functions of (q, p) on the spatio-frequency domain D.

Thus, when the generating function φ is non-quadratic, that is to say the symplectomorphism χ is non-linear as explained in the article " How Wigner Functions Transform Under Symplectic Maps " cited above, the representation P' of l The hologram deformed by the transformation T is not limited to a deformation of the spatio-frequency domain D by means of the symplectomorphism χ (term P(χ) in the formula above), but includes additional corrective terms which each depend on the wavelength λ, precisely which each include a factor equal to a power of the wavelength λ. In practice, as indicated below, we can use a single additional corrective term which includes a factor equal to the wavelength λ or, here, equal to the square λ² of the wavelength λ.

The Hamiltonian H _t and the Hamiltonian flow χ _t are linked by the Hamilton equations:

In practice, we can decompose the Hamiltonian flow χ _t into two Hamiltonian flows respectively linked to a translation and to a rotation, and therefore consider a Hamiltonian H ^trans linked to the translation and a Hamiltonian H ^rot linked to the rotation.

The Hamiltonian linked to the translation H ^trans simply corresponds to free propagation in a homogeneous medium with refractive index n ₀ (we can take in practice n ₀ = 1 for propagation in air): H ^trans = SQRT(n ₀ ²-|p|²).

Concerning the Hamiltonian linked to the rotation H ^rot , we can obtain the expression of its derivatives (which alone are necessary to evaluate the functions f _n defined above) by reporting the literal expression of the symplectomorphism χ _θ in Hamilton's formulas recalled above:

with R=SQRT(1 – ξ _x ² – ξ _y ²).

Higher order derivatives can be obtained on this basis by numerical methods (such as finite differences).

There represents an example of a digital holography device 10 using the teachings of the invention.

This digital holography device 10 comprises a processing module 2, a storage module 4, an acquisition module 6 and a holographic restitution module 8.

The digital holography device 10 is for example produced in the form of a display headset or augmented reality glasses.

The processing module 2 is for example a processor, here a microprocessor.

The acquisition module 6 comprises at least one movement sensor (for example of the accelerometer or gyroscope type) which transmits measurement signals (in analog or digital form) to the processing module 2, on the basis of which the processing module 2 can produce data representative of the position and/or orientation of the digital holography device 10.

The storage module 4 comprises for example a memory and/or a hard disk.

The storage module 4 notably stores computer program instructions designed so that the processing module 2 implements at least some of the steps of the method described below with reference to the when these instructions are executed by the processing module 2.

As explained below, the storage module 4 further stores a representation W of a digital hologram in the spatio-frequency domain D already mentioned. In practice, the storage module stores values W(L) respectively associated with different elements L of the spatio-frequency domain D (that is to say in practice with different elements of ℝ ⁴ ).

The holographic restitution module 8 here comprises a light modulator and a light source of wavelength λ, and allows the restitution of the digital hologram represented by the representation W to a user. We can for example refer to the patent application published as WO2019/001968 on this subject.

There represents an example of a method implemented by the processing module 2 of the digital holography device 10.

This process begins with a step E2 of receiving measurement signals from the acquisition module 6.

The processing module 2 then determines data representative of the position and/or orientation of the digital holography device 10 on the basis of the measurement signals received (step E4).

In order to present the digital hologram to the user taking into account the movement of the digital holography device 10 (and thus to simulate the reciprocal movement of the digital hologram relative to the user), the processing module 2 applies to the digital hologram (represented by the representation W as mentioned above) a transformation T defined in the three-dimensional propagation space E as a function of the aforementioned representative data.

To do this, processing module 2 determines in step E6 the symplectomorphism χ associated with this transformation T using the method proposed above. As already indicated, in a plane P_zparallel to the reference plane P_ref, the function χ (symplectomorphism) is such that the subset of the elements of the product space D x D of form (e, χ(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:

- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane P _z in the direction z perpendicular to the reference plane P _ref , and

- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables and, here, whose norm is predefined.

The processing module can thus determine in step E8 the representation W' of the transformed digital hologram (or digital hologram modified by application of the transformation T) on the basis of the representation W stored in the storage module 4 and the function χ (symplectomorphism) determined in step E6. Alternatively, the representation W (that is to say in practice data representative of the aforementioned values W(L)) could be received by the digital holography device 10 by means of a communication module (not shown) .

In accordance with what has been explained previously, the representation W' of the transformed digital hologram can be determined as follows:

for the different L of the spatio-frequency domain D used in the representation concerned, where N is a predetermined value for a given digital holography device (which corresponds to the desired degree of precision), λ is as already indicated the wavelength associated with (and therefore used to restore) the digital hologram, o is, as already indicated, the composition operator and where the functions f _n are defined above.

In practice, we can use only the first non-zero term (called " corrective term ") of the sum in the formula above; the processing module determines in this case the representation W' of the digital hologram modified as follows (the factor f ₁ being zero with the Weyl quantification used here):

with

where {.} ₃ is the Poisson bracket defined above.

Processing module 2 is thus designed to apply the function χ when applying the transformation T to the digital hologram. The function χ is particularly applied when composing this function with the representation W (term W o χ).

Furthermore, as indicated above, it is envisaged here that the distribution W' representing the digital hologram transformed by the transformation T is determined by the processing module 2 using at least one corrective term which depends on the wavelength. λ. This corrective term is non-zero because the function χ is non-linear (the generating function φ being non-quadratic as explained above).

The processing module 2 can then control the restitution of the transformed digital hologram (represented by the representation or distribution W') by means of the holographic restitution module 8.

To do this, the processing module 2 reconstructs for example in step E10 the digital hologram transformed from the representation W' obtained in step E8. This reconstruction step can in practice include a summation of light contributions respectively associated with the values W'(L) defining the representation W' for different elements L of the spatio-frequency domain D.

The processing module 2 then sends, in step E12, signals representative of the reconstructed digital hologram to the holographic restitution module 8 (in practice to the light modulator equipping this holographic restitution module 8) in order to allow the restitution of the transformed digital hologram to the user, here by illuminating, with the light source of wavelength λ, the light modulator configured on the basis of the signals representative of the reconstructed digital hologram.

Alternatively, the restitution of step E12 could be carried out on a stereoscopic system. Step E12 then comprises a stereoscopic conversion sub-step for converting the reconstructed hologram into stereoscopic signals and a display sub-step on a screen of the stereoscopic system from the stereoscopic signals.

There represents a method implemented in another context of use of the invention. This is a method of decoding a digital hologram H _n+1 from data received in a data stream.

This method is implemented in a decoding device. Such a decoding device comprises, for example, a processor and a memory storing computer program instructions designed to implement at least certain steps of the method of decoding. when these instructions are executed by the processor.

This decoding device may be a computer (for example a personal computer) or a digital holography device as described above with reference to the .

The process of begins with a step E20 of receiving a data stream. The decoding device receives for example this data stream coming from a remote server, typically by means of a communication module of the decoding device.

The process continues with a step E22 of obtaining a reference digital hologram H _n . The decoding device can obtain the reference digital hologram by decoding data received in the aforementioned data stream. These data are for example representative of a representation W _n of the digital hologram H _n (without reference to another digital hologram). Alternatively, the digital hologram H _n can be obtained by means of a process of the same type as that described here for obtaining the digital hologram H _n+1 .

The method also includes a step E24 of decoding a residue e _n from other data received in the data stream.

The method also includes a step E26 of obtaining a predictor PRED, for example by reading from a memory of the decoding device. Alternatively, the PRED predictor can be obtained by decoding data contained in the received data stream.

The method then comprises a step E28 of predicting a predicted digital hologram H' _n by applying to the reference digital hologram H _n a transformation T defined by the predictor PRED, using the techniques described above to transform the representation P in representation P', which therefore include the application of a function χ determined as described above.

As in the process of , the reference digital hologram H _n is for example represented by a representation W in the spatio-frequency domain D (that is to say by values W(L) respectively associated with different elements L of the spatio-frequency domain D).

In this case, the prediction step E28 may include a determination of a distribution W' representing the predicted digital hologram H' _n on the basis of the representation W of the reference digital hologram H _n in accordance with what is explained above for step E8 of the , that is to say as follows:

for the different L of the spatio-frequency domain D used in the representation concerned, where N is a predetermined value for a given decoding device (which corresponds to the desired degree of precision, with for example N = 1, in which case the formula contains a only corrective term λ.f _n (L)), λ is, as already indicated, the wavelength associated with digital holograms, o is, as already indicated, the composition operator and where the functions f _n are defined above.

The function χ is thus notably applied when composing this function with the representation W (term W o χ).

Furthermore, as indicated above, it is expected here that the distribution (or representation) W' representing the predicted digital hologram H' _n is determined by the decoding device using at least one corrective term which depends on the length d λ wave. This corrective term is non-zero because the function χ is non-linear as explained above.

The method can then comprise a step E30 of obtaining the decoded hologram H _n+1 by combining the residue e _n and the predicted digital hologram H' _n .

For example, for the different L of the spatio-frequency domain D used in the representation concerned, the representation (or distribution) W'' representing the decoded digital hologram H _n+1 is obtained as follows:

W’’(L) = W’(L) + E(L)

where E is the representation of the residue e _n in the spatio-frequency domain D.

When the decoding device comprises a holographic restitution module (for example when the decoding device is a digital holography device like that of the ), the method may further comprise a step of reconstructing the hologram to be restored on the basis of the representation W'' of the decoded hologram H _n+1 and/or a step of E32 of restoring the digital hologram decoded (and reconstructed) using the holographic restitution module.

There represents an example of a coding device 20 according to the invention.

This coding device is for example located within a server so as to deliver data representative of a sequence of digital holograms to a client electronic device, as described below.

The coding device 20 here comprises a first coding unit 22, a prediction unit 24, a residue determination unit 26, a second coding unit 28 and a communication unit 30.

The coding device 20 is for example produced in the form of an electronic device comprising a processor (such as a microprocessor); in this case, each of the aforementioned units can be implemented due to the execution, by this processor, of computer program instructions stored within the electronic device forming the coding device 20 (for example in a memory of this electronic device). However, as a variant, at least one of the aforementioned units can be produced by means of a dedicated electronic circuit, for example an integrated circuit with a specific application.

The first coding unit 22 is designed to produce data D1 which are representative of a digital hologram H _k , here a reference digital hologram, without reference to data relating to another digital hologram. We describe below with reference to the an example of a possible embodiment for the coding carried out by the first coding unit 22.

The prediction unit 24 is designed to predict a digital hologram H' _k , called " predicted digital hologram ", by transformation of the reference digital hologram H _k , using the techniques which have been presented above with reference to the Figures 1 to 4, in particular by applying the function χ defined above for determining the predicted digital hologram.

The residue determination unit 26 is designed to determine a residue e _{k +1} between the predicted digital hologram H' _k produced by the prediction unit 24 and the current digital hologram H _k+1 .

The second coding unit 28 is designed to produce data D3, D4 representative of the residue e _{k +1} produced by the residue determination unit 26.

Additional elements relating to the operation of these different units are given below with particular reference to the .

Finally, the communication unit 30 is designed to transmit the representative data produced by the first coding unit 22 and/or by the second coding unit 28 to the client electronic device.

There represents an example of a decoding device 40 according to the invention.

This decoding device 40 is for example an electronic device (such as the aforementioned client electronic device) designed to receive data representative of a digital hologram sequence coming from a server (such as the aforementioned server).

The decoding device 40 comprises a first decoding unit 42, a second decoding unit 44, a prediction unit 46, a combination unit 48, a reconstruction unit 49 and a communication unit 50.

The decoding device 40 is for example produced in the form of an electronic device comprising a processor (such as a microprocessor); in this case, each of the aforementioned units can be implemented due to the execution, by this processor, of computer program instructions stored within the electronic device forming the decoding device 40 (for example in a memory of this electronic device). However, as a variant, at least one of the aforementioned units can be produced by means of a dedicated electronic circuit, for example an integrated circuit with a specific application.

The first decoding unit 42 is designed to obtain a digital hologram H _k , here a reference digital hologram, from data D1 representative of this digital hologram. These representative data D1 are for example received by the communication unit 50 (coming from the coding device 20 in the examples described below). An example of a decoding method which can be implemented by the first decoding unit is described below with reference to the .

The second decoding unit 44 is designed to decode a residue e _{k +1} from data D3, D4 representative of this residue e _{k +1} . These data D3, D4 representative of the residue are here received by the communication unit 50 (for example from the coding device 20).

The prediction unit 46 is designed to predict a digital hologram H' _k , called " predicted digital hologram ", by transformation of the reference digital hologram H _k , using the techniques which have been presented above with reference to the Figures 1 to 4, in particular by applying the function χ defined above for determining the predicted digital hologram. Its operation is similar to that of the prediction unit 24 of the coding device 20.

The combination unit 48 is designed to obtain a digital hologram H _k+1 (called " decoded digital hologram ") by combining the residue e _{k +1} decoded by the second decoding unit 44, and the predicted hologram H' _k produced at the output of the prediction unit 46.

The reconstruction unit 49 is designed to reconstruct the decoded digital hologram H _k+1 on the basis of a representation of this digital hologram, this representation here using a decomposition into Gabor frames.

Additional elements relating to the operation of these different units are given below with particular reference to the .

There represents a method of exchanging data between the coding device 20 and the decoding device 40 with a view, for example, to the restitution, on a holographic restitution device, of a sequence of digital holograms coded by the coding device 20 , transmitted from the coding device 20 to the decoding device 40 (for example via a communications network), and decoded by the decoding device 40.

This method begins with a step E2 of initializing the index k which designates the current digital hologram H _k in the sequence of digital holograms processed. The index k is here initialized to 0.

The process continues with a step E4 of decomposing the current digital hologram H _k into Gabor frames W _k . As previously indicated, the current digital hologram H _k is then represented by a set of first coefficients (here complex) W _k (ζ) respectively associated with elements ζ of the spatio-frequency domain D. We recall that each of these elements of the spatio-frequency domain D is formed by a first spatial parameter x, a second spatial parameter y, a first frequency parameter η and a second frequency parameter ξ.

The coefficients W _k (ζ) can for example be calculated by the following formula:

W_k(ζ) = < (ζ)Φ^D ,H_k>

by using the notations of the formula [Math. 6] above and with < f , g > the scalar product between the functions f and g.

In practice, the representation of the current digital hologram H _k used is for example a parsimonious representation, that is to say that the coefficients W _k (ζ) are chosen to best represent the current digital hologram H _k while being in limited number. To do this, we use for example a basic pursuit algorithm (or " basis pursuit " according to the Anglo-Saxon terminology commonly used) to select the coefficients W _k (ζ) resulting from the decomposition of the current digital hologram H _k in Gabor frames to be used in the sparse representation. We can refer for example to the article " Basis pursuit ", by S. Chen and D. Donoho in Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (vol. 1, pp. 41-44), October 1994 regarding the basic tracking algorithm.

The process of then continues with a step E6 of coding the current digital hologram H _k , that is to say here of coding the coefficients W _k (ζ) representing the current digital hologram H _k . This coding step is here carried out by the first coding unit 22 mentioned above.

An example of possible implementation for step E6 is described below with reference to the .

Step E6 makes it possible to generate coded data D1 representative of the current digital hologram H _k .

These coded data D1 are transmitted, by the communication unit 30 of the coding device 20 and via a communication network, to the decoding device 40 (step E8).

Alternatively, the coded data D1 could be stored within the coding device 20 (for example for subsequent use in the case where the coding device 20 and the decoding device 40 are the same and unique electronic device).

In steps E10 to E14 now described, the digital hologram H _k of index k is used as a reference digital hologram for the differential coding (that is to say in the form of a residue) of the hologram digital hologram H _k+1 which follows the digital hologram H _k in the sequence of digital holograms.

During step E10, the prediction unit 24 determines a transformation T _k+1 (here a rigid transformation as already indicated) on the basis of the reference digital hologram H _k and the following digital hologram H _{k +1} , then determines a predicted digital hologram H' _k by transforming the reference digital hologram H _k using the function (or symplectomorphism) χ associated as explained above with the transformation T _k+1 determined.

The transformation T _k+1 is for example determined by movement analysis between the scene represented by the reference digital hologram H _k and the following digital hologram H _k+1 , as proposed in patent application WO 2021/004 797

The representation W' _k of the predicted digital hologram H' _k (by means of second coefficients W' _k (ε)) is then determined from the representation W _k of the reference digital hologram H _k (by means of the first coefficients W _k (ε)) using the function χ associated with the transformation T _k+1 in accordance with what has been previously described, that is to say that we have according to a simplified notation:

W' _k = W _k o χ +

where o is the function composition operator, λ is the recording wavelength of the digital holograms and P _n is the complex functions defined on the spatio-frequency domain D as explained previously.

A coefficient associated with an element (x ₀ , y ₀ , η ₀ , ξ ₀ ) and participating in the representation of the predicted digital hologram H' _k is therefore determined using the formula:

W _k (χ(x ₀ , y ₀ , η ₀ , ξ ₀ ) ) + (x ₀ , y ₀ , η ₀ , ξ ₀ ).

Thus, for the determination of a second coefficient W' _k (ε ₀ ) associated with an element ε ₀ formed by a first value x ₀ relating to the first spatial parameter, a second value y ₀ relating to the second spatial parameter, a third value η ₀ relating to the first frequency parameter and a fourth value ξ ₀ relating to the second frequency parameter, we use in particular the first coefficient W _k associated with the image, by the function χ, of the element ε ₀ formed by the first value x ₀ , the second value y ₀ , the third value η ₀ and the fourth value ξ ₀ .

According to the formula proposed above to calculate the coefficients W' _k (ε) representing the predicted digital hologram H' _k , we use a corrective term which depends on the acquisition wavelength λ of the digital hologram. This corrective term being a sum of functions, it is possible, according to a possible variant, to use a single term of this sum as a corrective term when determining the coefficients W' _k (ε).

According to another variant, to lighten the treatments, the corrective term can be ignored; in this case, any second coefficient W' _k (ε ₀ ) determined in step E10 and associated with an element ε ₀ formed by a first value x ₀ relating to the first spatial parameter, a second value y ₀ relating to the second spatial parameter , a third value η ₀ relating to the first frequency parameter and a fourth value ξ ₀ relating to the second frequency parameter, is equal to the first coefficient W _k associated with the image, by the function χ, of the element ε ₀ formed by the first value x ₀ , the second value y ₀ , the third value η ₀ and the fourth value ξ ₀ .

Furthermore, it is also possible to obtain the coefficients W' _k (ε) by multiplying the expression given above (with or without use of the corrective term), which then forms an intermediate value, by a spatial distribution function -frequency Δ with real values on the spatio-frequency domain D and variable as a function only according to the two frequency dimensions (that is to say that there exists a function f with two variables such that Δ(x,y,η ,ξ)=f(η,ξ) for any multiplet (x,y,η,ξ)), for example to obtain an attenuation of the digital hologram in a certain direction.

This spatio-frequency distribution function is for example of the form:

Δ(x,y,η,ξ) = cos θ + sin θ

with θ a predetermined value linked to the direction of attenuation, and the general formula which gives the expression of the coefficients W' _k representing the predicted hologram H' _k is then:

W' _k = [W _k o χ + ].Δ.

The method then comprises a step E12 of decomposing the digital hologram H _k+1 into Gabor frames W _k+1 .

This step is similar to step E4 carried out for the digital hologram H _k and makes it possible to obtain, to represent the digital hologram H _k+1 , a set of coefficients (here complex) W _k+1 (ζ) respectively associated with elements ζ of the spatio-frequency domain D.

The process of continues with a step E14 of determining a residue e _k+1 between the digital hologram H _k+1 and the predicted digital hologram H' _k , here by difference between the digital hologram H _k+1 and the predicted digital hologram H' _k . This step is here carried out by the residue determination unit 26.

In practice, this difference is made between the coefficients W _k+1 (ζ) representing the digital hologram H _k+1 and the coefficients W' _k (ζ) representing the predicted digital hologram H'k, and we thus note simplified way:

e _k+1 = W _k+1 – W' _k

More precisely, for all the elements ε of the spatio-frequency domain D for which at least one of the coefficients W _k+1 (ε) and W' _k (ε) is non-zero, the residue determination unit 26 determines a (complex) coefficient e _k+1 (ε) of the residue e _k+1 by the operation:

e _k+1 (ε) = W _k+1 (ε) – W' _k (ε)

The communication unit 30 can then transmit to step E16 data representative of D2, D3, D4 of the transformation T _k+1 and the residue e _k+1 to the decoding device 40 (via the aforementioned communication network ). Alternatively, this representative data could be stored in a memory of the coding device 20 for later use.

The data D2 representative of the transformation T _k+1 include values which characterize the transformation T _k+1 , for example the values of the components of the vector defining the translation involved and the angle values defining the different rotations involved.

We propose to code the residue e _k+1 as follows for its transmission to the decoding device 40. Alternatively, we could use the coding method described below with reference to the .

We treat here separately the real part and the imaginary part of each of the coefficients e _k+1 (ε).

We form images I _i,j which each group together the real part of the coefficients e _k+1 (ε) associated with elements whose first frequency parameter and the second frequency parameter are fixed.

In other words, a pixel in row p and column q of an image I _i,j has the value:

I _i,j (p,q) = Re[ e _k+1 ( (x _p , y _p , η _i , ξ _j ) ) ]

where Re[γ] designates the real part of the complex number γ and η _i and ξ _j are respectively the first frequency parameter and the second frequency parameter associated with the image I _i,j .

The sequence of images formed by the images I _i,j (when i and j vary so as to scan all the pairs first frequency parameter η _i - second frequency parameter ξ _j used in the representation) is then coded by means of a video encoder (for example in accordance with an AVC or HEVC type standard).

The aforementioned sequence of images can be formed by taking the images I _i,j in the following order: the different images I _1,j with j increasing, then the different images I _2,j with j decreasing, then the different images I _3,j with j ascending, etc.

The coded data D3 produced by the coding of the sequence of images I _{i, j} are sent to the decoding device 40 by the communication circuit 30.

In the same way, we form images J _{i, j} which each group together the imaginary part of the coefficients e _k+1 (ε) associated with elements whose first frequency parameter and the second frequency parameter are fixed.

In other words, a pixel in row p and column q of an image J _i,j has the value:

J _i,j (p,q) = Im[ e _k+1 ( (x _p , y _p , η _i , ξ _j ) ) ]

where Im[γ] designates the imaginary part of the complex number γ and η _i and ξ _j are respectively the first frequency parameter and the second frequency parameter associated with the image J _i,j .

The sequence of images formed by the images J _i,j (when i and j vary so as to scan all the first parameter η _i - second parameter ξ _j pairs used in the representation) is then coded using a video encoder (for example in accordance with an AVC or HEVC type standard).

The aforementioned sequence of images can be formed by taking the images J _i,j in the following order: the different images J _1,j with j increasing, then the different images J _2,j with j decreasing, then the different images J _3,j with j increasing, etc.

The coded data D4 produced by the coding of the sequence of images J _{i, j} are sent to the decoding device 40 by the communication circuit 30.

The process of continues with a step E18 of incrementing (by one unit) the index k for processing the following digital hologram.

The method then comprises a step E20 where the coding device 20 determines whether a reset of the differential coding is necessary, that is to say whether the new digital hologram H _k+1 to be coded must be entirely coded (as in step E6) or can also be coded by differential coding.

The criterion for determining whether a reset is necessary is for example a frequency criterion (so as to obtain a predetermined number of successive digital holograms coded by differential coding) or an error limitation criterion (by limiting to a predetermined threshold a measurement of error between the original digital hologram and the reconstructed digital hologram after decoding as explained below).

If the coding device 20 determines in step E20 that a reset is necessary, the process continues (arrow P) in step E22 for a new increment of the index k (by one unit) then loops at step E4 for processing the new digital hologram, now designated H _k .

If the coding device 20 determines in step E20 that a reset is not necessary, the process continues (arrow N) in step E10 for differential coding of the new digital hologram H _k+1 .

We now describe (still with reference to the ) decoding the data transmitted by the coding device 20 as explained above with a view to reconstructing the digital hologram sequence.

On receipt of the coded data D1 representative of the digital hologram H _k , the first decoding unit 42 decodes the coded data D1 (step E30) so as to obtain the coefficients W _k (ζ) forming the representation of the digital hologram H _k .

An example of a possible decoding method is described below with reference to the .

The reconstruction unit 49 can then reconstruct the digital hologram H _k in step E32 on the basis of the coefficients W _k (ζ) decoded in step E30.

To do this, we use the inverse transformation of that used in step E4 to obtain the representation of the digital hologram. In the example described here, we therefore use the inverse transformation to that used to obtain the decomposition into Gabor frames.

The digital hologram H _k is therefore reconstructed here according to the formula [Math. 6] presented above (using the decoded coefficients W _k (ζ) as values of in this equation).

The digital hologram H _k can then be rendered using a holographic rendering device (step E34). Such a holographic restitution device comprises for example a light modulator and a light source of wavelength λ. By respectively applying the values of the pixels of the digital hologram H _k to the pixels of the light modulator and by illuminating the light modulator by means of the light source, the digital hologram H _k is restored.

Subsequently, upon receipt of the coded data D2, D3, D4, the second decoding unit 44 decodes this data in step E40 so as to obtain the transformation T _k+1 and the residue e _k+1 .

The second decoding unit 44 determines the transformation T _k+1 from the coded data D2, here from the values of the components of the vector defining the characteristic translation of the transformation T _k+1 and the angle values defining the different rotations characteristics of the transformation T _k+1 .

The second decoding unit 44 also reconstructs the sequence of images I _{i, j} from the coded data D3 and can thus determine the respective real parts of the coefficients e _k+1 (ε) representing the residue e _k+1 .

Likewise, the second decoding unit 44 reconstructs the sequence of images J _{i, j} from the coded data D4 and can thus determine the respective imaginary parts of the coefficients e _k+1 (ε) representing the residue e _k+1 .

The second decoding unit 44 can thus determine the (complex) coefficients e _k+1 (ε), that is to say decode the residue e _k+1 , from the data D3, D4 received from the coding device 20 .

The prediction unit 46 can also predict the predicted hologram H' _k in step E42 by transforming a reference digital hologram, for example the digital hologram H _k decoded in step E30 or a digital hologram obtained during a previous passage to step E44 described below.

During this step E42, the coefficients W' _k (ε) of the predicted digital hologram H' _k are determined on the basis of the coefficients W _k (ε) representing the reference digital hologram H _k and the transformation T _{k +1} (decoded in step E40 on the basis of the data D2 received from the coding device 20), according to a process identical to that used in step E10 described above and which will therefore not be described again. The determination of the predicted digital hologram H' _k during decoding therefore also uses here the function χ associated as explained above with the transformation T _k+1 .

In step E44, the combination unit 48 can thus obtain the decoded digital hologram H _k+1 by combination of the residue e _k+1 (decoded in step E40) and the predicted digital hologram H' _k , here by summation (coefficient to coefficient) of the residue e _k+1 and the predicted digital hologram H' _k :

H _k+1 = H' _k + e _k+1 .

In other words, for any element ε of the spatio-frequency domain D corresponding to a coefficient in the representation of the residue e _k+1 or of the predicted digital hologram H' _k , the combination unit 48 determines the coefficient Wk+1( ε) associated with the element ε in the representation of the decoded digital hologram H _k+1 as follows:

H _k+1 (ε) = H' _k (ε) + e _k+1 (ε) .

The reconstruction unit 49 can then reconstruct the digital hologram H _k+1 in step E46 on the basis of the coefficients W _k+1 (ε) obtained in step E44, in the same way as was done for the digital hologram H _k in step E32.

The digital hologram H _k+1 can then be restored by means of the holographic restitution device (step E48), here by respectively applying the values of the pixels of the digital hologram H _k+1 to the pixels of the light modulator and by illuminating the light modulator by means of the light source.

There represents a variant embodiment of the process of according to which we consider several objects O _m (here M objects O ₁ , O ₂ , …, O _M ) in the three-dimensional scene represented by the digital holograms. The process of can also be implemented by the cooperation of the coding device 20 and the decoding device 40.

The process of also begins with the initialization (for example to the value 0) of an index k designating the digital hologram being processed (step E50).

The coding device 20 then performs a step E54 of decomposing the current digital hologram H _k into Gabor frames W _k . This step E54 is identical to step E4 described above with reference to and will therefore not be described in detail.

Step E54 thus makes it possible to obtain coefficients (here complex) W _k (ζ) respectively associated with elements ζ of the spatio-frequency domain D and representing the current digital hologram H _k .

The coding device 20 then proceeds to a step E56 of coding the coefficients W _k (ζ) representing the digital hologram H _k . This coding step is identical to step E6 described above and here carried out by the first coding unit 22 mentioned above. This coding step uses for example a coding as explained below with reference to the . The coding step E56 makes it possible to produce coded data D1 representative of the coefficients W _k (ζ).

The communication unit 30 can then send in step E58 the data D1' representing the coefficients W _k (ζ) to the decoding device 40. Step E58 here is identical to step E8 described above.

The method then comprises a step E60 (identical to step E12 of the method of ) decomposition of the digital hologram H _k+1 into Gabor frames W _k+1 . This step E60 makes it possible to obtain, to represent the digital hologram H _k+1 , a set of coefficients (here complex) W _k+1 (ζ) respectively associated with elements ζ of the spatio-frequency domain D.

The process of continues with a step E61 of partitioning the spatio-frequency domain D into a plurality of disjoint parts P _m respectively associated with the objects O _m .

An example of a method for obtaining such a partition is given below with reference to the .

The coding device 20 can then form, in step E62, disjoint sets E _m by grouping into a given set E _m the coefficients W _{k +1} (ζ) associated with elements ζ belonging to a part P _m of the space domain. frequency D associated with this given set E _m .

In other words, a coefficient W _{k +1} (ζ) belongs to a set E _m if and only if the element ζ belongs to the part P _m .

The coefficients W _{k +1} (ζ) of a set E _m thus represent a partial digital hologram H _{k +1 ,m} which corresponds to the object O _m .

In the example described here (and as explained below when describing the ), we propose to determine as follows to which set E _m a coefficient W _{k +1} (ζ) associated with an element ζ = (x,y,η,ξ) of the spatio-frequency domain D belongs.

We convert the frequency coordinates (η,ξ) into spatial coordinates (p _x ,p _y ) with:

p _x = 1/tan(arcsin(λ.η)) and p _y = 1/tan(arcsin(λ.ξ)),

then we determine to which subset G _{x,y, m} the couple (p _x , p _y ) belongs, for example by calculating on which side of each line (s _{x,y, m} ,i, s _{x,y, m,i+1} ) is the point (p _x , p _y ). (We recall that the subsets G _{x,y, m} , delimited by the segments [s _{x,y, m,i} , s _{x,y, m,i+1} ] are defined by the implementation of the method described below with reference to the during step E61.)

The coefficient W _{k +1} (ζ) associated with the element ζ = (x,y,η,ξ) will then be attributed to the set E _m associated with the subset G _{x,y, m} .

The process continues with a step E63 of initialization (here at the value 1) of an index m designating the object O _m (and therefore also the set E _m ) currently being processed.

The prediction unit 24 then predicts in step E64 a partial predicted digital hologram H' _k,m by transformation of a partial reference digital hologram H _k,m defined by the coefficients W _k (χ _m (ζ)) associated with the elements χ _m (ζ) for ζ varying in the part P _m associated with the object O _m .

As in step E10 of the embodiment of the , this step E64 uses a function χ _m associated with the transformation T _k+1,m associated with the object O _m in accordance with what is described above for the symplectomorphism χ. The transformation T _k+1,m is therefore determined here by means of an algorithm for estimating the movement of the object O _m . As already indicated, we can use the solution proposed in patent application WO 2021/004 797.

Here the prediction unit 24 determines the coefficients W' _k,m (ε) of the partial predicted hologram H' _k,m as follows:

W' _k,m (ε) = W _k,m ( χ _m (ε) ) + (ε)

for the elements ε belonging to the part P _m of the spatio-frequency domain D, that is to say so as to predict the coefficients W _{k +1} (ε) belonging to the set E _m (these coefficients are denoted W _{k +1,m} (ε) in the following).

The residue determination unit 26 can then determine a partial residue e _k+1,m between the partial digital hologram H _{k+1 ,m} and the partial predicted digital hologram H' _k,m , here by difference between 'partial digital hologram H _{k+1 ,m} and the partial predicted digital hologram H' _k,m ., i.e. for any element ε belonging to the part P _m of the spatio-frequency domain D (here for any coefficient W _{k +1, m} (ε) belonging to the set E _m ):

e _k+1,m (ε) = W _{k +1,m} (ε) - W' _k,m (ε).

The communication unit 30 then transmits to step E68 data D2' representative of the transformation T _k+1,m and the partial residue e _{k +1,m} .

To obtain these data D2', we use for example a coding identical to that described above for the coding of the transformation T _k and the residue e _k+1 . In particular, according to one possible embodiment, the partial residue e _k+1,m can be coded using the coding method described below with reference to the .

The coding device 20 then determines in step E70 whether all the objects O _m have been processed, that is to say here if am=M.

In the negative (arrow N), the index m is incremented by 1 (step E72) and the process loops to step E64 described above.

If so (arrow P), the process continues in step E74 now described.

In step E74, the index k is incremented by 1.

The method then comprises a step E76 (identical to step E20 described above) where the coding device 20 determines whether a reset of the differential coding is necessary, that is to say whether the new digital hologram H _k+1 to be coded must be fully coded or can also be coded by differential coding.

We can refer to the description of step E20 with regard to the criterion for determining whether a reset is necessary.

If the coding device 20 determines in step E76 that a reset is necessary, the process continues (arrow P) in step E78 for a new increment of the index k (by one unit) then loops at step E54 for processing the new digital hologram, now designated H _k .

If the coding device 20 determines in step E76 that a reset is not necessary, the process continues (arrow N) in step E60 for differential coding of the new digital hologram H _k+1 .

We now always describe with reference to the decoding the data D1', D2' transmitted by the coding device 20 as explained above with a view to reconstructing the digital hologram sequence.

This decoding method comprises a step E80 of decoding the data D1' as received by the communication unit 50 so as to obtain the disjoint sets E _m and the coefficients W _k (ζ) contained in each of these disjoint sets E _m .

The reconstruction unit 49 can then reconstruct the digital hologram H _k in step E82 on the basis of all the coefficients W _k (ζ) decoded in step E80 (that is to say the coefficients contained in all sets E _m ). Note that the information relating to the disjoint sets E _m is not used at this step, but will be used later.

This step E82 is similar to step E32 described in the context of the description of the and will therefore not be described in detail.

The digital hologram H _k can then be rendered in step E84 by means of a holographic rendering device of the same type as that used in step E34 described above.

We now describe the decoding of a digital hologram H _k+1 coded differentially with respect to a reference digital hologram H _k (such as for example the digital hologram decoded in step E80), here by means of the D2' data mentioned above.

The coding device 20 transmits, for example, signaling data to the decoding device 40 (within the data stream transmitted by the communication module 30 and received by the communication module 50) to indicate this differential coding of the next digital hologram of the sequence. This signaling data can for example indicate the number M of objects present in the three-dimensional scene.

The decoding device 40 then initializes an index m to 1 (step E86).

The decoding device 40 then decodes in step E88 the data D2' received so as to obtain the transformation T _k+1,m and the partial residue e _k+1,m .

The prediction unit 46 can then predict in step E90 the partial predicted digital hologram H' _k,m on the basis of the transformation T _k+1,m and the partial reference digital hologram H _k,m defined by the coefficients W _k (χ _m (ε)) such that the element ε belongs to the part P _m of the spatio-frequency domain D associated with the current object O _m . The processing carried out during this step E90 is identical to that carried out during step E64 during coding and will therefore not be detailed again here. These treatments make it possible to obtain the coefficients W' _k,m (ε) representing the partial predicted digital hologram H' _k,m .

The combination unit 48 then obtains the partial digital hologram H _k+1,m by combination of the partial residue e _k+1,m (decoded in step E88) and the partial predicted digital hologram H' _{k, m} , here by summation (coefficient to coefficient) of the partial residue e _k+1,m and the partial predicted digital hologram H' _k,m :

H _k+1,m = H' _k,m + e _k+1,m .

In other words, for any element ε belonging to the part P _m of the spatio-frequency domain D, the combination unit 48 determines the coefficient W _k+1,m (ε) associated with the element ε in the representation of the partial digital hologram H _k+1,m as follows:

W _k+1,m (ε) = W' _k,m (ε) + e _k+1,m (ε) .

The decoding device 40 then determines in step E94 whether all the objects O _m have been processed, that is to say here if am=M.

In the negative (arrow N), the index m is incremented by 1 (step E96) and the process loops to step E88 described above.

If so (arrow P), the process continues in step E74 with a step of merging all the coefficients W _k+1,m (ε) obtained during the different passages in step E92 and which represent (set) the decoded digital hologram H _{k+ 1} .

The reconstruction unit 49 can then reconstruct the digital hologram H _k+1 in step E100 on the basis of the coefficients merged in step E98, in the same way as was done for the digital hologram H _k in step E82.

The digital hologram H _k+1 can then be restored by means of the holographic restitution device (step E102), here by respectively applying the values of the pixels of the digital hologram H _k+1 to the pixels of the light modulator and by illuminating the light modulator by means of the light source.

We now describe with reference to the a process for partitioning the spatio-frequency domain D.

In the example described here, we partition the spatio-frequency domain D into parts P _m by partitioning, for each point of the digital hologram with coordinates (x,y), a set M _x,y containing all the pairs (u ,v) frequency variables used in the representation of the digital hologram concerned, that is to say all the directions associated with these pairs (u,v), or, which is equivalent, by partitioning a part of a PPN plane parallel to the plane of the digital hologram, as described below.

The process thus begins by initiating a loop which goes through all the points X with coordinates (x,y) of the digital hologram (step E122).

For each _point

In certain application cases, the objects, their positions and their shapes are known. This is the case, for example, when the digital hologram concerned was constructed from a representation of the three-dimensional scene including these objects.

In other application cases, the objects are reconstructed from the digital hologram concerned, for example using the techniques described in the article " Comparative analysis of autofocus functions in digital in-line phase-shifting holography ", ESR Fonseca, PT Fiadeiro, M. Pereira and A. Pinheiro, , Appl. Opt., AO, vol. 55, no. 27, p. 7663‑7674, September 2016.

For each object O _m , the method initiates in step E126 a loop which goes through all the points P of the object O _m . We assume that the object is discretized a finite number of points and we note (P _x , P _y , P _z ) the coordinates of the current point P.

For each point P of the object O _m , we determine in step E128 the point P' intersection of the segment [PX] and the plane PPN parallel to the (reference) plane of the digital hologram. Here we take as PPN plane the plane of equation z=1 (remembering that the reference plane is the plane of equation z=0).

Here, the coordinates of point P' are therefore ((P _x -x)/P _z ,(P _y -y)/P _z ,1).

We then determine in step E130 a point P'' of the digital hologram with coordinates (P'' _x , P'' _y ) such that no other point of the digital hologram is closer to the point P' than the point P''.

We can then fill in step E132 an element of a matrix G _x,y of the same dimensions as the digital hologram: we set the value m (corresponding to the current object O _m ) to the coordinate element ( P'' _x ,P'' _y ) in the matrix G _x,y . If this element has already been filled in during a previous passage to step E132, the previous value is overwritten (that is to say replaced by the current value of the variable m).

Once the loops on the objects Om and their points P mentioned above have been completed, we proceed to step E134 in which we define sets G _{x,y, m} : G _m = {(x',y'): G _x,y (x',y') = m}. In other words, a given set G _{x,y, m} is the set of pairs (x',y') for which the coordinate element (x',y') in the matrix G _x,y is worth m. The sets G _{x,y, m} are therefore disjoint sets and each correspond (thanks to the perspective projection of center (x,y,0) carried out in step E128) to the set of directions where the object O _m concerned (from the point of coordinates (x,y,0)), that is to say to the set of pairs of frequency coordinates (u,v) of the set M _x,y associated with these directions.

The definition of the sets G _x,y,m thus amounts to a partitioning of the set M _x,y and thus, when this is carried out for all the pairs (x,y), to a partitioning of the spatio-frequency domain D into parts P _m , with the following equivalence: (x,y,u,v) belongs to P _m if and only if the couple (1/tan(arcsin(λ.u)), 1/tan(arcsin(λ. v))) belongs to G _x,y,m .

The method continues with a step E136 of determining, for each set G _{x,y, m} , a set of segments [s _{x,y, m,i} ,s _{x,y, m,i+1} ] delimiting l 'set G _{x,y, m} . For example, we traverse the set of points bordering the set G _{x,y, m} (i.e. neighbors with a point which does not belong to the set G _{x,y, m} ) and we uses each of these points as one of the points s _{x,y, m,i} defining a boundary of a segment [s _{x,y, m,i} ,s _{x,y, m,i+1} ].

After step E136, the process possibly loops to step E124 for processing a new point X of the digital hologram, until all the points of the digital hologram have been processed.

We now describe with reference to the a method for coding coefficients associated respectively with elements of the spatio-frequency domain D and used to represent a digital hologram. In the example described here, this involves the coding of the coefficients W _k (ζ) representing a digital hologram H _k so as to obtain data D1 representative of this digital hologram H _k . In the context described above, this method is implemented by the first coding unit 22 of the coding device 20. Alternatively, however, this method could be used to code other types of coefficients, for example the coefficients forming the residue e _k+1 mentioned above.

As indicated above, the digital hologram H _k is here represented by a plurality of coefficients W _k (ζ) associated respectively with elements ζ which are part of a mesh Λ of the spatio-frequency domain D (see equation [ Math 5] above for the definition of the mesh Λ). We recall that the coefficients W _k (ζ) are complex numbers.

In the following, we note ( integer varying from 1 to N _x ) the indices respectively associated with the values of the first spatial parameter used in the representation of the digital hologram H _k , ( integer varying from 1 to N _y ) the indices respectively associated with the values of the second spatial parameter used in this same representation, ( integer varying from 1 to M _x ) the indices respectively associated with the values of the first frequency parameter used in the representation and ( integer varying from 1 to M _y ) the indices respectively associated with the values of the second frequency parameter used in the representation.

To facilitate the following presentation, we define blocks as follows (for all quadruplets ( , , , ) used in the representation):

( , ) = W _k ( ( , , , ) ) if the representation of H _k includes a coefficient associated with the element ( , , , )

( , ) = 0 otherwise.

In other words, each block includes (all) the coefficients associated with the elements having for the value of the first frequency parameter and for value of the second frequency parameter, organized in the form of a matrix according to the respective values of the first spatial parameter and the second spatial parameter.

The element defined by the position ( , ) in the block is zero when no coefficient is associated with the element ( , , , ) in the representation of the digital hologram H _k .

The method of Figure 10 begins with a step E150 of coding the positions of the non-zero coefficients in a first block with here = 1 and = 1.

The definition of a block (as proposed here) is a particular way of associating in a predefined manner a set of positions (in the block) to the coefficients associated respectively with the different multiplets of the form (x,y, , ).

In practice, the position used for coding can be defined within an ordered sequence of coefficients within the block concerned (here the block = ).

To do this, for the block concerned , we order for example the coefficients of the block by frame scanning order (commonly called " raster scan order "): we order the coefficients (1, ) in ascending order of index relating to the second spatial parameter ( increasing from 1 to N _y ), then the coefficients (2, ) in descending order of index ( decreasing from N _y to 1), then again in increasing order of index for the coefficients (3, ), etc.

We propose here to code the positions of the coefficients ( , ) non-zero by means of indicative data representing the difference between the positions of two successive non-zero coefficients in the ordered sequence of coefficients (except for the first coefficient whose position will for example be coded in absolute value).

So for example, the positions of the non-zero coefficients c ₁ , c ₂ , c ₃ , c ₄ in a sequence (0 c1 0 0 c2 c3 0 0 0 c4 0 0 0) will be coded by the numbers: 2, 3, 1, 4, where 2 represents the position of the first non-zero coefficient c ₁ , 3 the difference between the position of the second non-zero coefficient c ₂ and the position of the first non-zero coefficient c ₁ , 1 the difference between the position of the third non-zero coefficient c ₃ and the position of the second non-zero coefficient c ₂ and 4 the difference between the position of the fourth non-zero coefficient c ₄ and the position of the third non-zero coefficient c ₃ .

In the following, we note D11 the data indicative of the positions of the non-zero coefficients in the block .

The method then initiates a loop in step E152 so as to process all the blocks other than the block .

The different blocks can for example be processed in a frame scanning order by considering all the blocks as elements of a frame: we process the blocks in order of index ascending (from 2 to My in the example described here), then the blocks in order of index decreasing (here from My to 1), then the blocks in order of index ascending (here from 1 to My), then the blocks in order of index ascending (here from 1 to My), etc.

Whatever the order of processing of the blocks, we note in the following the block processed immediately before the current block (this block could be the block when processing the first block designated by the loop initiated in step 152, or a block previously processed in step E152 described below).

For all blocks other than the block (and here in the processing order defined above), the method carries out in step E154 the coding of the positions of the non-zero coefficients of this block (current block).

In the example described here, we propose to code on the one hand (within D12 data) the positions of the non-zero coefficients in the previous block for which the corresponding coefficient in the current block is zero, and on the other hand (within D13 data) the positions of the non-zero coefficients in the current block for which the corresponding coefficient in the previous block is zero.

In other words, the D12 data represent the positions of the coefficients which " disappear " from the representation when we move from the previous block to the current block , while the D13 data represent the positions of the coefficients which " appear " in the representation when moving from the previous block to the current block .

Thus, the data D13 designates at least one position associated with a non-zero coefficient in the current block , without however designating at least one other position associated with another non-zero coefficient when this other position is already indicated by coded indicative data (as part of the coding of the positions in the previous block ) for another pair of respective values of the first frequency parameter and the second frequency parameter.

The D12 data designate positions corresponding (each) to a zero coefficient in the current block , but for which the corresponding coefficient was non-zero in the block previously processed, these positions therefore being indicated in indicative data coded for another pair of respective values of the first frequency parameter and the second frequency parameter.

No data is therefore encoded in the data flow when, for a given position, the coefficients of the current block and the previous block are both zero, or are both non-zero.

The positions coded in step E154 are for example the absolute positions of the coefficients concerned within the ordered sequence (by frame scanning order) defined above within each block .

The method then comprises a step E156 of coding the (non-zero) coefficients associated with the positions indicated by the indicative data D11, D12, D13.

These coefficients are for example coded in the order of the processing carried out in steps E150 and E154, that is to say starting with the block (here the block ) then in order of frame scanning within the blocks , and, within each block, in the frame scanning order (described above in step E150).

For the coding of a coefficient W _k ( ( , , , ) ) given, we proceed on the one hand to the quantification of the real part of this coefficient and on the other hand to the quantification of the imaginary part of the coefficient, the quantified real part and the quantified imaginary part then being concatenated.

The quantizer used for quantification is for example a uniform scalar quantizer or a quantizer as described in the article “ Exact global motion compensation for holographic video compression ”, by Raees Kizhakkumkara Muhamad et al. in Applied Optics58, G204-G217 (2019).

In the example described here, the coding of the coefficients of step E156 is carried out after the coding of the positions of steps E150 and E154. Alternatively, the coding of the coefficients could be interspersed between steps of coding the positions: for example, the coding of the (non-zero) coefficients of a block could be carried out following the coding of the positions of the non-zero coefficients in this block as carried out in step E150 or in step E154.

All of the data obtained by the different coding steps mentioned above (step E150, E154, E156) can also be coded by an entropic coder, for example an adaptive arithmetic coder, in order to obtain the data D1 to be transmitted to the decoding device 40 (or alternatively to be stored for subsequent decoding).

Finally, markers can be inserted between the different types of data (here between the D11 data, the D12 data, the D13 data and the real and imaginary parts of quantized coefficients) to make it possible to find this data at the decoder level. However, other solutions can be used, such as using a fixed length for the data representing the real parts and the imaginary parts of the coefficients, for example.

The data stream D1 transmitted (from the coding device 20 to the decoding device 40 in the context described above) to represent the digital hologram H _k therefore comprises:

- data D11 indicative of the positions associated respectively with non-zero coefficients within the block (that is to say among a set of positions associated in a predefined manner with the coefficients associated respectively with the different multiplets defined by the value for the first frequency parameter and by the value for the second frequency parameter);

- then successively for each of the other blocks , data D12, D13 indicative of positions associated respectively with non-zero coefficients within the block concerned (that is to say among a set of positions associated in a predefined manner with the coefficients associated respectively with the different bytes defined by the value for the first frequency parameter and by the value for the second frequency parameter);

- for each block, data representative of the coefficients associated respectively with the positions indicated by said indicative data relating to the block concerned.

As already indicated, for each block, the position of the coefficients used for coding can be defined within an ordered sequence of coefficients within the block concerned, for example the frame scanning order already presented.

We describe below with reference to the a method of decoding such a data stream, as implemented for example by the decoding device 40 in step E30 described above.

For each block defined above in the context of the coding method of Figure 10, the decoding device (precisely the first decoding unit 42 in the context described with reference to Figure 6) decodes in step E160 the data indicative of the positions associated with non-zero coefficients in the block concerned, that is to say among a set of positions associated in a predefined manner (here by definition of the block) to the coefficients associated respectively with the different multiplets whose first frequency coordinate is the value associated with the index and whose second frequency coordinate is the value associated with the index .

The different blocks are for example transmitted (and therefore presented within the data stream) in an order predefined by convention between the coding device 20 and the decoding device 40, such as the already mentioned order of frame scanning within blocks arranged in the form of a block matrix. Alternatively, additional data could be placed in the header of the data relating to a block and indicate parameters designating the block concerned (these parameters being for example the indices And of the block concerned).

In the case of the block (here the block ), the data D11 indicative of the positions of the non-zero coefficients (here within an ordered sequence of coefficients of the block ) represent a difference between a position associated with a non-zero coefficient and another position associated with another non-zero coefficient, as explained during the description of the coding method.

In the case of blocks other than the block , the data D12, D13 indicative of the positions of the non-zero coefficients (here within an ordered sequence of coefficients of the block ) designate either the positions of the zero coefficients in the current block while the coefficient at the same position was non-zero in the block previously decoded (data D12), i.e. the positions of the non-zero coefficients in the current block while the coefficient at the same position was zero in the block previously decoded (D13 data).

Thus, the data D13 designates at least one position associated with a non-zero coefficient without designating at least one other position associated with another non-zero coefficient when this other position has been indicated by indicative data decoded in relation to another block , that is to say for another pair of respective values of the first frequency parameter and the second frequency parameter (this pair of values being associated with the coefficients forming the other block).

When the data indicative of the positions of the non-zero coefficients have been decoded for a given block , the decoding device 40 (precisely here the first decoding unit 42) can decode the coefficients associated respectively with the positions indicated by these indicative data.

The decoding device 40 thus finds the non-zero coefficients ( , ), that is to say the non-zero coefficients W _k ( ( , , ) ).

By going through all the blocks (for which the data are successively transmitted in the data stream), the decoding device 40 thus obtains all of the non-zero coefficients W _k of the representation of the digital hologram H _k .

Note that, as already indicated in the context of the coding method of Figure 10, the decoding device 40 can receive all the data indicative of the positions of the non-zero coefficients for the set of blocks , then the coefficients associated with these positions, or, alternatively, by successively considering the different blocks , the data indicative of the positions of the non-zero coefficients of this block and the coefficients associated with these positions.

Claims (14)

Method for decoding a digital hologram (H_k+1) propagating in a three-dimensional propagation space from received data (D3, D4), comprising the following steps:
- obtaining (E30; E80) a reference digital hologram (H_k) defined in a reference plane (P_ref) and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;
- decoding (E40; E88) of a residue (e_k+1; e_k+1,m) from the data received;
- prediction (E42; E90) of a predicted digital hologram (H’_k) by transformation of the reference digital hologram;
- obtaining (E44; E92) the decoded digital hologram (H_k+1) by combination of the residue and the predicted hologram,
characterized in that the step of predicting (E42; E90) the predicted digital hologram comprises, for the determination of the predicted digital hologram at the level of a plane parallel to the reference plane, the application of a function χ having as starting set the space-frequency domain D and as arrival set the space-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e )) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:
- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and
- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables. Decoding method according to claim 1, in which the representation of the reference digital hologram (H _k ) comprises a set of first coefficients (W _k ) respectively associated with elements of the spatio-frequency domain D. Method according to claim 2, in which the predicted digital hologram is represented by a set of second coefficients (W _{k +1} ) respectively associated with elements of the spatio-frequency domain D and in which the prediction step (E42; E90 ) of the predicted digital hologram comprises, for the determination of a second coefficient (W _{k +1} ) associated with a given element of the spatio-frequency domain D, the application of the function χ to said given element in order to obtain a other element of the spatio-frequency domain D and the use of the first coefficient (W _k ) associated with this other element. Decoding method according to claim 3, in which the prediction step comprises a sub-step of obtaining the second coefficient (W _{k +1} ) associated with the given element by multiplication by a spatio-frequency distribution function with values real on the spatio-frequency domain D. Decoding method according to one of claims 2 to 4, in which said representation results from a decomposition of the reference digital hologram into Gabor frames, said Gabor frames corresponding respectively to said first coefficients (W _k ). Decoding method according to one of claims 2 to 5, comprising, for each of a plurality of objects (O_m) of the three-dimensional scene represented by the reference digital hologram, the application of a function χ_mfrom the spatio-frequency domain D towards itself such that the subset of the elements of the space produces D x D of form (e, χ_m(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ_mis a non-quadratic generating function with four variables defined as the scalar product of said first vector and a vector resulting from the application to said second vector of a transformation (T_{k+1, m}) associated with the object concerned (O_m). Decoding method according to one of claims 2 to 6, in which the step of predicting the predicted digital hologram comprises a step of determining a corrective term as a function of the acquisition wavelength of the digital hologram. Method according to claim 7, in which the coefficient associated with the element (x₀,y₀,η₀, ξ₀) participating in the representation of the predicted digital hologram is determined using the formula:
W_k (χ(x₀,y₀,η₀, ξ₀) ) + (x₀,y₀,η₀, ξ₀)
where W_k(ε) is the first coefficient associated with an element ε of the spatio-frequency domain D, is the acquisition wavelength of the digital hologram and is a complex function defined on the spatio-frequency domain D. Method according to claim 7 taken in dependence on claim 4, in which the coefficient associated with the element (x ₀ , y ₀ , η ₀ , ξ ₀ ) participating in the representation of the predicted digital hologram is determined using the formula :
[W _k (χ(x ₀ , y ₀ , η ₀ , ξ ₀ )) + (x ₀ , y ₀ , η ₀ , ξ ₀ )].Δ(x ₀ , y ₀ , η ₀ , ξ ₀ )
where W _k (ε) is the first coefficient associated with an element ε of the spatio-frequency domain D, is the acquisition wavelength of the digital hologram, is a complex function defined on the spatio-frequency domain D and Δ is the spatio-frequency distribution function. Method according to one of claims 1 to 9, in which the reference digital hologram or the residue is represented by a set of coefficients (W _k ; e _k+1 ; W _k ; e _k+1,m ) respectively associated to elements of the spatio-frequency domain D each corresponding to a multiplet formed by a first spatial parameter, a second spatial parameter, a first frequency parameter and a second frequency parameter, and in which the method comprises the following steps, carried out for at least a pair of respective values of the first frequency parameter and the second frequency parameter:
- decoding (E160) of data indicative of positions associated respectively with non-zero coefficients among a set of positions associated in a predefined manner with the coefficients associated respectively with the different bytes comprising said pair of values of the first frequency parameter and the second frequency parameter;
- decoding (E162) of the coefficients associated respectively with the positions indicated by said indicative data. Method for coding a digital hologram (H_k+1) propagating in a three-dimensional propagation space, comprising the following steps:
- prediction (E10; E64) of a predicted digital hologram (H’_k) by transformation of a reference digital hologram (H_k) defined in a reference plane (P_ref) and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;
- determination (E14; E66) of a residue (e_k+1; e_k+1,m) between said digital hologram (H_k+1) and the predicted digital hologram (H’_k),
- coding of the determined residue (E16; E68);
characterized in that the prediction step (E10; E64) of the predicted digital hologram (H’_k) includes, for the determination of the predicted digital hologram (H’_k) at the level of a plane parallel to the reference plane (P_ref), the application of a function χ having as starting set the spatio-frequency domain D and as arrival set the spatio-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:
- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and
- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables. Device for decoding (40) a digital hologram (H_k+1) propagating in a three-dimensional propagation space from received data (D2, D3), comprising:
- a first decoding unit (42) designed to obtain a reference digital hologram (H_k) defined in a reference plane (P_ref) and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;
- a second decoding unit (44) designed to decode a residue (e_k+1) from the data received (D2, D3);
- a prediction unit (46) designed to predict a predicted digital hologram (H’_k) by transformation of the reference digital hologram (H_k) ;
- a combination unit (48) designed to obtain the decoded digital hologram (H_k+1) by combination of the residue (e_k+1) and the predicted hologram (H’_k),
characterized in that the prediction unit (46) is designed to apply, for the determination of the predicted digital hologram (H’_k) at the level of a plane parallel to the reference plane, a function χ having as starting set the spatio-frequency domain D and as arrival set the spatio-frequency domain D, and such that the subset of the elements of the product space D x D of form (e, χ(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:
- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and
- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables. Device for coding (20) a digital hologram (H_k+1) propagating in a three-dimensional propagation space, comprising the following steps:
- a prediction unit (24) designed to predict a predicted digital hologram (H’_k) by transformation of a reference digital hologram (H_k) defined in a reference plane (P_ref) and represented by a representation in a spatio-frequency domain D with two spatial dimensions and two frequency dimensions;
- a unit for determining (26) a residue (e_k+1) between said digital hologram (H_k+1) and the predicted digital hologram (H’_k),
- a coding unit (28) designed to code the determined residue (e_k+1) ;
characterized in that the prediction unit (24) is designed to apply, for the determination of the predicted digital hologram (H’_k) at the level of a plane parallel to the reference plane (P_ref), a function χ having as starting set the spatio-frequency domain D and as arrival set the spatio-frequency domain D, and such that the subset of the elements of the space produces D x D of form (e , χ(e)) is the subset defined by the elements of the form (x, y, , , , ,η_x, η_y) for (x, y, η_x,η_y) varying in ℝ⁴, where φ is a four-variable non-quadratic generating function defined as a dot product:
- a first vector, the first two coordinates of which correspond respectively to the first two variables of the four variables and the third coordinate of which corresponds to the coordinate of said parallel plane in a direction perpendicular to the reference plane, and
- a vector resulting from the application of the transformation to a second vector, whose first two coordinates correspond respectively to the last two variables of the four variables. Computer program comprising instructions designed to implement a method according to one of claims 1 to 11 when these instructions are executed by a processor.