CN113467211B - Holographic encoding method based on gradient reduction of spectral loss function - Google Patents
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- 239000011159 matrix material Substances 0.000 claims description 8
- 238000001093 holography Methods 0.000 claims description 3
- 230000003287 optical effect Effects 0.000 claims description 3
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- 238000011478 gradient descent method Methods 0.000 claims description 2
- 238000004364 calculation method Methods 0.000 claims 2
- 238000009792 diffusion process Methods 0.000 claims 2
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- 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
- G03H1/0841—Encoding method mapping the synthesized field into a restricted set of values representative of the modulator parameters, e.g. detour phase coding
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- 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
- G03H1/0841—Encoding method mapping the synthesized field into a restricted set of values representative of the modulator parameters, e.g. detour phase coding
- G03H2001/085—Kinoform, i.e. phase only encoding wherein the computed field is processed into a distribution of phase differences
Abstract
The invention discloses a holographic coding method based on gradient decline of a spectrum loss function, which divides a frequency spectrum of a pure phase hologram into a signal frequency spectrum region and a noise frequency spectrum region, extracts the signal frequency spectrum region by filtering the frequency spectrum of a complex amplitude hologram, and then uses a gradient decline method to enable the signal frequency spectrum of the pure phase hologram to be approximately equal to the signal frequency spectrum of the complex amplitude hologram, wherein the noise frequency spectrum region of the pure phase hologram can be regarded as an optimized free variable and ignored. Compared with the common bi-phase encoding method and GS algorithm, the signal-to-noise ratio of the method is higher than that of the bi-phase encoding method, and the method is more suitable for holographic display of three-dimensional objects than the GS algorithm.
Description
Technical Field
The invention belongs to the field of three-dimensional display technologies, and particularly relates to a holographic encoding method based on gradient reduction of a spectrum loss function.
Background
Three-dimensional display technology is an important research and development direction of display technology. In recent years, with the popularization and application of virtual reality technology and augmented reality technology, research on head-mounted display devices has been increasing. Among them, holographic near-eye displays are receiving a lot of attention.
Holographic near-eye displays can carry depth information and have potential advantages in future near-eye display applications. To display the hologram, it is preferable to adjust the amplitude and phase of the light at the same time. But the spatial light modulator can only show amplitude or phase. A feasible method in experiments is to calculate the pure phase hologram that can be displayed in the phase-type spatial light modulator by using an encoding method. Currently, there are two commonly used holographic encoding algorithms: the double-phase encoding (double-phase) algorithm and the GS (Gerchberg-Saxton) algorithm.
Bi-phase encoding is the encoding of a complex amplitude value into two pure phase values. For the display of the three-dimensional object, firstly, a chromatography method or a point cloud method is used for firstly calculating the complex amplitude hologram of the three-dimensional object, and then a pure phase hologram is directly calculated by using double-phase encoding. However, bi-phase encoding suffers from coding noise, which causes the displayed image to appear noisy. The GS algorithm is an iterative method with the amplitude of the display image as the optimization target and the phase of the display image as the free variable. But such algorithms are difficult to compute holograms of three-dimensional objects. Moreover, when the phase of the displayed image is taken as a free variable, each pixel value of the actually reconstructed image is inevitably partially overlapped, and the phase values are different, so that interference occurs to cause noise.
Therefore, it is an important research direction in the field of holographic display to improve the display quality by proposing a new encoding method.
Disclosure of Invention
The invention aims to solve the problem of display quality caused by an encoding algorithm in a holographic near-eye and is suitable for displaying a three-dimensional object.
The purpose of the invention is realized by the following technical scheme: a holographic coding method based on gradient descent of a spectrum loss function divides the frequency spectrum of a pure phase hologram into a signal frequency spectrum region and a noise frequency spectrum region, extracts the signal frequency spectrum region by filtering the frequency spectrum of a complex amplitude hologram, and then enables the signal frequency spectrum of the pure phase hologram to be approximately equal to the signal frequency spectrum of the complex amplitude hologram by using a gradient descent method, wherein the noise frequency spectrum region of the pure phase hologram can be regarded as an optimized free variable and ignored; a filter is needed to be used for filtering the noise spectrum in an actual optical path; assuming a pure phase hologram, a complex amplitude hologram and an original size of m x n pixels (typically 1920 x 1080 pixels), the method comprises the steps of:
1) calculating the frequency spectrum F of a complex amplitude hologram by fast Fourier transform for a two-dimensional or three-dimensional original image light field0，F0Is a matrix of m x n, and F0Expanding the vector into an m x n dimensional vector;
2) filtering the frequency spectrum of the complex amplitude hologram, namely selecting a rectangular region with the length and the width of the center position of the frequency spectrum being half of the original size respectively, and extracting the rectangular region as a signal frequency spectrum region; representing the signal spectrum as
Wherein
Representing a Hadamard product, wherein P represents an m-n-dimensional vector, and if a certain element of P is in a signal spectrum region, the value of the certain element is 1, otherwise, the value of the certain element is 0;
3) for the initial pure phase hologram, performing fast Fourier transform to calculate a frequency spectrum F, selecting a rectangular region with a certain position length and width respectively being half of the original size as a signal frequency spectrum region, and taking the rest region as a noise frequency spectrum region; representing the signal spectrum as
4) The loss function is calculated for the signal spectrum of a phase-only hologram, since the spectrum is complex, the loss function C is defined as the form of the real part Re plus the imaginary part Im:
5) calculating the gradient of the loss function according to the loss function in the step 4), and updating the pure phase hologram by using the following gradient descent formula:
wherein theta iskA vector of m x n dimensions, C, which is expanded for the matrix formed by the phase values of the current phase-only hologramkFor the current loss function, η is the step size of the iteration, θk+1A vector of m x n dimensions is expanded for the matrix formed by the phase values of the updated phase-only hologram, k representing the kth iteration,
is a vector of dimensions m x n, the formula is:
wherein
Is a matrix with m x n rows and columns, F(k)Is the frequency spectrum F after the number k of iterations,
the FFT is a fast fourier transform, which is used to convert the optical field into a frequency spectrum,
is formed by thetakObtaining a m x n-dimensional complex vector;
6) and repeating the steps 4) and 5) for iteration until the loss function is converged to obtain the final pure phase hologram.
Furthermore, the amplitude value of the light field of the original image is obtained by calculating the gray value of the original image, and the phase value of the light field is set to be a constant value at the same depth, so that the noise caused by interference due to different phases of adjacent pixels of the reconstructed image can be avoided.
Further, the initial phase-only hologram may be generated using random values, or may be generated using a bi-phase hologram as the initial phase-only hologram, which may reduce the number of iterations.
Further, for practical purposes, the spatial light modulator used to display phase-only holograms can be formulated to have errors if crosstalk effects are present
Adding correction to crosstalk effect; the crosstalk effect can be expressed as a point spread model, and the phase values on the phase-only hologram can be rolled up into a point spread function in iteration; the formula for calculating the frequency spectrum can be rewritten as:
wherein
Representing a convolution, and a represents a point spread function of a pixel subject to crosstalk effects.
Further, if the size of the point spread function is in the order of one pixel, then the phase-only hologram needs to be oversampled.
Further, if off-axis holography is to be realized, a phase grating cannot be added in the last step like a common method, because the added phase grating is also influenced by the crosstalk effect, the position of a signal frequency spectrum region of the pure-phase hologram can be deviated from the center of a frequency spectrum in an iteration process instead of being arranged in the center of the frequency spectrum, and therefore off-axis holography is realized.
The invention has the beneficial effects that: compared with the common bi-phase coding method and the GS algorithm, the signal-to-noise ratio of the method is higher than that of the bi-phase coding method, and the method is more suitable for holographic display of three-dimensional objects than the GS algorithm.
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FIG. 1 is a schematic diagram of a signal spectrum region and a noise spectrum region according to an embodiment of the present invention, in which the abscissa and ordinate fx,fyWhen the spatial distance of the spectral plane in the actual experiment is shown, f is the focal length of the lens for Fourier transform, λ is the wavelength of light, and p is the pixel size of the spatial light modulator, the spectral region is
The spectral region of the signal can be selected as
Fig. 2 is a schematic diagram of simulation results of a reconstructed image of bi-phase encoding according to an embodiment of the present invention, (a) is an original image which is a two-dimensional picture of A, B, C letters, (b) is a simulation result of a reconstructed image of bi-phase encoding, and (c) is a simulation result of a reconstructed image of a method of the present invention;
fig. 3 is a schematic diagram showing simulation results of a reconstructed image by GS encoding according to an embodiment of the present invention, (a-c) shows an original image composed of 3-layer two-dimensional pictures, wherein the 1 st layer (a) is an alphabetic picture, the 2 nd and 3 rd layers (b-c) are random snowflake pictures, respectively, (d) shows simulation results of a reconstructed image by GS encoding, and (e) shows simulation results of a reconstructed image by the method of the present invention.
Detailed Description
The invention is described in further detail below with reference to the figures and specific examples.
Example 1
As shown in fig. 2, the present embodiment provides a comparison with the simulation result of a bi-phase encoded reconstructed image, where (a) is the original image, which is a two-dimensional picture of A, B, C letters. (b) The peak signal-to-noise ratio PSNR is 20.579, which is a simulation result of a reconstructed image of bi-phase encoding. (c) Is the simulation result of the reconstruction graph of the method of the invention, and the peak signal-to-noise ratio PSNR is 24.236. The initial pure phase hologram of the method of the invention adopts a biphase encoded hologram, and the total iteration times are 5 times. The light intensity and gray level conversion formula of holographic coding and reconstruction simulation uses:
where I is the normalized light intensity and gray is the normalized gray level. The diffraction distance between the reconstructed image plane and the holographic plane is 80 mm. The observation and simulation results show that the method has higher display quality and less noise, and the signal-to-noise ratio of the method is higher compared with that of the double-phase coding.
Example 2
As shown in fig. 3, this example provides a comparison with the simulation result of the reconstructed image of the GS algorithm, where (a-c) is an original, the original is composed of 3-layer two-dimensional pictures, the 1 st layer (a) is an alphabetic picture, the diffraction distance from the hologram surface is 80mm, the 2 nd and 3 rd layers (b-c) are random snowflake pictures, respectively, (b) the diffraction distance from the hologram surface is 81.5mm, and (c) the diffraction distance from the hologram surface is 83 mm. (d) The simulated result of the reconstructed image encoded by GS showed a diffraction distance of 80mm from the hologram surface. (e) The simulation result of the reconstructed image of the method of the invention is that the diffraction distance from the holographic surface is 80 mm. Because the GS algorithm can only optimize two-dimensional original images, 3 holograms of the GS algorithm are respectively calculated for 3 layers of original images, and then the simulated 3 reconstructed images are directly superposed, which is equivalent to simulating a time division multiplexing multi-layer GS algorithm. The GS algorithm uses a total number of iterations of 30. The initial pure phase hologram of the method of the invention adopts a biphase encoded hologram, and the total iteration times are 5 times.
It can be seen that the defocus patterns of the second and third layers of snowflakes in the GS-encoded reconstructed image are very noisy, because the GS-encoded reconstructed image has the phase of the reconstructed image as a free variable, and when the image is out of focus, adjacent pixels are overlapped and interfered, and the phase difference causes noise. The snowflake defocusing patterns in the simulation result of the method have no obvious noise, and the phase of the original image can be set to be a constant value, so that the obvious noise can not appear in a certain defocusing range. Therefore, the method is more suitable for displaying the three-dimensional object than the GS algorithm.
The above-described embodiments are intended to illustrate rather than to limit the invention, and any modifications and variations of the present invention are within the spirit of the invention and the scope of the appended claims.
Claims (6)
1. A holographic coding method based on gradient descent of a spectrum loss function is characterized in that a frequency spectrum of a pure phase hologram is divided into a signal frequency spectrum region and a noise frequency spectrum region, the signal frequency spectrum region is extracted by filtering the frequency spectrum of a complex amplitude hologram, then the signal frequency spectrum of the pure phase hologram is approximately equal to the signal frequency spectrum of the complex amplitude hologram by a gradient descent method, and the noise frequency spectrum region of the pure phase hologram is regarded as an optimized free variable and ignored; setting the size of pure phase hologram, complex amplitude hologram and original image as m pixel, the method includes the following steps:
1) calculating the frequency spectrum F of the complex amplitude hologram by using fast Fourier transform for the two-dimensional or three-dimensional original image light field0，F0Is a matrix of m x n, and F0Expanding the vector into an m x n dimensional vector;
2) filtering the frequency spectrum of the complex amplitude hologram, namely selecting a rectangular region with the length and the width of the center position of the frequency spectrum being respectively half of the original size, and extracting the rectangular region as a signal frequency spectrum region; representing the signal spectrum as
Wherein
Representing a Hadamard product, wherein P represents an m-n-dimensional vector, and if a certain element of P is in a signal spectrum region, the value of the certain element is 1, otherwise, the value of the certain element is 0;
3) for the initial pure phase hologram, performing fast Fourier transform to calculate a frequency spectrum F, selecting a rectangular area with the length and width of a certain position being half of the original size as a signal frequency spectrum area, and taking the rest area as a noise frequency spectrum area; representing the signal spectrum as
4) The loss function is calculated for the signal spectrum of a phase-only hologram, since the spectrum is complex, the loss function C is defined as the form of the real part Re plus the imaginary part Im:
5) calculating the gradient of the loss function according to the loss function in the step 4), and updating the pure phase hologram by using the following gradient descent formula:
wherein theta iskA vector of m x n dimensions, C, which is expanded for the matrix formed by the phase values of the current phase-only hologramkFor the current loss function, η is the step size of the iteration, θk+1A vector of m x n dimensions is expanded for the matrix formed by the phase values of the updated phase-only hologram, k representing the kth iteration,
is a vector with dimensions of m x n, and the formula is as follows:
wherein
Is a matrix with m x n rows and columns, F(k)Is the frequency spectrum F after the number k of iterations,
the FFT is a fast fourier transform, which is used to convert the optical field into a frequency spectrum,
is formed by thetakObtaining a m x n-dimensional complex vector;
6) and repeating the steps 4) and 5) for iteration until the loss function is converged to obtain the final pure phase hologram.
2. The method of claim 1, wherein the amplitude value of the light field of the original image is calculated by using the gray value of the original image, and the phase value of the light field is set to be constant at the same depth, so as to avoid the noise caused by the interference of adjacent pixels of the reconstructed image due to different phases.
3. The method of claim 1, wherein the initial phase-only hologram or bi-phase hologram is generated using random values as the initial phase-only hologram, and wherein the number of iterations is reduced using the bi-phase hologram.
4. Method according to claim 1, characterized in that for practical situations the spatial light modulator used for displaying the phase-only hologram will cause display errors if there are crosstalk effects, in the formula
Adding correction to crosstalk effect; the crosstalk effect is expressed as a point diffusion model, and the phase value of the phase-only hologram is rolled up into a point diffusion function in iteration; the formula for the calculation of the spectrum is rewritten as:
wherein
Representing a convolution, and a represents a point spread function of a pixel subject to crosstalk effects.
5. The method according to claim 4, characterized in that if the size of the point spread function is in the order of one pixel, a supersampling calculation of the phase-only hologram is required.
6. Method according to claim 4, characterized in that the location of the signal spectral region of the phase-only hologram is displaced in an iterative process not in the center of the spectrum but by a distance, so that off-axis holography is achieved.
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