Classifications

• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
  • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
  • G03H1/04—Processes or apparatus for producing holograms
  • G03H1/08—Synthesising holograms, i.e. holograms synthesized from objects or objects from holograms
  • G03H1/0808—Methods of numerical synthesis, e.g. coherent ray tracing [CRT], diffraction specific
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
  • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
  • G03H1/04—Processes or apparatus for producing holograms
  • G03H1/08—Synthesising holograms, i.e. holograms synthesized from objects or objects from holograms
  • G03H1/0891—Processes or apparatus adapted to convert digital holographic data into a hologram
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03H—HOLOGRAPHIC PROCESSES OR APPARATUS
  • G03H1/00—Holographic processes or apparatus using light, infrared or ultraviolet waves for obtaining holograms or for obtaining an image from them; Details peculiar thereto
  • G03H1/04—Processes or apparatus for producing holograms
  • G03H1/08—Synthesising holograms, i.e. holograms synthesized from objects or objects from holograms
  • G03H1/0808—Methods of numerical synthesis, e.g. coherent ray tracing [CRT], diffraction specific
  • G03H2001/0816—Iterative algorithms

Definitions

• the present invention relates to the technical field of digital holography.
• the digital hologram is decomposed on a dictionary comprising a large number of decomposition functions.
• the denser the dictionary i.e. the greater the number of functions, the finer the decomposition. This results in the spatio-frequency parameters being selected from a large number of discrete values.
• the dictionary is sufficiently dense, we can then obtain a redundant decomposition which makes it possible not to lose information during the transmission.
• a method for transmitting data representative of a digital hologram comprising the following steps:
• each raw value being determined from among a discrete set of predetermined values, said raw values being determined such that a first scalar product between said hologram and a first decomposition wavelet characterized by said raw values are maximum;
• a digital hologram can be decomposed on the basis of a dictionary comprising a limited number of functions which are here decomposition wavelets. Indeed, during the decomposition, the discrete values determined during the successive iterations are adjusted. Consequently, it is not necessary to start from a large number of discrete values to find the most adequate values, as in the state of the art. In other words, calculating refinement values for the parameters characterizing the decomposition functions makes it possible to limit the discrete set of predetermined values.
• the digital hologram can be finely decomposed since the refinement values allow an optimization of the decomposition function which is determined at each iteration.
• the amount of information to be transmitted to reconstruct the digital hologram is also reduced. On the one hand because the decomposition functions precisely representing the hologram once optimized, a limited number of decomposition functions is sufficient. On the other hand, as will be described later, because the transmission of only part of the data resulting from the decomposition makes it possible to reconstruct the hologram. The amount of information transmitted is then reduced even further.
• the high quality of the spatio-frequency decomposition according to the invention makes it possible to improve many functionalities linked to digital holograms such as for example the analysis of holographic signals, hologram editing, motion estimation or the generation of dynamic holograms.
• This transmission method can also be used in the case of holographic streaming depending on the position of an observer: indeed, the refinement step makes it possible to provide more precision on the directions of light emission, and therefore improve the quality of the view reconstructed from a sub-hologram corresponding to the observation window corresponding to the position of the observer.
• the transmission step comprises the transmission iii) of data representative of said raw values;
• said group comprises at least one frequency parameter, said method comprising a step of transmitting, for each frequency parameter, metadata representative of said discrete set of values;
• said group comprises at least one frequency parameter
• said data representative of said refinement values comprise, for each frequency parameter, a real part of said derivation scalar product
• said group comprises at least one spatial parameter
• said method comprises a step of transmitting data representative of the raw value of each spatial parameter
• said method comprises a step of quantifying said refinement values, and a step of obtaining said fine values on the basis of said quantified refinement values;
• said method comprises a step of transmitting a decision relating to a comparison between the modulus of said first scalar product and the modulus of said second scalar product;
• said method comprises step of encoding said data representing said refinement values and said coefficient in a data stream;
• said method comprises a step of determining other raw values, characterizing another decomposition wavelet, of the group of parameters, and said data representative of the raw value of each spatial parameter comprise, for each spatial parameter, the raw value or the other raw value and a difference between the raw value and the other raw value.
• the present invention also relates to a device for transmitting a digital hologram comprising:
• a breakdown module programmed to perform the following steps: a) determination of raw values of a group of parameters, each raw value being determined from a discrete set of predetermined values, said raw values being determined such that a first scalar product between said hologram and a first decomposition wavelet characterized by said raw values is maximum; b) for each parameter of said group, calculation of a refinement value as a function of a derivation scalar product determined on the basis of a scalar product between said hologram and a derivative of said first wavelet with respect to said setting ; c) determination of a coefficient on the basis of a second scalar product between said hologram and a second decomposition wavelet characterized by fine values obtained by a combination of said raw values and of said refinement values;
• a transmission module designed to transmit: i) data representative of said refinement values; (ii) said coefficient.
• the present invention also relates to a method for constructing a digital hologram comprising the following steps:
• This particularly advantageous method makes it possible to construct a digital hologram from a reduced number of data representative of this hologram. Indeed, it is not necessary to receive all the data from the decomposition to build the hologram.
• said method comprises, for at least one parameter among the group of parameters, a step of determining, on the basis of said data representative of said refinement values, a raw value relating to this parameter among a discrete set of values;
• said method comprises, at least for said parameter, a step determining said construction value on the basis of said data representative of said refinement values and of said raw value relating to said parameter, said parameter being in particular a frequency parameter;
• said method comprises a step of receiving a decision and said decomposition wavelet is also determined on the basis of said decision;
• said method comprises a step of decoding said data stream and extracting said data representative of refinement values and of said coefficient;
• said method comprises a step of receiving metadata and a step of determining said discrete set of values on the basis of the metadata;
• the reception step comprises the reception iii) of data representative of at least one raw value relating to at least one parameter from among the group of parameters.
• the present invention also relates to a device for constructing a digital hologram comprising:
• reception module designed to receive: i) data representative of refinement values of a group of parameters, ii) a coefficient associated with a decomposition wavelet;
• decoding module programmed to perform the following steps: a) determination of construction values on the basis of said data representative of said refinement values, each construction value being associated with a parameter of said group; b) determination of said decomposition wavelet characterized by said construction values;
• a construction module programmed to execute the following step: c) construction of said hologram by weighting said decomposition wavelet by said coefficient.
• the present invention also relates to a system for transmission and construction of a digital hologram comprising a transmission device as described above and a construction device as described above.
• the transmission device comprises an encoding module programmed to encoding said data representative of said refinement values and said coefficient into a data stream;
• said reception module receives the data stream comprising said data representative of refinement values and said coefficient
• said decoding module is programmed to decode said data stream and extract said data representative of refinement values and said coefficient.
• the present invention finally relates to a data stream comprising:
• said data stream comprises a third set of data representative of said raw values, said raw values being determined such that a scalar product between the hologram and said first decomposition wavelet is maximum;
• said data stream comprises a fourth set of data representative of discrete sets of predetermined values, each raw value being determined as one of said predetermined values.
• Figure 1 is a schematic representation of a light beam generated by the diffraction of an incident light wave and corresponding to a decomposition wavelet;
• Figure 2 is a schematic representation of a device for transmitting a digital hologram according to the invention
• Figure 3 is a block diagram representing a sequence of steps allowing the transmission of data representative of a digital hologram according to the invention
• Figure 4 is a schematic representation of a device for constructing a digital hologram according to the invention.
• Figure 5 is a block diagram representing a sequence of steps allowing the construction of a digital hologram according to the invention.
• Figure 6 is a block diagram showing a sequence of sub-steps of the block diagram of Figure 5;
• Figure 7 is a schematic representation of a data stream transmitted by the transmission device of Figure 2 and received by the construction device of Figure 4.
• the transmission device 100 comprises a decomposition module 10, a scheduling module 20, an encoding module 30 and a transmission module 40.
• the decomposition module 10 is designed to determine the data representative of the digital hologram H. For this, the decomposition module 10 is based on a dictionary D of decomposition wavelets.
• the ordering module 20 is designed to order the data representative of the digital hologram H.
• the encoding module 30 is designed to encode the data representative of the digital hologram H, here in the order defined by the scheduling module 20, into a data stream to be transmitted.
• the encoding module 30 also designed here to encode metadata relating to the dictionary D of decomposition wavelets.
• the transmission module 40 is designed to transmit the data stream F prepared by the encoding module 30 over a communication network. As described later, the transmission module 40 here transmits the data stream F on the communication network to a device 200 for constructing a digital hologram H.
• the aforementioned modules 10, 20, 30, 40 can in practice be implemented by the cooperation of at least one hardware element, such as a processor and a communication circuit, in particular for the transmission module 40, and of software elements, such as computer program instructions executable by the aforementioned processor.
• a hardware element such as a processor and a communication circuit, in particular for the transmission module 40
• software elements such as computer program instructions executable by the aforementioned processor.
• These computer program instructions are here recorded on a memory included in the transmission device 100 and to which the processor has access.
• These computer program instructions may in particular be such that the transmission device 100 implements at least part of the steps described below with reference to FIG. 3 when these instructions are executed by the processor of the transmission device 100 .
• a module 10, 20, 30, 40 performs an action within the framework of a step of the method, this means that the processor executes the computer program instructions dedicated to carrying out this step or part of this step.
• the processor, and therefore the aforementioned modules, are thus programmed to perform the steps described below with reference to Figure 3.
• FIG. 3 represents the main steps of the method for transmitting data representative of the digital hologram H according to the first embodiment.
• step E1 the digital hologram H is selected from a holographic database, for example recorded on a remote server.
• the digital hologram H is for example received by transmission device 100 from the remote server.
• the digital hologram H is here recorded on the memory of the transmission device 100.
• the digital hologram H here consists of a two-dimensional matrix of complex numbers.
• the digital hologram H is here an uncompressed hologram.
• a discretized dictionary D of decomposition wavelets is selected from a database, for example recorded on a remote server.
• the dictionary D is for example received by transmission device 100 from the remote server.
• the dictionary is pre-recorded on the memory of the transmission device 100.
• a digital hologram H can be sparsely decomposed on a set of decomposition wavelets.
• the decomposition wavelets are discretized g Y n Gabor wavelets.
• Gabor wavelets are indeed particularly suitable for the decomposition of digital holograms since they present the best compromise of localization both in the spatial domain and in the frequency domain.
• the decomposition wavelets can however be other types of wavelets such as wavelets by Morlet.
• Each Gabor wavelet g Y n is characterized by a group of parameters, called space-frequency parameters.
• This group of parameters includes spatial translation parameters, called spatial parameters, here a first spatial parameter x n , and a second spatial parameter y n and frequency parameters, here a rotation parameter 0 n , a spatial frequency parameter f n , and a dilation parameter o n .
• the dilation parameter o n is also often referred to as the dispersion parameter or the variance parameter.
• the Gabor wavelet g yn is defined by the following formula:
• a Gabor wavelet g Y n thus corresponds to a diffracted light ray R, coming from an incident light wave I, at the level of the point of coordinates (x n , y n ) in the plane (xOy ) of the digital hologram H and with a direction defined by an azimuth angle equal to the rotation parameter 0 n , a diffraction angle cp n , and an angular dispersion determined by the dilation parameter On.
• the spatio-frequency parameters of the roundel therefore present a duality with the diffraction parameters of the diffracted light ray R.
• the spatial parameters here relate to the position of the decomposition wavelet in a holographic plane, i.e. to the position of emergence of the light ray R diffracted by this wavelet.
• the frequency parameters here relate to the shape and direction of a light ray R diffracted by the decomposition wavelet.
• the discretized dictionary D of decomposition wavelets comprises, for each parameter of the group of parameters y n , a discrete set of values.
• the discrete set of values of the spatial parameters x n , y n is for example all the values integers between 1 and respectively N x and N y .
• Nz the number of decomposition wavelets in the discretized dictionary D of decomposition wavelets.
• f xn f n cos(0n)
• f yn f n sin(0 n ).
• the three frequency parameters are then the dilation o n , the spatial frequency along the direction xf xn and the spatial frequency along the direction yf yn .
• the transmission device 100 it is possible to finely decompose the digital hologram H while using a dictionary D comprising a small number of decomposition wavelets.
• a sparse dictionary D saves storage space and reduces the calculation time needed to decompose the digital hologram H.
• each discrete set for the spatial parameters comprises between 1 and 4096 values.
• the dictionary D here comprises both the analytical functions which define the decomposition wavelets and which are recorded in the memory of the transmission device 100 in the form of computer instructions, and the metadata making it possible to determine the discrete sets of values of the spatio-frequency parameters.
• the digital hologram H could however in practice comprise other chromatic components, each of the chromatic components then being broken down and transmitted as described below.
• the embodiment described here uses, as indicated above, a decomposition using Gabor wavelets. It would however be possible as a variant to use other decompositions, for example using Morlet wavelets, as described for example in the article "Morlet Wavelet transformed holograms for numerical adaptive viewbased reconstruction", by K. Viswanathan , P. Gioia and L. Morin, in Proc. SPIE 9216, Optics and 15 Photonics for Information Processing VIII, August 2014, San Diego, USA.
• steps E3 to E8 an iterative decomposition algorithm of the matching pursuit type (better known as “matching pursuit”).
• matching pursuit better known as “matching pursuit”.
• the first iteration of this algorithm according to the first embodiment is described here in detail.
• the decomposition module 10 determines raw values for the aforementioned group of parameters.
• the raw values of the parameters are noted here with an index c.
• the spatial parameters are then denoted x c and y c and the frequency parameters o c , f xc , fyc.
• the group of parameters is then denoted y c and the Gabor wavelet that they characterize, called the first wavelet, is denoted g yc .
• the decomposition module 10 searches, among all the discrete values contained in the dictionary D, for the values which maximize the modulus of the scalar product between the digital hologram H and the first wavelet g yc .
• the dot product operator is described in more detail in step E4.
• the decomposition module 10 performs this scalar product a number of times equal to the number of combinations between the discrete sets of each parameter and selects the scalar product whose modulus is the highest. This scalar product is called the first scalar product.
• the raw values thus define the decomposition wavelet from the dictionary D which is the most representative of the digital hologram H.
• Step E3 is a conventional step of an iterative matching-pursuit type decomposition algorithm.
• the decomposition module 10 proceeds to a step E4 of refining these raw values.
• This refinement is based on an optimization of the scalar product between the digital hologram H and the first wavelet g YC .
• the optimum of this scalar product is determined by finding the points where the derivatives with respect to each of the parameters cancel each other out. These points are here called the fine values Xf, yf, Of, f X f, fyf of the parameters, denoted with an index f.
• the fine values Xf, yf, Of, f X f, fyf of the parameters characterize a second wavelet g Y f.
• the second wavelet g Y f is such that:
• the frequency parameter refinement formulas are as follows: [Math. 13] and
• the notation R[.] represents the real part of the elements between square brackets.
• the refinement formulas here involve scalar products of derivation Dxc, D yc , D CT C, Dfxc, Dfy C .
• Each derivation scalar product here represents a scalar product between the hologram H and a derivative of the first wavelet with respect to one of the parameters.
• the refinement formulas thus involve a scalar product of derivation D xc for the first spatial parameter, a scalar product of derivation D xy for the second spatial parameter, a scalar product of derivation D CTC for the dilation parameter, a scalar product of derivation Df XC for the frequency parameter along x and a derivation scalar product Dfxy for the frequency parameter along y.
• Dyc represents one of the derivation inner product
• the derivative ôy c represents the derivative with respect to one of the parameters
• g yc represents the first wavelet defined by the raw values x c , y c , o c , fxc, fyc.
• the notations D yc and yc both represent the scalar product of derivation, however D yc is calculated numerically, while yc is calculated theoretically.
• the decomposition module 10 numerically calculates the scalar derivative products D xc , D yc , Doc, Dfxc, Dfy C , according to the above formula. To do this, the decomposition module 10 uses for example analytical formulas of the derivatives of the first wavelet with respect to the parameters which are recorded on the memory of the transmission device 100. The decomposition module 10 applies the raw values determined to the step E3 to these saved formulas, then performs the scalar products with the digital hologram H.
• the decomposition module 10 calculates, on the basis of the refinement formulas, the fine values. It first calculates the fine value Of of the dilation parameter. Then, it calculates the fine values of the two other frequency parameters f X f, f y f and of the two spatial parameters Xf, yf.
• the decomposition module 10 more specifically calculates the fine values on the basis of the real parts of the scalar products of derivation.
• the decomposition module 10 determines the fine values by combining the raw values with refinement values.
• the refinement values are determined here according to the real part of the scalar product of derivation associated with the parameter concerned, as visible in the refinement formulas above.
• the refinement values for the parameters other than the dilation parameter are here also determined according to the fine value of the dilation parameter. These refinement values are therefore the terms added to the right of the sum in the refinement formulas, i.e. the terms added to the raw values to calculate the fine values.
• the decomposition module 10 therefore determines the second decomposition wavelet g Y f which is generally more representative of the digital hologram H than the first wavelet g yc .
• the determination of the first wavelet g yc is limited by the discrete values, taken from the dictionary D, which the raw values can take, whereas the fine values are determined on continuous sets.
• this refinement step has several advantages, in particular to obtain fine decompositions, that is to say here of high quality, on the basis of a dictionary can dense and thus to reduce the necessary calculation time to decomposition.
• the decomposition module 10 can numerically calculate only the scalar derivative products D xc , D yc , D CTC for the spatial parameters and for the dilation parameter. Then, it calculates the real parts of the scalar products of derivation Df XC , Df yc of the frequency parameters thanks to the following relations:
• the notation l[.] represents the imaginary part of the elements between square brackets.
• the decomposition method here comprises, once the decomposition module 10 has calculated the scalar products of derivation, a step of quantifying the fine values and the refinement values.
• step E5 the quantification module 20 quantifies in particular the real parts of the scalar products of derivation, which are here generally denoted R[D yc ].
• step E4 the fine values are calculated on the basis of the real parts of the quantized derivative scalar products R[D yc ].
• Step E5 thus constitutes an anticipated quantification, with a view to transmission, of certain data representative of the digital hologram H.
• the quantization module 20 implements a Lloyd Max quantizer, for example on 16 levels or on 32 levels.
• the decomposition module 10 determines a fine coefficient Cf on the basis of a second scalar product between the hologram digital H and the second wavelet g Y f.
• the decomposition module 10 compares the modulus of the raw coefficient c c with the modulus of the fine coefficient Cf.
• the decomposition module 10 therefore compares the second scalar product between the digital hologram H and the second wavelet g Y f with the scalar product between the digital hologram H and the first wavelet QYC.
• the decomposition module 10 then generates a decision d relating to this comparison.
• the decision d is equal to 1 when the modulus of the fine coefficient Cf is greater than or equal to the modulus of the raw coefficient c c and equal to 0 when the modulus of the fine coefficient c is less than the modulus of the raw coefficient c c .
• a modulus of a fine coefficient Cf greater than a modulus of a gross coefficient c c means that the second wavelet g Y f is more representative of the digital hologram H than the first wavelet g YC , and vice versa. In practice, in the majority of cases, the modulus of the fine coefficient Cf greater than the modulus of the raw coefficient c c .
• the decomposition module 10 determines a stored coefficient c m and a stored wavelet g vm .
• the stored coefficient c m is equal to the fine coefficient Cf and the stored roundlet g vm is equal to the second wavelet g Y f.
• the stored coefficient c m is equal to the raw coefficient c c and the stored wavelet g Y m is equal to the first wavelet g YC .
• the fine coefficient Cf, the raw coefficient c c and the stored coefficient c m are here complex numbers.
• the decision d therefore makes it possible to keep in memory which decomposition wavelet, among the first wavelet and the second wavelet, is actually used to decompose the digital hologram H during an iteration.
• the decomposition module 10 determines a residual r on the basis of the digital hologram H, of the stored coefficient c m and of the stored wavelet g Y m.
• the calculation of the residual r is a classic step in a matching pursuit type algorithm. During successive iterations of steps E3 to E8, which are not described in detail here, the operations implemented during these steps are performed on the residue r instead of the digital hologram H and the residue is updated to the end of each step E8.
• step E4 For example, for a residue r, the first equation described in step E4 is generalized by the following equation:
• steps E3 to E8 are iterated a predetermined number of times.
• steps E3 to E8 are iterated here L times, L being a positive integer.
• the iteration number L is determined either by a throughput criterion or by a quality criterion.
• bit rate criterion the iterations are stopped when a predetermined maximum bit rate of the stream to be encoded is reached.
• quality criterion the iterations are stopped when the norm of the residual is smaller than a predetermined threshold value.
• steps E3 to E8 could be iterated a predetermined number of times.
• steps E3 to E8 could be iterated for a predetermined duration or until the last stored coefficient is lower than a predetermined threshold value.
• the decomposition module 10 After performing L iterations, the decomposition module 10 has in particular determined the following data which are representative of the digital hologram H: L groups of raw values, L groups of fine values, L stored coefficients and L decisions.
• steps E3 to E8 allows precise decomposition of the digital hologram H even on the basis of a sparse dictionary D.
• the values of the parameters being refined this also makes it possible to reduce the number of wavelets to be extracted (and therefore the computation time) for a given quality of decomposition.
• a method for encoding data representative of the digital hologram H according to the first embodiment is now described.
• This method which is notably implemented by the encoding module 30, is particularly suited to the breakdown presented above.
• This method makes it possible to encode and transmit a data stream F containing the minimum information necessary for the construction of the digital hologram H by the construction device 200, which is also suitable for this decomposition and which is represented in FIG.
• the scheduling module 20 Prior to encoding, in order to limit the quantity of data to be transmitted, the scheduling module 20 orders the data representative of the digital hologram H resulting from each iteration according to a line-by-line scanning scheme, better known by the name scheduling English in “raster scan”. Here, the scheduling module 20 only orders the data which is encoded and transmitted.
• the scheduling is performed based on the spatial parameters. Then, the scheduling indices resulting from the scheduling are applied to the frequency parameters, the coefficients and the decisions.
• Step E10 data representative of the hologram H are encoded and transmitted. Step E10 makes it possible to generate a data stream containing sufficient information to construct the digital hologram H.
• the information sufficient to construct the digital hologram H is described below.
• the encoding module 30 quantifies the data which is intended to be encoded.
• the encoding module 30 implements a Lloyd Max quantizer.
• the encoding process is characterized by an encoding rate.
• the encoding bit rate is here in particular the sum of a bit rate to encode the values of the parameters, of a bit rate to encode the stored coefficients and of a bit rate to encode the decisions.
• the encoding rate corresponds here to a number of bits necessary for the encoding, here binary, of the data.
• the encoding module 30 encodes, for the frequency parameters, only the real parts of the scalar derivative products.
• the groups of values coded for the frequency parameters are represented by the notation ⁇ R[D CTC ], R[Df XC ], R[Dfy C ] ⁇ K, where the index K corresponds to one iterations of the decomposition.
• the frequency parameters it is possible, for the frequency parameters, to determine from a triplet of real parts of the scalar products of derivation the triplet of raw values which was used to obtain it during the decomposition and therefore the fine value triplet associated with this triplet of real parts of the scalar products of derivation.
• the refinement values of the spatial parameters are negligible compared to the raw values.
• the fine values Xf, yf of the spatial parameters are substantially equal to the raw values x c , y c .
• the refinement formula for the dilation parameter can be approximated by calculating [3 as follows:
• the encoding module 30 therefore encodes here in a data stream F: i) the stored coefficients, grouped together under the notation C m ,K, ii) positions, grouped together under the notation PK, consisting raw values of the spatial parameters, and iii) the real parts of the derivation scalar products for the frequency parameters, grouped under the notation ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ K, iv) the decisions, grouped under the notation dK.
• the scheduling module 20 the encoding rate of the positions is reduced by encoding the positions of the wavelets in a differential way after the scheduling in raster scan.
• the encoding module 30 here encodes in the data stream F metadata M representative of the sets of discrete values for the frequency parameters. It is in fact not necessary to transmit metadata representative of the sets of discrete values for the spatial parameters since the raw values of the spatial parameters are transmitted.
• These metadata M include for example the formulas, presented during step E2, making it possible to determine the discrete sets, as well as the analytical formulas of the Gabor wavelets and their derivatives with respect to the parameters.
• the encoding module could not transmit metadata.
• the construction device receiving the data stream could for example be designed to specifically construct decomposed digital holograms based on the dictionary used by the decomposition device.
• the encoding module 30 implements an entropy coding to generate the data stream F.
• the data stream F is here a sequence of bits.
• the transmission module 40 transmits the data stream F, here intended for the construction device 200.
• the transmission is carried out by means of a communication network, for example by Internet, by a wired connection or by a wireless connection. son.
• the steps of the construction method are here implemented by the construction device 200, which is here specifically designed to construct the digital hologram H based on the data stream F.
• the construction device 200 includes a receiving module 50, a decoding module 60 and a building module 70.
• the reception module 50 is designed to receive the data stream F and send it to the decoding module 60.
• the decoding module 60 is designed to decode the data stream F and extract the data representative of the digital hologram H.
• the construction module 70 is adapted to construct the digital hologram H on the basis of the data representative of the digital hologram H extracted by the decoding module 60.
• the aforementioned modules 50, 60, 70 of the construction device 200 can in practice be implemented by the cooperation of at least one hardware element, such as a processor and a communication circuit, in particular for the module of reception 50, and software elements, such as computer program instructions executable by the aforementioned processor.
• a hardware element such as a processor and a communication circuit, in particular for the module of reception 50
• software elements such as computer program instructions executable by the aforementioned processor.
• These computer program instructions are here recorded on a memory included in the construction device 200 and to which the processor can have access.
• These computer program instructions may in particular be such that the construction device 200 implements at least part of the steps described below with reference to FIG. 5 when these instructions are executed by the processor of the construction device 200 .
• a module 50, 60, 70 of the construction device 200 performs an action within the framework of a step of the method, this means that the processor executes the computer program instructions dedicated to the realization of this stage or part of this stage.
• the processor, and therefore the aforementioned modules, are thus programmed to perform the steps described below with reference to Figure 5.
• FIG. 5 represents the main steps of the method for constructing the digital hologram H according to the first embodiment.
• the construction method begins with a step E11 during which the reception module 50 receives the data stream F and transmits it to the decoding module 60.
• the decoding module 60 decodes the data representative of the digital hologram H.
• the decoding module 60 here decodes the data which is encoded in step E10 of the transmission method. As shown in FIG. 5, this means here that the decoding module 60 extracts from the data stream F: i) the stored coefficients, ii) the positions, iii) the real parts of the scalar products of derivation for the frequency parameters, iv ) decisions, and iv) metadata M.
• the data extracted by the decoding module 60 are grouped together in packets, one packet containing all the data resulting from the same iteration of the decomposition method.
• the decomposition modulus determines which stored coefficient, position and decision are associated with a given triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ .
• the decoding module 60 determines from which raw values the real parts of the scalar products of derivation of the frequency parameters are derived.
• the decoding module 60 determines, for each triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ , the triplet of raw values ⁇ o c , f xc , f y c ⁇ which served as the basis, during an iteration of step E4 of the decomposition method, for determining this triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D ac ], R[Df XC ], R[Df yc ] ⁇ . Since the decomposition was performed in L iterations, the decoding module 60 therefore determines here L triplets of raw values.
• the decoding module 60 determines, on the basis of the metadata M and for each frequency parameter, the discrete set of values from the dictionary D.
• the decoding module 60 uses for example the formulas presented in step E2 and whose parameters are transmitted via the metadata M.
• the decoding module could be designed to process only digital holograms decomposed on the basis of the dictionary D.
• the discrete sets could for example be pre-recorded on the memory of the construction device.
• step E12 we describe here in detail how the decomposition module 60 performs step E12, as well as a following step E13, for a triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ K issued from iteration K, among the L iterations, of the decomposition process.
• the decomposition module 60 performs steps E12 and E13 for all the triplets, T1 that is to say for the L triplets, of real parts of the scalar products of derivation for the frequency parameters.
• the decoding module 60 first determines a triplet of raw values associated with this triplet of real parts of the scalar derivative products for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ K.
• the decomposition module 60 tests, iteratively, all the triplets of raw values which can be defined on the basis of the discrete sets of values.
• the decomposition module 60 therefore iteratively tests Nz triplets of raw values which correspond to all the composition wavelets initially contained in the dictionary D.
• FIG. 6 represents all the tests carried out for a triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D(jc], R[Df XC ], R[Dfyc] ⁇ K.
• the decomposition module 60 determines, on the basis of the triplet of raw values considered and the scalar products of derivation for the frequency parameters, test values ot, fxt, f y t.
• the decomposition module 60 first calculates, on the basis of the raw value considered for the dilation parameter and the scalar derivative products for the frequency parameters, a test value for the dilation parameter ot. It calculates this test value for the dilation parameter ot according to the refinement formula presented in step E4 and the approximate formula of [3 presented in step E10.
• the decomposition module 60 calculates, on the basis of the raw value considered for the spatial frequency parameter along the direction x, of the test value for the dilation parameter ot and of the scalar derivative products for the frequency parameters, a value test fxt for the spatial frequency parameter along the x direction. It calculates this test value fxt for the spatial frequency parameter along the direction x in accordance with the refinement formula presented in step E4. The decomposition module 60 finally calculates a test value fyt for the spatial frequency parameter along the y direction in the same way as for the spatial frequency parameter along the x direction.
• the decomposition module 60 verifies whether the calculated test value triplet ⁇ ot, fxt, fyt ⁇ satisfies the following verification equations:
• the decoding module 60 tests another triplet of raw values.
• this first case is represented by the arrow going from step E51 to step E50.
• the decoding module 60 evaluates that the triplet of raw values considered ⁇ o c , f xc , fyc ⁇ K is the one from which the triplet of real parts of the derivation scalar products for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Dfy C ] ⁇ K.
• the decoding module 60 ends its iterative test for this triplet of real parts of the scalar derivative products for the frequency parameters ⁇ R[D ac ], R[D fxc ], R[Df yc ] ⁇ K.
• the decoding module 60 retains this triplet of raw values considered ⁇ o c , f xc , f yc ⁇ K for step E13.
• this second case is represented by the arrow going from step E50 to step E13.
• step E13 When no calculated test value triplet ⁇ ot, fxt, f y t ⁇ verifies the verification equations, the considered raw value triplet ⁇ o c , f xc , fyc ⁇ K which is kept for step E13 is that which satisfies the verification equations best, that is to say the one for which the differences between the members of the equalities are the smallest. Step E13
• the decoding module 60 determines a third decomposition wavelet g V K called construction wavelet.
• This wavelet is characterized by a group YK of construction values XK, YK, ÜK, fxK, fyK assigned to the spatial and frequency parameters.
• the decoding module 60 here takes into account the decision dK associated with this triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D CTC ], R[D fxc ], R[Df yc ] ⁇ K.
• the module decoding 60 determines the construction values OK, fxK, fyK of the frequency parameters as equal test values ⁇ ot, fxt, f y t ⁇ K which verified the verification equations in step E12.
• This triplet of test values ⁇ ot, fxt, f y t ⁇ K is the one resulting from the considered raw value triplet ⁇ o c , f xc , f yc ⁇ K determined in step E12.
• the decoding module 60 finds the fine values of the frequency parameters resulting from the iteration K of the decomposition method and characterizes the construction wavelet g V K on the basis of these fine values.
• the construction values OK, f X K, fyK for the triplet of real parts of the scalar products of derivation for the frequency parameters ⁇ R[D CTC ], R[Df XC ], R[Df yc ] ⁇ K are then substantially equal to the fine values Of, f X f, f y f from iteration K.
• substantially equal means that these values are equal to within the approximation on the parameter [3 described above.
• the module decoding 60 determines the construction values OK, f X K, fy K as equal to the triplet of raw values considered ⁇ o c , f xc , f yc ⁇ K determined in step E12. In other words, the decoding module 60 then uses the raw values of the frequency parameters from iteration K of the decomposition process and characterizes the construction wavelet g V K on the basis of these raw values.
• the decoding module 60 determines the construction values XK, YK of the spatial parameters on the basis of the position PK.
• the construction values XK, YK of the spatial parameters are more specifically equal to the position PK. This means that the decoding module 60 uses the raw values of the spatial parameters, determined during the decomposition, to characterize the construction wavelet g V K.
• the decoding module 60 has therefore determined L wavelets of construction g V K.
• the construction wavelets g V K are sent to the construction module 70.
• the construction module 70 calculates the digital hologram H by summing the contribution of the various construction wavelets weighted by their associated memorized coefficient.
• the memorized coefficient associated with a construction wavelet is that which is associated with the triplet of real parts of the scalar products of derivation for the frequency parameters and with the decision which were used to determine the construction wavelet.
• the construction module 70 thus determines the digital hologram H by performing the sum Zc m ,K g Y K for K ranging from 1 to L, ie for the L construction wavelets.
• the construction module 70 therefore constructs the digital hologram H by performing the sum of the construction wavelets g V K weighted by their associated stored coefficient c m ,K.
• the build device 200 here includes a display module 80 designed to display the hologram H constructed as shown above by the build module 70.
• the display module 80 comprises for example a spatial light modulator, or SLM for “Spatial Light Modulator”, which is optionally integrated into a visiohead, for example an augmented reality visiohead.
• SLM spatial light modulator
• SLM spatial Light Modulator
• a visiohead for example an augmented reality visiohead.
• a second embodiment of the decomposition, transmission, reception and construction of the digital hologram H is now described.
• the transmission device 100 and the construction device 200 comprise the same modules as in the first embodiment.
• the steps for selecting the digital hologram (corresponding to step E1) and for summing the construction wavelets (corresponding to step E14) are also identical to those of the first embodiment.
• this second embodiment makes it possible to further reduce the amount of data to be processed and transmitted.
• each Gabor wavelet is defined by the following formula:
• the Gabor wavelets g v n are defined by a group of parameters which includes four spatio-frequency parameters: the first spatial parameter, the second spatial parameter, the rotation parameter, and a second dilation parameter, denoted s n .
• the dilation parameter o n (often called the dispersion or variance parameter in the literature) which is described in the first embodiment is itself called the first dilation parameter
• the discrete sets of values are for example given by the formulas described in the first embodiment.
• 3(n) T*n where n is a strictly positive integer and T a number strictly positive real.
• the step of determining the raw values (corresponding to step E3) is identical to that of the first embodiment except that, in this second embodiment, the values raw for the group of parameters mentioned above.
• the raw values are noted 0 C for the rotation parameter and s c for the second dilation parameter.
• the raw values are determined by maximizing the modulus of the inner product between the digital hologram and the first wavelet which is characterized by the raw values.
• each raw value is determined as one of the predetermined values belonging to its set of discrete values.
• the step of determining the fine values also includes the refinement of the raw values.
• the mathematical formulas on which the decomposition module 10 is based to calculate the fine values are here also the Math formulas. 2 to Math. 8. With reference to the refinement formulas below, the fine values are noted 0f for the rotation parameter and Sf for the second dilation parameter.
• derivation scalar product for the rotation parameter denoted Dec
• D sc derivation scalar product for the second dilation parameter
• the refinement formula for the second dilation parameter is as follows:
• the refinement formula above for the second dilation parameter is an approximate formula obtained after simplification of the theoretical formulas allowing the calculation of the refinement values (Math. 5 to Math. 8).
• a higher-order, i.e. less simplified, refinement formula can for example be obtained through the formulas of the Math formulas. 31 and Math. 32.
• the spatial parameter refinement formulas are as follows:
• the decomposition module 10 first calculates the scalar products of derivation then the fine values are calculated thanks to the refinement formulas.
• a hybrid refinement is preferably applied corresponding to a refinement according to one or other of the frequency parameters, that is to say either according to the rotation parameter or according to the second dilation parameter .
• the decomposition module 10 performs the comparison between: i) a scalar product, denoted P s , between the hologram (or the residue during the iterations following the first iteration) and a wavelet characterized by the fine value of the second parameter dilation and the raw values for the other parameters (spatial parameters and rotation parameter); and ii) a scalar product, denoted Pe, between the hologram (or the residue during the iterations following the first iteration) and a wavelet characterized by the fine value of the rotation parameter and the raw values for the other parameters (spatial parameters and second dilation parameter).
• the wavelets considered here are defined by the raw values of the spatial parameters. Indeed, as detailed in the first embodiment, the refinement values of the spatial parameters are negligible compared to the raw values.
• the values transmitted by the transmission module 40 for the spatial parameters are the raw values, they are thus already taken into account during the breakdown.
• the decomposition modulus 10 determines, for this iteration of the decomposition, the second wavelet as the wavelet characterized by the fine value of the second dilation parameter and the raw values for the other parameters.
• the decomposition module 10 determines, for this iteration of the decomposition, the second wavelet as the wavelet characterized by the fine value of the rotation parameter and the raw values for the other parameters.
• the decomposition module 10 also generates a choice relating to this comparison.
• the choice is thus representative of the result of this comparison in the sense that it makes it possible to encode the fact that the modulus of P s is strictly greater or not than the modulus of Pe.
• the choice here is representative of the sign (positive, negative or zero) of the difference between the modulus of P s and the modulus of Pe.
• the choice is equal to 1 when the modulus of P s is strictly greater than the modulus of Pe and equal to 0 when the modulus of P s is less than or equal to the modulus of Pe.
• This hybrid refinement (either according to the second dilation parameter or according to the rotation parameter) makes it possible, thereafter, to transmit only the real part of the scalar product of derivation for the parameter according to which the refinement was carried out.
• the quantity of data to be transmitted is reduced (the choice can be encoded on only 1 bit).
• the quantization module 20 also performs an anticipated quantization (corresponding to step E5), here of the real part of the derivation scalar product for the frequency parameter according to which the refinement was carried out.
• the quantization modulus 20 quantifies the real part of the scalar product of derivation for the second dilation parameter.
• the quantization module 20 quantizes the real part of the derivation scalar product for the rotation parameter. In both cases, it is not necessary to quantize the other real part of the scalar product of derivation since it will not be transmitted.
• the fine value of the frequency parameter according to which the refinement is carried out is calculated on the basis of the quantized real part of its derivative scalar product.
• the fine value of the second dilation parameter is equal to the raw value of the second dilation parameter for the calculation of the fine value.
• the fine value of the rotation parameter is then calculated according to Math. 30 as described later with reference to the method of construction.
• the quantization module 20 implements a Lloyd Max quantizer, for example on 16 levels or on 32 levels.
• the decomposition module 10 calculates, in the same way as in the first embodiment, on the basis of the second wavelet determined at the refinement step: the fine coefficient, the raw coefficient, the decision, the stored coefficient and plump memorized. This calculation is performed analogously to steps E6 and E7 of the first embodiment.
• the decomposition module 10 determines the residual r in the same way as in step E8 of the first embodiment.
• the steps presented above are iterated L times.
• the decomposition module 10 After performing L iterations, the decomposition module 10 has in particular determined the following data which are representative of the digital hologram: L groups of raw values, L groups of fine values, L stored coefficients, L decisions and L choices.
• the scheduling module 20 orders the data representative of the digital hologram resulting from each iteration according to a line-by-line scanning scheme, in the same way as in step E9 of the first embodiment .
• step E10 data representative of the hologram are encoded and transmitted.
• This step includes generating a data stream containing enough information to construct the digital hologram.
• the encoding module 30 therefore encodes the following variables here in the data stream: i) the stored coefficients, ii) the positions made up of the raw values of the spatial parameters, iii) the raw values of the frequency parameters, iv) decisions, grouped under scoring, v) choices, grouped under scoring.
• the refinement values of the spatial parameters are negligible compared to the raw values. Only the raw values of the spatial parameters are thus transmitted.
• the encoding module 30 also encodes the following variables into the data stream: vi) the real parts of the derivation scalar products for the frequency parameters.
• the encoding module 30 encodes more particularly, among the real parts of the derivation scalar products for the two frequency parameters, only that for the frequency parameter according to which the hybrid refinement was carried out at said iteration (i.e. either the second expansion parameter or the rotation parameter).
• the positions and the raw values of the frequency parameters are data representative of the raw values.
• the real parts of the scalar products for the frequency parameters are data representative of the refinement values.
• the encoding module 30 encodes, for each iteration, the real part of the scalar product of derivation for the frequency parameter according to which the refinement was carried out, that is to say either the real part of the scalar product of derivation according to the parameter of rotation is the real part of the scalar product of derivation according to the second parameter of dilation.
• the raw values of the frequency parameters are here encoded using a scalar uniform quantizer, for example on 8 levels.
• the data stream of this second embodiment thus presents a very good compromise between the quantity of data transmitted which is for example lower than in the first embodiment, and the quality of the decomposition of the digital hologram which remains very representative of the digital hologram thanks to refinement.
• the data stream F is more particularly constructed in the following way.
• the data stream F firstly comprises a main header EP which indicates the start of the data stream F and which is representative: of the metadata M, of the types of encoding used (quantification levels, number of contexts used by the entropic coder ) and the encoding order of the variables, for example as shown in Figure 7.
• the metadata M include in particular lists with, for each of the variables, the values that they can take. These lists include in particular the discrete sets of values for the frequency parameters.
• the metadata M also includes the resolution of the digital hologram H, from which the decoding module 60 determines the discrete sets of values for the spatial parameters.
• the variables are here encoded in the data stream F in the form of indices, for example from 1 to 32 for the real parts of the scalar products for the frequency parameters which have been quantified by a Lloyd Max quantizer on 32 levels.
• the data stream F also includes an end marker MF delimiting the end of the data stream F.
• the data stream F then includes MO wavelet markers.
• the wavelet marker denoted MOK delimits the part of the data stream F encoding for the variables i) to vi) resulting from iteration K, that is to say for the decomposition roundlet determined during the iteration K.
• the variables are thus grouped by iteration, from the first iteration, delimited by the marker denoted MOi, to the last.
• the data stream F then comprises variable markers MV.
• the transmission module 40 transmits it to the construction device 200.
• the construction method begins with a reception step (corresponding to step E11) during which the reception module 50 receives the data stream F and transmits it to the decoding module 60.
• the module decoding 60 decodes, in particular here thanks to the information contained in the main header EP, the data representative of the digital hologram.
• the decoding module 60 extracts from the data stream: i) the stored coefficients, ii) the positions, iii) the raw values of the frequency parameters, iv) the decisions, v) the choices, vi) the real parts of the derivation scalar products for the frequency parameters.
• the decoding module 60 For each iteration, the decoding module 60 more specifically extracts from the data stream, among the real parts of the scalar products of derivation for the two frequency parameters, only that for the frequency parameter according to which the hybrid refinement was carried out at said iteration ( either the second dilation parameter or the rotation parameter).
• the data extracted by the decoding module 60 are grouped together in packets, one packet containing all the data resulting from the same iteration of the decomposition method.
• the decoding module 60 determines the construction values of the spatio-frequency parameters.
• the decoding module 60 determines a group of four construction values for each iteration of the decomposition: one for each spatial parameter and one for each frequency parameter. The decoding module 60 therefore determines four construction values for each wavelet marker of the data stream.
• the decoding module 60 thus performs L times (for the L iterations of the decomposition) the determination of four construction values and therefore determines L construction wavelets, each characterized by a group of four construction values.
• L times for the L iterations of the decomposition
• L construction wavelets each characterized by a group of four construction values.
• the decoding module 60 determines the construction values as equal to the raw values of the spatio-frequency parameters. The decoding module 60 therefore considers in particular the raw values of the frequency parameters resulting from the iteration K of the decomposition process and characterizes the construction round on the basis of these raw values.
• the decoding module 60 determines, on the basis of the choice hK, which frequency parameter was used for the hybrid refinement during the iteration K. This corresponds here to the case where the decision dK is equal to 1 .
• the construction value for the second expansion parameter is determined, based on the raw value of the second expansion parameter and the real part of the derivation inner product for the second expansion parameter, in accordance with the refinement formula for the second expansion parameter , that is, in accordance with Math. 23.
• the other construction values that is to say those of the two spatial parameters and that of the rotation parameter, are all three determined by the decoding module 60 as equal to the raw values of these parameters resulting from iteration K of the decomposition.
• the construction value for the rotation parameter is determined, based on the raw value of the rotation parameter and the real part of the inner derivative product for the rotation parameter, according to the refinement formula for the rotation parameter, i.e. say in accordance with Math. 29, in which it is considered that the fine value of the second expansion parameter is equal to the raw value of the second expansion parameter.
• the construction value 9K for the rotation parameter is determined according to the following formula:
• the other construction values that is to say those of the two spatial parameters and that of the second dilation parameter, are all three determined by the decoding module 60 as equal to the raw values of these parameters. from iteration K of the decomposition.
• the construction wavelets are sent to the construction module 70.
• the construction module construction 70 calculates the digital hologram by summing the contribution of the various construction wavelets weighted by their associated memorized coefficient.
• this second embodiment it is possible to perform the refinement both according to the second dilation parameter and according to the rotation parameter. In other words, it is possible not to apply the hybrid refinement but a refinement according to all the frequency parameters, as in the first embodiment.
• the real parts of the scalar derivative products for these two parameters are then quantified, preferably in advance, for example by using a Lloyd Max quantizer, for example on 16 levels or on 32 levels.
• the data stream then includes the real parts of the derivation scalar product for the rotation parameter and of the derivation scalar product for the second dilation parameter but does not include the choices.
• each construction value of the frequency parameters is then determined by performing the combination of the raw value with the part or parts real values of the scalar products of derivation according to the refinement formulas, that is to say according to Math. 23 and Math. 29.
• the constructed hologram is therefore more representative of the digital hologram but the quantity of data to be transmitted, that is to say the size of the data stream, is greater.
• the decomposition module implements the following formulas for a and y to determine the fine value of the second dilation parameter:
• D'ec represents the second-order derivation scalar product between the digital hologram and a second derivative of the first wavelet with respect to the second dilation parameter.
• D'ec's formula is as follows:
• the fine value of the second expansion parameter is determined with greater precision. Consequently, the fine value of the rotation parameter is also determined with more precision.
• the second determined wavelet is more representative of the digital hologram than that determined in the general case of the second embodiment (i.e. by implementing the formulas Math. 24 and Math. 29).

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