METHOD AND DEVICE FOR RECONSTRUCTING A DIGITAL HOLOGRAM, METHOD FOR DISPLAYING A DIGITAL HOLOGRAM AND ASSOCIATED SYSTEM

Abstract

A digital hologram is represented by a set of coefficients respectively associated with a plurality of definition wavelets each defined by a tuple of coordinates in a multidimensional space. A method for reconstructing the digital hologram in order to display it by a display, includes the following steps: depending on at least one data item representative of a characteristic of the display, determining a transformation of the multidimensional space; and generating a reconstructed hologram by assigning each coefficient of at least some of the coefficients to a reconstruction wavelet defined by an image reconstruction tuple by the predetermined transformation of the tuple of coordinates defining the definition wavelet associate with the coefficient in question. An associated display method, reconstruction device and system are also described.

Claims

1. A method for reconstructing a digital hologram for the digital hologram to be displayed by a display device, the digital hologram being represented by a set of coefficients respectively associated with a plurality of definition wavelets, each of the definition wavelets being defined by a tuple of coordinates in a multidimensional space, the method comprising: determining a transformation of said multidimensional space, based on at least one data item representative of a characteristic of the display device; and generating a reconstructed hologram by assigning each of the coefficients of at least some of said coefficients to a reconstruction wavelet defined by a reconstruction tuple image, by the determined transformation, of the tuple of coordinates defining the definition wavelet associated with the respective coefficient.

2. The method according to claim 1, wherein said characteristic of the display device is a construction characteristic of the display device.

3. The method according to claim 1, wherein said characteristic of the display device is a position or direction characteristic of the display device.

4. The method according to claim 1, wherein said transformation is determined based on a first data item representative of a construction characteristic of the display device and a second data item representative of a position or direction characteristic of the display device.

5. The method according to claim 1, wherein the generating the reconstructed hologram comprises: assigning each of the coefficients associated with the respective definition wavelet, defined by the respective tuple of coordinates to the reconstruction wavelet defined by the respective reconstruction tuple image of the respective tuple of coordinates by the determined transformation, and selecting the coefficients assigned to reconstruction wavelets defined by the tuples of coordinates verifying a predetermined criterion.

6. The method according to claim 1, wherein the generating the reconstructed hologram comprises: determining a criterion modified based on the determined transformation and a predetermined criterion, and selecting coefficients whose associated definition wavelet is defined by a tuple of coordinates verifying the modified criterion.

7. The method according to claim 1, wherein the generating the reconstructed hologram comprises scanning a binary tree having leaves that correspond to the coefficients of said set of coefficients.

8. A method for displaying a digital hologram, the method comprising: reconstructing the digital hologram by the method according to claim 1; and displaying the reconstructed hologram by said display device.

9. A reconstruction device for reconstructing a digital hologram for the digital hologram to be displayed by a display device, the reconstruction device comprising: at least one processor configured to store a representation of the digital hologram comprising a set of coefficients respectively associated with a plurality of definition wavelets, each of the definition wavelets being defined by a tuple of coordinates in a multidimensional space, determine a transformation of said multidimensional space as a function of at least one data item representative of a characteristic of the display device, and; generate a reconstructed hologram by assigning each of the coefficients of at least some of said coefficients to a reconstruction wavelet defined by a reconstruction tuple image, by the determined transformation, of the tuple of coordinates defining the definition wavelet associated with the respective coefficient.

10. The reconstruction device according to claim 9, wherein the at least one processor is further configured to receive said representative data item, and transmit the assigned coefficients, and, for each of the assigned coefficients, information indicating the reconstruction wavelet to which the respective selected coefficient is assigned in the reconstructed hologram.

11. A system comprising: the reconstruction device according to claim 9; and said display device.

12. The method according to claim 2, wherein the generating the reconstructed hologram comprises: assigning each of the coefficients associated with the respective definition wavelet, defined by the respective tuple of coordinates, to the reconstruction wavelet defined by the respective reconstruction tuple image of the respective tuple of coordinates by the determined transformation, and selecting the coefficients assigned to reconstruction wavelets defined by the tuples of coordinates verifying a predetermined criterion.

13. The method according to claim 2, wherein the generating the reconstructed hologram comprises: determining a criterion modified based on the determined transformation and a predetermined criterion, and selecting coefficients whose associated definition wavelet is defined by a tuple of coordinates verifying the modified criterion.

14. The method according to claim 2, wherein the generating the reconstructed hologram comprises scanning a binary tree having leaves that correspond to the coefficients of said set of coefficients.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Moreover, various other features of the invention will be apparent from the appended description made with reference to the drawings that illustrate non-limiting embodiments of the invention, and wherein:

[0037] FIG. 1 schematically shows a system for displaying a digital hologram;

[0038] FIG. 2 schematically shows an example of display device usable in the system of FIG. 1;

[0039] FIG. 3 is a flowchart showing an example of a method for reconstructing a digital hologram according to the invention;

[0040] FIG. 4 shows an alternative embodiment for certain steps of the method of FIG. 3;

[0041] FIG. 5 schematically shows a plane mirror and a converging lens used to model a parabolic mirror; and

[0042] FIG. 6 shows angles used in the following description.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0043] FIG. 1 shows a system according to a possible embodiment of the invention.

[0044] The system of FIG. 1 comprises an electronic device 100 and a display unit 200.

[0045] As will be understood from the following description, the electronic device 100 forms a device for reconstructing a digital hologram as proposed by the invention.

[0046] In the example described here, the electronic device 100 and the display unit 200 are distant from each other and exchange data between each other via a communication network I (or, as an alternative, several interconnected communication networks), as will be described in more detail hereinafter. The electronic device 100 can then form a server capable of providing data to the display unit 200 (that then forms a client) for displaying a digital hologram, as described hereinafter.

[0047] The electronic device 100 comprises a storage module 102, a reception module 104, a transformation determination module 106, a processing unit 108 and a transmission module 114.

[0048] In practice, the above-mentioned modules and unit 102, 104, 106, 108, 114 can be implemented by the cooperation of at least one hardware element (such as a processor of the electronic device 100 and/or, in particular for the reception module 104 and/or the transmission module 114, a communication circuit) and software elements (such as computer-program instructions executable by the above-mentioned processor).

[0049] These computer-program instructions can in particular be such that the electronic device 100 implements a part at least of the steps described hereinafter with reference to FIGS. 3 and 4 when these instructions are executed by the processor of the electronic device 100.

[0050] The storage module 102 (made in practice by means of a memory or a drive disk) stores a representation of a digital hologram H comprising a set of coefficients c.sub.θ,ξ,x,y associated with a plurality of definition wavelets ϕ.sub.θ,ξ,x,y respectively, each defined by a tuple of coordinates (θ, ξ, x, y) in a multidimensional space E (here a 4-dimensional space).

[0051] The definition wavelets ϕ.sub.θ,ξ,x,y are here Gabor-Morlet wavelets, which are each defined by a quadruplet (θ, ξ, x, y) comprising:

[0052] a first coordinate 6 that defines the direction of a diffracted ray associated with the wavelet ϕ.sub.θ,ξ,x,y in question;

[0053] a second coordinate that defines the spatial frequency of this diffracted ray;

[0054] a third coordinate x and a fourth coordinate y that define the position of this diffracted ray in the plane of the digital hologram H.

[0055] Such a representation of the digital hologram H comprises in practice a predetermined number of coefficients c.sub.θ,ξ,x,y in relation with a discretization of the multidimensional space E of the definition parameters (θ, ξ, x, y) of the wavelets ϕ.sub.θ,ξ,x,y.

[0056] For example, the representation of the digital hologram H comprises coefficients θ.sub.θ,ξ,x,y associated with the definition wavelets ϕ.sub.θ,ξ,x,y respectively, defined by tuples of the form (θ.sub.k, ξ.sub.I, x.sub.m, y.sub.n) with: [0057] θ.sub.k=2πk/N.sub.θ for k an integer between 0 and N.sub.θ−1, [0058] ξ.sub.I=I.Δ.sub.ξ for I an integer between −N.sub.ξ and N.sub.ξ, [0059] x.sub.m=m.Δ.sub.x for m an integer between −N.sub.x and N.sub.x, [0060] y.sub.n=n.Δ.sub.y for n an integer between −N.sub.y and N.sub.y,

[0061] where N.sub.θ, N.sub.ξ, N.sub.x, N.sub.y are predefined integer numbers (that affect the number of coefficients c.sub.θ,ξ,x,y used in the representation) and where Δ.sub.ξ, Δ.sub.x, Δ.sub.y are predefined discretization pitches.

[0062] The so-represented digital hologram H can be written:

H=Y.sub.k,l,m,nϕ.sub.θk,ξl,xm,yn.c.sub.θk,ξl,xm,yn.

[0063] The reception module 104 is adapted to receive data via the above-mentioned communication network I, in particular data C, P representative of characteristics of a display device 202 of the display unit 200.

[0064] As explained more precisely hereinafter, these data representative of characteristics of the display device 202 may be data C representative of a construction characteristic of the display device 202 and/or data P representative of a position or direction characteristic of the display device 202.

[0065] The transformation determination module 106 is designed to determine a transformation σ of the above-mentioned multidimensional space E as a function of the data C, P representative of the characteristics of the display device 202 received by the reception module 104.

[0066] Different examples of determination of such a transformation a are given hereinafter in the present disclosure. Moreover, as explained hereinafter, this transformation a is used to generate a reconstructed (digital) hologram H′ based on the digital hologram H by taking into account the characteristics of the display device 202.

[0067] The processing unit 108 comprises a selection module 110 and a reassignment module 112.

[0068] The selection module 110 is designed to select the coefficients c.sub.θ,ξ,x,y whose associated definition wavelet ϕ.sub.θ,ξ,x,y is defined by a tuple of coordinates (θ, ξ, x, y) whose image (θ′, ξ′, x′, y′) by the transformation σ (determined by the transformation determination module 106) verifies a predetermined criterion. (With the above notation, we hence have: (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)].

[0069] The predetermined criterion is for example the fact that the image (θ′, ξ′, x′, y′) belongs to a predefined sub-set of the multidimensional space E, as explained hereinafter.

[0070] The selection module 110 thus makes it possible to select only the coefficients c.sub.θ,ξ,x,y that are relevant for a display on the display device 202 (by taking into account the characteristics of the display device 202 via the transformation σ).

[0071] The reassignment module 112 is designed to assign a given coefficient c.sub.θ,ξ,x,y to a wavelet ϕ.sub.θ,ξ,x,y called hereinafter “reconstruction wavelet”, different from the definition wavelet ϕ.sub.θ,ξ,x,y associated with this coefficient c.sub.θ,ξ,x,y and defined by a reconstruction tuple (θ′, ξ′, x′, y′) image by the transformation a of the tuple coordinates (θ, ξ, x, y) defining this definition wavelet ϕ.sub.θ,ξ,x,y (i.e. we have: (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)]).

[0072] As explained hereinafter, by taking into account the characteristics of the display device 202 thanks to the transformation a, this new assignment of the coefficients makes it possible to generate a reconstructed hologram H′ whose display by the display device 202 will best recreate the digital hologram H for the user.

[0073] The transmission module 114 is capable of transmitting data on the communication network I (in particular to the display unit 200). The transmission module 114 can transmit in particular the coefficients c.sub.θ,ξ,x,y selected by the selection module 110, as well as possibly, for each selected coefficient c.sub.θ,ξ,x,y information indicative of the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ to which the selected coefficient c.sub.θ,ξ,x,y is assigned by the reassignment module 112 (and hence in the reconstructed hologram H′). As an alternative, the reconstruction wavelet ϕ.sub.θ,ξ,x,y to which a transmitted coefficient c.sub.θ,ξ,x,y is assigned could be derived from the position of this coefficient c.sub.θ,ξ,x,y in the flow of data. In other words, the flow of data comprises in this case the coefficients ordered in an order determined by the reconstruction wavelets to which the coefficients are assigned (the reconstruction wavelets being themselves ordered in a predefined order).

[0074] The display unit 200 comprises the already-mentioned display device 202, a position and/or direction sensor 204, a transmission module 206, a reception module 208 and a control module 210. An example of display device 202 is described hereinafter with reference to FIG. 2.

[0075] In practice, the above-mentioned modules 206, 208, 210 can be implemented by the cooperation of at least one hardware element (such as a processor of the display unit 200 and/or, in particular for the reception module 208 and/or the transmission module 206, a communication circuit) and software elements (such as computer-program instructions executable by the above-mentioned processor).

[0076] These computer-program instructions can in particular be such that the display unit 200 implements a part at least of the steps described hereinafter with reference to FIG. 3 when these instructions are executed by the processor of the display unit 200.

[0077] The position and/or direction sensor 204 is linked to the display device 202 and provides data P representative of the position and/or direction of the display device 202.

[0078] Precisely, the position of the display device 202 is for example defined by a translation T and the direction of the display device 202 by a rotation R. The position and/or direction sensor 204 comprises for example an Inertial Measurement Unit, or IMU.

[0079] The use of such a position and/or direction sensor 204 is interesting in particular when the display device 202 is of the portable type (as this is the case in the example described herein, as explained hereinafter). Such a position and/or direction sensor could however be omitted in embodiments of the invention in which it is not searched to know the position or direction of the display device (the invention making it possible, in this case, to take into account other characteristics of the display device, such as for example construction characteristics).

[0080] The transmission module 206 is capable of transmitting data on the communication network I, in particular to the electronic device 100 (via the reception module 104 of the electronic device 100).

[0081] The transmission module 206 can hence transmit (to the electronic device 100) data C representative of a construction characteristic of the display device 202 (these data C being for example provided directly by the display device 202) and/or data P representative of a position or direction characteristic of the display device 202 (here the data P produced by the position and/or direction sensor 204).

[0082] On its side, the reception module 208 is capable of receiving data on the communication network I, in particular from the electronic device 100 (precisely from the transmission module 114 of the electronic device 100).

[0083] The reception module 208 can hence receive in particular the coefficients c.sub.θ,ξ,x,y transmitted by the transmission module 114 of the electronic device 100, as well as possibly, for each transmitted coefficient c.sub.θ,ξ,x,y, information indicative of the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ to which this transmitted coefficient c.sub.θ,ξ,x,y is assigned in the reconstructed hologram H′.

[0084] The control module 210 can then calculate the reconstructed hologram H′ by summing the different reconstruction wavelets ϕ.sub.θ′,ξ′,x′,y′ with weighting of each reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ by the coefficient c.sub.θ,ξ,x,y assigned to this reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ in the reconstructed hologram H′, i.e. by denoting c′.sub.θ′,ξ′,x′,y′ the coefficient assigned to the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ (i.e. with c′.sub.θ′,ξ′,x′,y′ =.sub.θ,ξ,x,y):

H′=Σ.sub.θ′,ξ′,x′,y′ϕ.sub.θ′,ξ′,x′,y′.c.sub.θ′,ξ′,x′,y′

[0085] The control module 210 can hence control the display of the reconstructed hologram H′ by the display device 202. In practice, when the display device 202 comprises a light modulator 4 formed of pixels identified by a pair of coordinates, the control module 210 controls the pixel of coordinates (i,j) as a function of the value H′(i,j) associated with these coordinates (i,j) in the reconstructed hologram H′.

[0086] FIG. 2 schematically shows an example of display device 202 usable in the display unit 200 of FIG. 1.

[0087] The display device 202 is here a portable display device, for example a Head Mounted Display, or HMD.

[0088] The display device 202 comprises a light source 2 (monochromatic, of wavelength A), the above-mentioned light modulator 4, a lens 6 and a mirror 8, here parabolic. Such an optical system is for example described in the book “Introduction to Matrix Methods in Optics”, Anthony Gerrard, James M. Burch, Wiley, 1975, ISBN 0471296856, see in particular Chapter 18: “Matrix Methods in Paraxial Optics”.

[0089] The light modulator 4 is for example a Spatial Light Modulator, or SLM.

[0090] In practice, the mirror 8 can be made by means of a semi-transparent blade in order to superimpose the object displayed by the display device 202 to the real environment of the user.

[0091] The lens 6 is a converging lens, of focal distance f.sub.1, making it possible to make a system such as a Fourier Transform Optical System, or FTOS.

[0092] The parabolic mirror 8 is modeled hereinafter by the combination of an inclined flat mirror 81 and a converging lens 82 of focal distance f.sub.2 located at a distance d from the above-mentioned converging lens 6. The flat mirror 81 and the converging lens 82 are shown in FIG. 5.

[0093] FIG. 3 is a flowchart showing an example of method for reconstructing a digital hologram that may be implemented in the system of FIG. 1.

[0094] The method of FIG. 3 starts by a step E10 of taking into account construction characteristics (or intrinsic parameters) of the display device 202.

[0095] This step can be implemented in practice by the transmission of data C representative of construction characteristics of the display device 202 from the display device 202 to the transmission module 206 during an initialization phase of the display unit 200.

[0096] In conceivable embodiments, step E10 may comprise the transmission of the data C representative of construction characteristics of the display device 202 by the transmission module 206 to the electronic device 100 (thanks to the reception module 104 of the electronic device 100). In these embodiments, the data C representative of construction characteristics of the display device 202 are received by the reception module 104 of the electronic device 100 at step E12.

[0097] The method continues with a step E14 of measuring the position and direction of the display device 202 by means of the position and/or direction sensor 204. The position and/or direction sensor 204 thus produces data P representative of a position or direction characteristic of the display device 202. These data P here define a translation T representative of the position of the display device 202 and a rotation R representative of the direction of the display device 202.

[0098] The method then comprises a step E16 of transmitting, by the transmission module 206 and to the electronic device 100, the data P representative of a position or direction characteristic of the display device 202.

[0099] Step E16 here further comprises the transmission, by the transmission module 206 and to the electronic device 100, of the data C representative of construction characteristics of the display device 202.

[0100] In the example described hereinabove in which the display device is of the type shown in FIG. 2, these data C representative of construction characteristics of the display device 202 comprise the focal distance f.sub.1 of the lens 6, as well as the focal distance f.sub.2 and the distance d associated with the mirror 8 as explained hereinabove, and possibly the wavelength λ of the light source 2 used for displaying the hologram.

[0101] The method then comprises a step E18 of receiving, by the reception module 104 of the electronic device 100, the data P representative of a position or direction characteristic of the display device 202 and, as the case may be, the data C representative of construction characteristics of the display device 202.

[0102] The method continues with a step E20 of determining a transformation a of the above-mentioned multidimensional space E based on the data P representative of a position or direction characteristic of the display device 202 and the data C representative of construction characteristics of the display device 202.

[0103] Different examples of construction of the transformation σ as a function of the data P, C representative of characteristics of the display device 202 will now be described.

[0104] In these examples, a function F is used that, to any tuple (θ,ξ,x,y) of the multidimensional space E associates the tuple of spatial frequency coordinates (f.sub.x, f.sub.y, X, Y), with: [0105] f.sub.x=ξ cos θ [0106] f.sub.y=ξ sin θ [0107] X=x [0108] Y=y

[0109] where f.sub.x and f.sub.y are the spatial frequencies (respectively along the abscissa axis x and the ordinates axis y) associated with a diffracted ray of direction θ and spatial frequency ξ.

[0110] The electronic device 100 (here, the transformation determination module 106) determines on the one hand an intrinsic transformation ρ.sub.i corresponding to light path changes due to the construction characteristics of the display device 202. This intrinsic transformation ρ.sub.1 is here determined in the space-frequency coordinate space (f.sub.x, f.sub.y, X, Y).

[0111] In practice, when the construction characteristics of the display device 202 are fixed, the intrinsic transformation ρ.sub.1 is determined only once.

[0112] The intrinsic transformation ρ.sub.i may be obtained by composition of several transformations respectively associated with elements (in particular, optical elements) of the display device 202.

[0113] In the case of FIG. 2, for example, the intrinsic transformation is determined as follows: ρ.sub.i=ρ.sub.4 o ρ.sub.3 o ρ.sub.2 o ρ.sub.1

[0114] where o is the composition operator, ρ.sub.1 is the transformation associated with the travel (of light) through the lens 6, ρ.sub.2 is the transformation associated with the propagation of light from the lens 6 to the mirror 8, ρ.sub.3 is the transformation associated with the equivalent plane mirror 81 and ρ.sub.4 is the transformation associated with the equivalent lens 82 (see hereinabove as regards the plane mirror 81 and the lens 82 equivalent to the hyperbolic mirror 8).

[0115] In paraxial approximation, these different transformations ρ.sub.1, ρ.sub.2, ρ.sub.3, ρ.sub.4 are linear (in the space-frequency coordinate space) and can thus be written as matrices M.sub.1, M.sub.2, M.sub.3, M.sub.4, respectively, with

[00001]$M_1=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ \frac{-1}{\lambda f_1}& 0& 1& 0\\ 0& \frac{-1}{\lambda f_1}& 0& 1\end{array}\right)$$M_2=\left(\begin{array}{cccc}1& 0& \lambda d& 0\\ 0& 1& 0& \lambda d\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$M_3=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)$$M_4=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ \frac{-1}{\lambda f_2}& 0& 1& 0\\ 0& \frac{-1}{\lambda f_2}& 0& 1\end{array}\right)$

[0116] (the matrices M.sub.1 and M.sub.2 being expressed with respect to the reference frame (x′,y′,z′) represented in FIG. 2, the matrix M.sub.3 connecting the position and angles expressed in the reference frame (x″,y″,z″) to those expressed in the reference frame (x′″,y′″,z′″) and the matrix M.sub.4 corresponding to the travel through the lens of the reference frame (x′″, y′″, z′″) into itself).

[0117] The intrinsic transformation ρ.sub.1 is in this case a linear transformation (in the space-frequency coordinate space) defined by the matrix M:

[00002]$M=M_4{M}_3{M}_2{M}_1=\left(\begin{array}{cccc}\frac{d}{f_1}-1& 0& -\lambda d& 0\\ 0& 1-\frac{d}{f_1}& 0& \lambda d\\ \frac{1}{\lambda f_1}-\frac{\frac{d}{f_1}-1}{\lambda f_2}& 0& \frac{d}{f_2}-1& 0\\ 0& -\frac{1-\frac{d}{f_1}}{\lambda f_2}-\frac{1}{\lambda f_1}& 0& 1-\frac{d}{f_2}\end{array}\right)$

[0118] The transformation determination module 106 can thus define the intrinsic transformation ρ.sub.1 on the basis of the data C representative of construction characteristics of the display device 202.

[0119] According to a conceivable alternative, the intrinsic transformation ρ.sub.i could be determined on the basis of the data C representative of construction characteristics of the display device 202 within the display unit 200. In this case, the display unit can transmit to the electronic device 100 data representative of the intrinsic transformation ρ.sub.i, for example the elements of the matrix M in the example envisaged hereinabove.

[0120] In other embodiments, in order to process rays having a significant inclination with respect to the optical axis and to take into account the existence of non-linear phenomena, the paraxial approximation is not used. Each transformation ρ.sub.1, ρ.sub.2, ρ.sub.3, ρ.sub.4 can hence be defined, as well as the intrinsic transformation ρ.sub.i, by means of a limited development (or Taylor development) with several variables of the type:

[00003]${\rho }_m\left(V\right)={\rho }_m\left(V_0\right)+\underset{k=0}{\overset{n}{.Math.}}\frac{1}{k!}{D_k\left(V-V_0\right)}^k$

[0121] where V and Vo are elements of the space-frequency coordinate space and D.sub.k matrices formed by means of partial derivatives of the function ρ.sub.m in question with respect to the different variables of this function.

[0122] The electronic device 100 (here, the transformation determination module 106) determines on the other hand an extrinsic transformation ρ.sub.e corresponding to light ray changes due to the position and direction of the display device 202 (represented by the data P).

[0123] In the example described herein, as already indicated, the data P here define a translation T representative of the position of the display device 202 and a rotation R representative of the direction of the display device 202.

[0124] Let's denote t.sub.x, t.sub.y, t.sub.z the components of the translation T according to the three coordinate axes x, y, z (shown in FIG. 2), respectively, i.e. T=(t.sub.x, t.sub.y, t.sub.z).

[0125] Let's also denote R.sub.x, R.sub.y, R.sub.z the rotations about the three coordinate axes x, y, z, respectively, which, combined to each other, form the rotation R.

[0126] For small rotation angles (denoted a.sub.x, a.sub.y, a.sub.z, about the three coordinate axes x, y, z, respectively), each rotation can be approximated by an affine transformation and be written as a matrix, as follows:

[00004]$R_z=\left(\begin{array}{ccc}\cos a_Z& -s\mathrm{in}{a}_Z& 0\\ \sin a_Z& \cos a_Z& 0\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{cc}{R}_z^{\prime }& 0\\ 0& 1\end{array}\right)$$R_y=\left(\begin{array}{ccc}\cos a_y& 0& \sin a_y\\ 0& 1& 0\\ -s\mathrm{in}{a}_y& 0& \cos a_y\end{array}\right)$$R_x=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos a_x& -s\mathrm{in}{a}_x\\ 0& \sin a_x& \cos a_x\end{array}\right)$

[0127] and R=R.sub.xR.sub.yR.sub.z.

[0128] As described in the article “Global motion compensation for compressing holographic videos”, by D. Blinder, C. Schretter and P. Schelkens, vol. 26, Optics Express 2018 (20), pp. 25524-25533, the associated transformation in the 5-dimensional phase space is given by the matrix:

[00005]$A^A=\left(\begin{array}{ccc}{R}_z^{\prime }& 2\frac{t_z}{\lambda }{R}_Z^{\prime }& b_{\tau }\\ 0& R_Z^{\prime }& b_{\omega }\\ 0& 0& 1\end{array}\right)\mathrm{with}$$b_{\tau }=\left(\begin{array}{c}{t}_x\\ t_y\end{array}\right)$$b_{\omega }=\left(\begin{array}{c}{\theta }_x\\ {\theta }_y\end{array}\right)$

[0129] By keeping only the first four lines of the matrix A′, a matrix A is obtained:

[00006]$A=\left(\begin{array}{ccc}{R}_z^{\prime }& 2\frac{t_z}{\lambda }{R}_z^{\prime }& b_{\tau }\\ 0& R_z^{\prime }& b_{\omega }\end{array}\right)$

and it may be written:

[00007]$\left(\begin{array}{c}{X}^{\prime }\\ Y^{\prime }\\ f_x^{\prime }\\ f_y^{\prime }\end{array}\right)=A\left(\begin{array}{c}X\\ Y\\ f_x\\ f_y\\ 1\end{array}\right).$

[0130] The matrix A thus defines the extrinsic transformation ρ.sub.e.

[0131] In the embodiments in which the angles a.sub.x, a.sub.y, a.sub.z, the use of which is contemplated, do not allow the linear approximation used hereinabove, the extrinsic transformation ρ.sub.e can be determined as follows.

[0132] A ray diffracted at the point of coordinates (x,y) and whose direction is defined by a polar angle θ and an azimuth angle φ (as shown in FIG. 6) is modified by a rotation R and a translation T into a ray defined by the coordinates

[00008]$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ {\theta }^{\prime }\\ {\phi }^{\prime }\end{array}\right)=F\left(\begin{array}{c}x\\ y\\ \theta \\ \phi \end{array}\right)$

[0133] where F is determined as follows.

[0134] Let's define the point q of the three-dimensional space having for coordinates (x, y, 0) in the reference frame linked to the initial hologram H, as well as the three-dimensional vector p having for coordinates in said reference system (sin(φ) cos(θ), sin(φ)sin(θ),cos(φ)).

[0135] Let's define the point q′ of the three-dimensional space by

q′=R.sup.−1q−R.sup.−1T

[0136] (with R and T the matrices defined hereinabove) and the vector p′ by

p′=R.sup.−1p

Let's define the point q″ by

[00009]$q^{″}=q^{\prime }-\frac{q_z^{\prime }}{p_Z^{\prime }}{p}^{\prime }$

q.sub.z′ and p.sub.z′ denoting the third coordinate of q′ and p′, respectively. The case in which p.sub.z′ is null is not described because it corresponds to a direction incompatible with the intended applications (display system behind the plane of the initial hologram).

[0137] The transformation F is then given by

[00010]$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{c}{q}_x^{″}\\ q_y^{\prime }\end{array}\right)\mathrm{and}$$\left(\begin{array}{c}{\theta }^{\prime }\\ {\phi }^{\prime }\end{array}\right)={G^{-1}}^{°}{R}^{\circ }G\left(\begin{array}{c}\theta \\ \phi \end{array}\right)$

where G is the function

[00011]$G\left(\begin{array}{c}\theta \\ \phi \end{array}\right)=\left(\begin{array}{c}\sin \varphi \cos \theta \\ \sin \varphi \sin \theta \\ \cos \varphi \end{array}\right)$

whose inverse is given by

[00012]$G^{-1}\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}\mathrm{ar}\cos \left(z\right)\\ \mathrm{arc}\tan \left(y\right)\end{array}\right)$

[0138] In the example described herein, the transformation determination module 106 can thus determine, at step E20, a transformation σ of the multidimensional space E that takes into account the intrinsic transformation ρ.sub.i and the extrinsic transformation ρ.sub.e.

[0139] The intrinsic ρ.sub.1 and extrinsic ρ.sub.e transformations having here been determined in the space-frequency coordinate space, the function Γ defined hereinabove is used to take these transformations in the multidimensional space E into account. In other words: σ=Γ.sup.−1 o ρ o Γ, where ρ is the transformation of the space-frequency coordinate space obtained by combination of the inverse ρ.sub.1.sup.−1 of the intrinsic transformation ρ.sub.i, and the extrinsic transformation ρ.sub.e: ρ=ρ.sub.1.sup.−1 o ρ.sub.e.

[0140] It can be observed that the extrinsic transformation ρ.sub.e is here defined from the plane of the initial hologram to the plane of the eye and that the above combination thus uses the function ρ.sub.1.sup.−1 that goes from the plane of the eye to the plane of the light modulator 4 (the intrinsic function ρ.sub.1 defined hereinabove going from the plane of the light modulator 4 to the plane of the eye).

[0141] The method of FIG. 3 then continues with a step E22 of reassignment of each coefficient c.sub.θ,ξ,x,y (stored in the storage module 103 in association with a definition wavelet ϕ.sub.θ,ξ,x,y) to a reconstruction wavelet ϕ.sub.θ,ξ,x,y defined by a reconstruction tuple (θ′,ϵ′, x′, y′) such that: (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)].

[0142] In practice, for a set of definition tuples (θ, τ, x, y), the image σ[(θ,ξ, x, y)] of the tuple concerned by the transformation σ is determined, then the coefficient c.sub.θ,ξ,x,y associated with the definition wavelet ϕ.sub.θ,ξ,x,y defined by this definition tuple is assigned to the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ defined by the determined image σ[(θ, ξ, x, y)] (or, when a predetermined set of reconstruction wavelets is considered, to the reconstruction wavelet whose definition parameters are the closest to the image σ[(θ, ξ, x, y)]).

[0143] In the example described herein, this reassignment step E22 is implemented by the reassignment module 112.

[0144] The method of FIG. 3 then comprises a step E24 of selecting coefficients c.sub.θ,ξ,x,y whose consideration affects the display by the display device 202. The coefficients c.sub.θ,ξ,x,y assigned (after step E22) to a reconstruction wavelet ϕ.sub.θ,ξ,x,y relevant for the display device 202, i.e. defined by a tuple (θ′, ξ′, x′, y′) verifying a predetermined criterion, are thus selected.

[0145] In the example described herein, the coefficients c.sub.θ,ξ,x,y assigned to the reconstruction wavelets ϕ.sub.θ′,ξ′,x′,y′ defined by a tuple (θ′, ξ′, x′, y′) such that:

[0146] Γ(θ′, ξ′, x′, y′) ∈ S are selected,

[0147] where S is the sub-set of the space-frequency coordinate space defined as follows: [−S.sub.x/N.sub.x, S.sub.x/N.sub.x]×[−S.sub.y/N.sub.y, S.sub.y/N.sub.y]×[−N.sub.x/2, N.sub.x/2]×[−N.sub.y/2, N.sub.y/2], and where N.sub.x and N.sub.y are the horizontal and vertical resolutions, respectively, of the light modulator 4 and S.sub.x and S.sub.ythe horizontal and vertical dimensions, respectively, of the light modulator 4.

[0148] The maximum diffraction angles θ.sub.x and θ.sub.y that may be obtained (in the horizontal and vertical directions, respectively) are indeed given by:

[0149] θ.sub.x=arcsin(λN.sub.x/2S.sub.x) and

[0150] θ.sub.y=arcsin(λN.sub.y/2S.sub.y),

[0151] which correspond to the frequencies 2S.sub.x/N.sub.x and 2S.sub.y/N.sub.y, respectively.

[0152] As an alternative, rather than carrying out a reassignment step for all the coefficients c.sub.θ,ξ,x,y of the digital hologram H, then a selection step as just explained above, it is possible to carry out the reassignment step for only the coefficients c.sub.θ,ξ,x,y associated with a definition wavelet ϕ.sub.θ,ξ,x,y defined by a definition tuple (θ, ϵ, x, y) whose image (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)] verifies the predetermined criterion, i.e. whose image (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)] verifies Γ(θ′, ξ′, x′, y′) ∈S with the notations used hereinabove.

[0153] The method continues with step E26 in which the transmission module 114 of the electronic device 100 transmits on the communication network I the coefficients c.sub.θ,ξ,x,y selected at step E24 and, for each selected coefficient c.sub.θ,ξ,x,y information indicative of the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ to which this selected coefficient c.sub.θ,ξ,x,y has been assigned at the reassignment step E22.

[0154] The reception module 208 of the display unit 200 receives the transmitted coefficients c.sub.θ,ξ,x,y (as well as, here, the information indicative of the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ to which each coefficient c.sub.θ,ξ,x,y is assigned) at step E28.

[0155] The set of coefficients c.sub.θ,ξ,x,y respectively assigned to the reconstruction wavelets ϕ.sub.θ′,ξ′,x′,y′ defines the reconstructed hologram H′.

[0156] The reconstructed hologram H′ can hence be calculated (here by the control module 210) at step E30 by summing the different reconstruction wavelets ϕ.sub.θ′,ξ′,x′,y′ with weighting of each reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ by the coefficient c.sub.θ,ξ,x,y assigned to this reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ in the reconstructed hologram H′, i.e. by denoting c′.sub.θ′,ξ′,x′,y′ the coefficient assigned to the reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ (i.e. with c′.sub.θ′,ξ′,x′,y′=c.sub.θ,ξ,x,y):

H′=Σ.sub.θ′,ξ′,x′,y′ϕ.sub.θ′,ξ′,x′,y′c.sub.θ′,ξ′,x′,y′

The method then continues with step E32 in which the reconstructed hologram H′ is displayed by the display device 202.

[0157] The method then loops to step E14 for taking into account (as the case may be) a new position or direction of the display device 202.

[0158] An alternative embodiment of a part of the just-described method will now be described with reference to FIG. 4.

[0159] According to this alternative, steps E22 and E24 are replaced by steps E40 and E44 described now.

[0160] In this alternative, step E40 carries out a step of determining a criterion modified as a function of the transformation σ determined at step E20 of the above-mentioned criterion.

[0161] In the example described herein, the sub-set S′ of the multidimensional space E corresponding to the antecedent of the above-mentioned sub-set S by the transformation σ is determined: S′=σ.sup.−1 o Γ.sup.−1 (S)=Γ.sup.−1 o ρ.sup.−1 (S), where ρ is the transformation already introduced hereinabove: ρ=ρi.sup.−1 o ρ.sub.e. The so-determined sub-set S′ is called hereinafter the “modified sub-set”.

[0162] The extrinsic transformation ρ.sub.e depending on the data P representative of the position and/or direction characteristics of the display device 202, the sub-set S′ also varies as a function of these data P.

[0163] The alternative then continues with a step E42 of selecting coefficients c.sub.θ,ξ,x,y whose associated definition wavelet ϕ.sub.θ,ξ,x,y is defined by a tuple of coordinates (θ, ξ,x,y) verifying the modified criterion, i.e. herein belonging to the modified sub-set S′.

[0164] The method continues in this case with a step E44 of reassigning each selected coefficient c.sub.θ,ξ,x,y to a reconstruction wavelet ϕ.sub.θ′,ξ′,x′,y′ defined by a reconstruction tuple (0′, ξ′, x′, y′) image of the definition tuple (θ, ξ,x,y) associated with this selected coefficient c.sub.θ,ξ,x,y, i.e. such that: (θ′, ξ′, x′, y′)=σ[(θ, ξ, x, y)].

[0165] A conceivable possible design for storing the coefficients will now be described; this possible design is applicable in the different examples of implementation described hereinabove.

[0166] In the case where a great number of coefficients c.sub.θ,ξ,x,y associated with the definition wavelets ϕ.sub.θ,ξ,x,y are zero or negligible with respect to a given relevance threshold, it can be advantageous to organize these coefficients c.sub.θ,ξ,x,y as a kd tree (technique whose principle is known by the person skilled in the art), in order to reduce the time of search for a coefficient c.sub.θ,ξ,x,y corresponding, by the transformation σ, to the reconstruction tuple (θ′, ξ′, x′, y′).

[0167] In this alternative, the definition coefficients c.sub.θ,ξ,x,y are associated with the leaves of a binary tree (or kd tree) created by recursively subdividing the set E of the tuple of coordinates associated with the coefficients in each dimension successively: the set E is first subdivided into two sub-sets E.sub.1.sup.0 and E.sub.2.sup.0 represented by a threshold value θ.sub.0 in such a way that a tuple (θ, ξ, x, y) is associated with a coefficient in E.sub.1.sup.0 if and only if θ<θ.sub.0. The second stage of the tree is filled in the same way but considering the variable ξ, and so on by proceeding cyclically on the variables θ, ξ, x and y until the partitioned sets contain only one element.

[0168] During the selection step E24, the tuples (θ′, ξ′, x′, y′) defining reconstruction wavelets and corresponding to the set S are scanned, and for each of them, the antecedent (θ, ξ, x, y) of (θ′, ξ′, x′, y′) by the transformation a is obtained and a recursive search is made in the above-described tree. The reached leaf provides the coefficient c.sub.θ,ξ,x,y corresponding to the corresponding definition wavelet ϕ.sub.θ,ξ,x,y.