Method and display system

Abstract

A method of displaying a computer-generated hologram (CGH) is disclosed. An image array for display as a CGH is initialised using an initialising complex array to give an initialised array, wherein th

Description

Technical Field The present invention relates to methods and apparatus for displaying computer generated holograms (CGH). Background In CGH methods, target image data is typically initialised by multiplication with a complex phase array, forming an array of complex data used to compute a hologram for display. Various different phase arrays have been proposed, including random phase and linear or “flat” phase. Random phase acts to spread information across the Fourier space, maximizing the area where the image can be viewed (the “eyebox”) and providing defocus which is similar to the real-world. This in turn triggers natural depth &focus cues. Random phase is also well suited to methods that quantise for display with phase-only modulation. However, the resulting uncorrelated phases in the image plane interfere when the hologram is illuminated by a suitably coherent source required for display, resulting in “speckle” noise when viewing the image. Solutions have been proposed to reduce the speckle introduced when using random phase. These tend to be computationally intensive, for example following the initialisation with random phase with a phase-retrieval algorithm, such as Gerchberg-Saxton, to approximate a phase profile which lessens the unwanted interference, or displaying multiple holograms time-sequentially with independent random phase profiles at a very high rate (which also requires computing multiple holograms and display hardware capable of very high frame rates). Alternatively, the image may be physically blurred in the optical path, which requires additional hardware and reduces resolution. Linear phase profiles have been proposed as an alternative to random phase when initialising image data for display as a CGH. Linear phase profiles comprise an array with values of e'q><xyL where ^(vy) = ax + by with a and b real constants. The argument of the linear phase profile gives an inclined plane of phase values which passes through the origin. Applying a linear phase profile induces a highly uneven amplitude distribution in Fourier space, the majority of the intensity is concentrated in one spot (referred to as a “spike” because it resembles a spike when the amplitude is plotted in the Fourier domain, having a concentrated spot of high intensity). Such displays can avoid phase-induced image speckle entirely but bring their own problems. The effective eyebox is very small because of the spike-like amplitude distribution in the pupil. Apparent defocus is also extremely limited, so a viewer receives fewer accommodation-driven depth cues when viewing the image. This reduces one of the advantages of CGH images: the ability for the eye to focus naturally at different depths. Moreover, the spike-like amplitude distribution is more sensitive to obstructions and irregularities at or near the pupil, such as eye floaters within a viewer’s eyeball and eyelashes. These disturb the converging rays associated with the spike and induce distracting visual artefacts. It would be useful to provide an improved method for CGH display. Summary Examples discussed herein propose a new technique for use when initialising images for CGH displays. Rather than initialising with a phase profile, this technique initialises images for CGH display with a complex array, which comprises a sum of weighted affine phase profiles. An affine phase profile is a linear phase profile with an additional constant offset in its argument, so that the argument may not pass through the origin, having the form where <p(x, y) = ax + by + c with a, b and c real constants. A weighted affine phase profile is an affine phase profile multiplied by a real valued weight. The values of the constants a and b determine the position of the spike in the Fourier domain. The constant c acts as a phase offset and, together with the real valued weight, two additional, independent degrees of freedom per spike in the Fourier domain are provided. This provides more control over the amplitude of the sum of the phase profiles so that amplitude variation may be reduced. By using a sum of weighted affine phase profiles, the distribution of the intensity in the Fourier domain is concentrated in multiple discrete spots, referred to as "spikes". This results in multiple high points of intensity reaching the viewer's pupil, improving defocus. Moreover, the affine nature of the phase profiles results in a structured amplitude variation pattern in the displayed CGH image which may not even be noticed by a viewer at all. If it is visible, the deterministic repeating nature of the amplitude variation enables it to be reduced with lower computational complexity compared to prior random phase and flat phase approaches. According to a first aspect of the present invention, there is provided a method of displaying a computer generated hologram, CGH. The method comprises: initialising an image array for display as a CGH using an initialising complex array to give an initialised array, wherein the initialising complex array comprises a sum of a plurality of weighted affine phase profiles, wherein each weighted affine phase profile comprises a real weight, a linear phase gradient and a phase offset, and corresponds to a spike in a Fourier domain; determining a modulation pattern, |H, for display by a spatial light modulator using the initialised array, wherein the Fourier Transform of H, F(H), comprises multiple spikes; and displaying the modulation pattern using a spatial light modulator and an optical system. By using a sum of weighted affine phase profiles for the initialisation and the Fourier Transform of H, F(H), comprising multiple spikes, speckle noise may be reduced or absent compared to random phase methods and a larger eye-box and improved defocus is achieved compared to linear phase methods. Some amplitude variation will be introduced, but this may be acceptable for the improved defocus. For example, the amplitude variation may have a regular repeating pattern, which may be more acceptable than speckle noise. Some of the weighted affine phase profiles may be weighted with a real weight of 1 and a phase offset of zero, and so effectively have no weight or phase offset applied. The image array may be, for example, an array of amplitude values, such as an array of amplitude values for display at a particular depth. The image array might be a layer of a layer-based hologram, or a vehicle heads-up display image for display at a single depth. The image array may also be an array of amplitude values with associated respective depths (such as a point cloud or similar). It will be appreciated that the modulation pattern may be determined in many ways from the initialised array, depending on the properties of the final hologram and the spatial light modulator (SLM) used to display the CGH. Examples include encoding schemes involving quantising and optionally spatial filtering in an optical path after the SLM. A non-exhaustive list includes a binary quantisation scheme, for instance when using a Digital Mirror Device (DMD), potentially in combination with a spatial filter as described in WO2023 / 002175; amplitude-only schemes such as described in WO2023 / 180693, and phase-only schemes, such as Double Phase Amplitude Control, which is discussed on page 4 of WO2023 / 180693. Independent of the particular encoding scheme used, there may be reduced speckle when compared with random phase initialisation and there may be improved defocus and / or expanded viewing area when compared with linear phase initialisation. In some examples, at least one of the plurality of weighted affine phase profiles has a non-zero phase offset. This may change the pattern of amplitude variation so that amplitude is less noticeable to a viewer. For example, the non-zero phase offset may be predetermined to reduce amplitude variation compared to the case where all of the plurality of weighted affine phase profiles have a zero phase offset. Alternatively, or additionally, at least two of the plurality of weighted affine phase profiles may have different real weights from each other. This may change the pattern of amplitude variation so that amplitude variation is less noticeable to a viewer. For example, the different real weights may be predetermined to reduce visible amplitude variation compared to the case where all of the plurality of weighted affine phase profiles have a same weight. By adjusting the phase offset and / or the real weights, the amplitude variation introduced by the initialising complex array in the displayed image may be reduced. The phase offset and / or real weights can be determined numerically, such as by adjusting and observing the overall impact on the amplitude pattern, by an optimisation algorithm, such as one looking to minimise the variance of the amplitude pattern , or in any other suitable way. A set of initialised arrays for display sequentially in time may be determined in some examples. Each initialised array may be determined using a respective initialising complex array, wherein the respective initialising complex arrays are predetermined such that a time-averaged visible amplitude variation is reduced compared to a visible amplitude variation when a single initialising complex array is used. This may allow averaging over time in the viewer’s eye to reduce the impact of amplitude variations. As the amplitude pattern is structured and deterministic, a set of initialising complex arrays can be determined that when displayed in sequence reduce perceptible amplitude variations. For example, amplitude peaks introduced by a subsequent initialising complex array may be generally aligned with an amplitude trough in a previous initialising complex array. The use of weighted affine phase profiles may be particularly advantageous when averaging over time, because two degrees of freedom may be exploited, namely the real weights and the phase offsets. These may (i) allow the position of amplitude peaks and troughs to be adjusted and / or (ii) reduce amplitude variation so that fewer time-sequential frames are required. When a set of initialised arrays is used, they may be formed from the same or different images for display; the amplitude variation introduced is independent of the image so the time-averaging works regardless of whether the image array is the same or different. For example, a same image array may be used for relatively slowly changing information (such as a vehicle heads up display) or relatively lower frame rate video or and different images may be used for relatively faster changing information and relatively higher frame rate video. Each respective initialising complex array may correspond to a same positioning of spikes in the Fourier domain, with respective real weights and / or phase offsets applied. By having the spike positions in the Fourier domain remain the same, it may be easier to determine the respective real weights and / or phase offsets (or equivalently, a complex weighting comprising a real weight and phase offset) so that the average amplitude variation induced by the set of initialising complex arrays is reduced. The respective initialising complex arrays may form a set of initialising complex arrays. The set of initialising complex arrays may comprise at least two subsets of initialising complex arrays, where each subset comprises initialising complex arrays whose corresponding pluralities of spikes in the Fourier domain share geometric similarity. “Similarity” is used in its mathematical sense. In this way, each subset shares a distinct geometry. At least one of the subsets may comprise at least two initialising complex arrays. By using at least two subsets, more time-averaged complex geometries can be implemented, while managing the complexity of the amplitude variation. The number of spikes in each subset may be less than the total number of spikes, and it is less complex to determine weightings and / or phase offsets the fewer spikes that the initialising complex array defines in the Fourier space. When subsets are used, the members of each subset may be displayed in a predetermined order such that timeaveraging acts to reduce visible amplitude variations. Any suitable subsets of initialising complex arrays can be used. Examples of geometric similarities in the Fourier domain include: a square with vertices in the Fourier Domain made up of four spikes and four rotations; a fixed set of spikes in the Fourier Domain and cycling through different phase offsets; and a regular pentagon with vertices in the Fourier domain made up of five spikes and where the radius of the pentagon varies between frames. When the numbers of initialising complex arrays in each subset are the same, a first initialising complex array from each subset may be used to determine a sequence of initialised arrays, followed by a second initialising complex array from each subset, and so on. For example, initialised arrays formed from the first complex initialising array in each subset may be displayed in sequence, initialised arrays formed from the second complex initialising array in each subset may then be displayed in sequence, and so on. The method may comprise determining a plurality of initialised arrays for display sequentially in time, each initialised array determined using a respective initialising complex array, each respective initialising complex array (i) corresponding to a respective region of a Fourier plane and (ii) corresponding to a plurality of higher intensity spots in the respective region of the Fourier plane; and wherein the displaying comprises spatially filtering in a Fourier plane corresponding to the different respective regions. The spikes in the Fourier domain result in the distribution of intensity in a Fourier plane being concentrated in multiple discrete spots of high intensity. These higher intensity spots generally correspond to spikes in the Fourier domain. Fourier planes may be created within the optical system. The initialised arrays may be formed from a same or different images. A “region” of the Fourier plane as referred to here is a space containing a plurality of higher intensity spots, such as a filtered area of the Fourier plane. Spatial filtering in a Fourier plane of an optical system is an effective way of reducing quantisation noise in some examples. By time multiplexing different regions of the Fourier plane in conjunction with the spatial filtering, a greater area of the Fourier plane can be used, increasing a visible eyebox. For example, the various regions may be arranged as described in WO 2023 / 002175 incorporated herein by reference for all purposes. The spatial filtering may be by shuttering different physical regions of the Fourier plane, or by using multiple illumination sources so that a same spatial filter is located at a different position on a Fourier plane depending on which of the multiple illumination sources is used. The optical system may be configured such that a viewer’s pupil receives a plurality of spikes in the Fourier domain when viewing the image. When a viewer’s pupil receives a plurality of spikes, perceived defocus may be improved. A viewer’s pupil can receive a plurality of spikes by positioning the spikes so that a viewer’s pupil receives more than one spike simultaneously, such as by using an optical system that forms a Fourier plane at or close to a viewer’s pupil and the spikes being positioned within the area of the viewer’s pupil. The displaying may comprise spatially filtering the output of the spatial light modulator using a spatial filter positioned in a Fourier plane of the optical system, wherein the spatial filter defines at least one simply connected region containing at least two higher intensity spots in the Fourier plane. “Simply connected” is used in its mathematical sense. Simply connected regions containing at least two higher intensity spots may improve defocus within a region that can then be made large, thereby achieving greater resolution in the image domain. The plurality of weighted affine phase profiles may correspond to a plurality of spikes which are distributed substantially uniformly in the Fourier Domain. This may make it easier to determine any weightings and / or phase offsets to reduce visible amplitude variation. This may also result in a more uniform intensity distribution at the viewer’s pupil, resulting in more uniform defocus. A substantially uniform distribution means that the spikes are spread over the Fourier domain and not concentrated in any particular place. Some examples may adopt a regular spacing of the spikes in the Fourier Domain, but an irregular spacing can also give substantially uniform distribution. The initialising complex array may comprise at least four weighted affine phase profiles, these then correspond to at least four spikes in the Fourier domain. In other examples, the number of weighted affine phase profiles (and hence the number of spikes in the Fourier domain) may be at least five, at least six, at least seven, at least eight or at least nine. Increasing the number of spikes increases the defocus benefits and may be particularly advantageous when combined with a substantially uniform distribution of the spikes. In general, an initialising complex array will apply to a single colour or wavelength for display. Different colours (wavelengths) for display may have different numbers of weighted affine phase profiles making up the initialising complex array. The determining a modulation pattern may comprise multiplying the initialised array by at least one non-affme phase profile. The non-affine phase profile can be, for example, a depth transformation to set the initialised data to a perceived depth, a correction for display optics, such as aberrations introduced by the optical system, and so on. The image array and the initialising complex array may have the same dimensions and the initialising may then comprise a pointwise multiplication of the image array by the initialising complex array. When the image array and the phase array have the same dimensions, they can be applied efficiently by pointwise multiplication. In some examples, if the image array is smaller or larger than the initialising complex array, techniques such as interpolation on the image array may be used to match their dimensions before the pointwise multiplication. The method may be applied to a CGH display system comprising: an at least partially coherent illumination source; a spatial light modulator arranged to be illuminated by the at least partially coherent illumination source; an optical system for relaying an image formed by the spatial light modulator to a viewer; and a processor configured to control the spatial light modulator to display a computer generated hologram according to the method discussed above, with or without optional features also described. The CGH display system may form part of a head-mounted display or a heads-up display in a vehicle. Further features and advantages of the invention will become apparent from the following description of preferred embodiments of the invention, given by way of example only, which is made with reference to the accompanying drawings. Brief Description of the Drawings Figure 1 shows a diagrammatic representation of an example CGH display system; Figure 1A shows a diagrammatic representation of the spatial filter of Figure 1 positioned substantially in a Fourier plane and the resulting spike locations in the filtered Fourier domain; Figure 2 a diagrammatic representation of spike locations in a Fourier domain for use with the CGH display system of Figure 1; Figure 3 shows a diagrammatic representation of an example amplitude pattern introduced by an initialising complex array corresponding to Figure 2; Figure 4 shows a diagrammatic representation of sine waves and the magnitude of their sum corresponding to the amplitude pattern of Figure 3; Figure 5 shows a diagrammatic representation of phase offset sine waves and the magnitude of their sum; Figure 6 shows a diagrammatic representation of an example plurality of phase offsets corresponding to Figure 5 applied to the spike locations of Figure 2; Figure 7 shows a diagrammatic representation of an example amplitude pattern introduced by an initialising complex array corresponding to Figure 6; Figure 8 shows an overview of a method according to the present disclosure; Figure 9 is a diagrammatic representation of a set of pluralities of phase offsets required for producing amplitude patterns which sum to a low variation; Figure 10 shows an overview of a method of applying the set of pluralities of phase offsets of Figure 9 onto to sequential images for display; Figure 11 is a diagrammatic representation of two alternating sets of pluralities of phase offsets, using the set of Figure 9 and an additional set; and Figure 12 is a diagrammatic representation of multiple sets of pluralities of phase offsets applied to different regions of a Fourier plane of a display system. Detailed Description Underlying Theory The underlying theory will first be described followed by its application to a monochrome flat image, followed by its use in example CGH display systems. Flat images can be used in holography on their own (for example in a heads-up display) or as component parts of a multi-layer image in layer-based holography. Likewise monochrome images may be used on their own or as part of colour displays by display of red, green and blue images simultaneously or sequentially in time. As used herein, “monochrome” includes binary images and those including gradations, such as different amplitude levels, of a single-colour image. A monochrome flat image may be described by the discrete function A(x,y\ where each element of the array A encodes the amplitude at a particular point of the image. A complex wavefront for this image is first initialised by computing an initialising complex array, resulting from adding independent weighted affine phase profiles as defined in Equation 1 below: y} = --- V rk exp (2%i(pfc;r 4- qky + ty}), (pk, qk) € R2, >0 .1 h == — >wk exp (27ri(p£.x + ' wk “ h 155v (Equation 1) followed by pointwise multiplication - i.e. for every point in the image: 5[A](.r,? / ) = A(x,y)g(x,y) (Equation2) Here (Pk, Qk) are frequency-space pairs, with the standard Fourier coordinate convention (i.e. modulo 1, with (0, 0) at the top-left, horizontal component increases to the right, second component increases downwards). Each image plane product with a linear gradient corresponds to a spatial shift by (pk, qk) in the Fourier domain. Because the Fourier transform is linear, the Fourier transform of S[A] will look like a weighted sum of Fourier transforms for A. The complex weights (wk) in Equation 1 comprise a "real weight" n and a phase offset tk', multiplying a linear phase gradient by such a complex weight induces a phase offset in the gradient, resulting in an affine phase gradient, and an amplitude change, together resulting in a weighted affine phase gradient. As will be explained in more detailed below, at least one of rk and tk may be optional, and they allow for control over the amplitude distribution of the initialising complex arrays (and hence that of the complex image S[A]) whilst preserving the spikes they result in. The weights rk, and phase offsets tk, may be chosen with information of later processing of the data, such as quantising for display. When data is quantised, information is lost. Therefore quantisation may affect the plurality of spikes. Quantisation may in some cases remove the spikes from the Fourier plane introduced by the affine phase gradients. For example: the spike .n positioned at (1 / 2 1 / 2) in the Fourier domain, if complex weighted by e'z., corresponds to a purely imaginary initialising complex array. In this case, if the initialised array is quantised to its real part, that spike would be removed. Alternatively, with the spike positioned at (1 / 4, 1 / 2) (weight of 1), if the initialised image S[A] is directly quantised to absolute value, the Fourier domain will have a spike at (1 / 2, 0) instead. Alternatively, with two spikes at (1 / 4, 1 / 2) and (1 / 2, 1 / 4) (weights of 1), if the initialised image is quantised to absolute value, the Fourier domain will have spikes at (p / 4, q / 4) for every integer p, q. The skilled person will be aware of how to consider these quantisation constraints as part of their choice of weighted affine gradients and / or selection of the filtered region in the Fourier domain. Equation 1 also includes an approximate ‘normalising’ factor $7 , dependent on the number of spikes, so that the intensity of the resulting image remains roughly consistent across different spike arrangements. For example, the normalising factor may approximately preserve the same average intensity level across your different spikes. Some example may not require this normalising factor. In other examples, W = V A may be used, assuming the moduli of the complex weights do not themselves differ wildly from 1). In other examples where a subset of M spikes 5 reaches the viewer at any given time (such as when a viewer’s pupil receives M of the N spikes), a normalising factor of i]N = y[M may be used. In addition, or alternatively, the skilled person may also consider a non-uniform normalisation across the spikes (equivalently, a restrictive distribution of the real weights n) in order to balance the position-dependent power contribution of a physical source for each spike. For 10 example, if the power of a physical source in a Fourier plane is modelled as a Gaussian centred around this source, the weighting n may be chosen as the inverse of this Gaussian multiplied by a target constant. The function G, used as an initialising complex array to initialise a monochrome flat image for display as a CGH, does not in general have uniform amplitude, so this 15 operation affects the pixel values directly, and may be visible on the final hologram. This visible impact will be discussed in more detail below, but for an image of sufficiently high resolution, it may not be noticeable or may be more acceptable to a viewer than the speckle associated with the use of random phase. Having initialised the image, with the initialising complex array, the resulting 20 initialised array comprising complex values is propagated to the desired depth from the viewer. Depth propagation is applied in Fourier space &is non-affine, so it affects the plurality of spikes in Fourier space of the initialising complex array differently depending on position. The resulting hologram (or ‘sub-holograms’ for layer-based holography) from each of the plurality of the spikes are distinct and perspective-correct: (Equation 3) wherep is device pixel-pitch, d is target depth (in dioptres), X is wavelength, and (xo, yo) is the focus centre. This modified complex image can then be passed onto a hologram-generation algorithm producing an image-plane hologram H, the details of may be factored into the search for suitable initialising complex arrays. 014 1 algorithm O [Aj ~ >rt (Equation 4) It will be understood that a wide variety of algorithms to produce H can be applied. This may also consider the operation of the display system which will display the CGH to provide an overall encoding scheme. Example encoding schemes involving quantising and optionally spatial filtering in an optical path after the SLM. A non-exhaustive list includes a binary quantisation scheme, for example when using a Digital Mirror Device (DMD), potentially in combination with a spatial filter as described in WO2023 / 002175; amplitude-only schemes such as described in WO2023 / 180693, and phase-only schemes, such as Double Phase Amplitude Control, which is discussed on page 4 of WO2023 / 180693.. Note that the real part of S[A] encodes both phase and amplitude information, and hence so will H The multiplication by G in Equation 2 may have some impact on the final perceived image. For example, a pixel pattern may be perceived as overlaid on top of the image. The replay of the H may be denoted by R, where R is the image (i.e. amplitude) perceived by the viewer: (wtina) (Equations) By abuse of notation, the perceptual pixel pattern induced by G and the rest of the display method / algorithm (e.g. quantisation) may be denoted by R / A (“R modulo A”). The perceptual pixel pattern is reduced (i.e. is less perceptible to the viewer) if R / A is approximately uniform. Example phase distribution and weighting In this example, with reference to Figure 1, an initialising complex array comprising a sum of a plurality of weighted affine phase profiles is used in a CGH display system comprising an amplitude modulating SLM 100. The SLM 100 is illuminated by an at least partially coherent illumination source 102, such as a laser or suitably coherent LED. Light from the illumination source 102 passes through a condenser lens 104 and on to a reflective beam splitter 106 where it is reflected towards the SLM 100. The SLM 100 is controlled to form a quantised version of the hologram for display (in this quantised to amplitude levels which can be formed by the SLM). Modulated light is then reflected back from the SLM, through the beam splitter 106 and onto a Fourier lens 108. The Fourier lens forms a Fourier plane 110 at its focal distance. A spatial filter 112 is positioned in or near the Fourier plane (shown in section view in Figure 1) and spatially filters the Fourier plane so that only the targeted region is allowed to pass. A further Fourier lens 114 is positioned at a focal distance from the spatial filter 112 along the optical axis and forms an image of SLM 100, possibly via optional relay optics (not shown), onto a viewer’s pupil 116. A diagrammatic representation of the spatial filter is shown in Figure 1 A. The spatial filter comprises a square diamond shape positioned with the centre of its upper left side at the zero order 118 of the Fourier plane. As is well known, the Fourier plane comprises multiple unit squares extending from the zero order. The spatial filter shown in Figure 1A extends one unit 120 from the zero order in all directions. A simply connected region 122 allows light to pass, as shown this region 122 includes multiple amplitude spikes 124 positioned in the Fourier plane. The remainder 126 of the spatial filter 112 is configured to block the passage of light. In this example, the positioning of the region 122 that allows light to pass may mean that conjugate and higher order terms of that region are blocked in the Fourier plane, reducing quantisation noise (more information can be found in WO 2023 / 002175, incorporated herein by reference for all purposes). As can be seen in Figure 2, in this example the filtered Fourier region takes a diamond shape comprising a five-spike distribution, with one spike 202 at the centre of the re-imaged Fourier plane and four spikes 204 in a diamond shape around it. The zero order from Figure 1A is shown at 206. The position of the five spikes, 202,204 is given by: ffi 1\ (1 1) fl 1) fl 1) fl 3n 1'2 ' 2 ' ’ 4 ' 2 ' ■ 2 ’ 4 ' 4 ’ 2 ' ’ ' 2 ’ 4 / (Equation 6) The particular values to give this distribution of spikes in this example are: k Pk qk 1 V2 ' / 2 2 % y2 3 ’ / 2 % 4 y2 ¾ 5 ¾ y2 In a particular example (with quantisation for display on an amplitude SLM and where W v ^'), the perceptual result R / A can be modelled by taking the absolute value of the real part of the array: (Equation 7) Without weighting or offsets in the individual affine phase gradients (i.e. n = 1, tk = 0, so that Wk = 1) - or equivalently, using a sum of linear phase gradients, Figure 3 shows the amplitude pattern which is observable on the replayed hologram with lighter areas indicating a higher amplitude(forming an overlay on the amplitude desired for the image to be displayed). The pattern repeats at the pixel level, an individual pixel dimension is indicated at 302 in Figure 3. Thus, depending on the pixel size in the replay field, the pattern may hardly be noticeable and be acceptable in some applications. However, the phase offsets tk can be chosen to reduce the amplitude variation in R / A by considering how the individual cosine waves interact and sum to create the amplitude variation. Figure 4 depicts a portion along a row in Figure 3, showing the individual sine waves 402, 404, 406 associated with three horizontal spike positions in the Fourier Domain. The solid line 408 shows the continuous resulting amplitude (which cannot be negative) and the dots 410 represent the pixel values (i.e. sampled at integer values). Three cosine waves are shown because the cosine waves induced by the middle column of spikes are identical along this row. Hence to compute 408 and 410, this cosine wave (404) is accounted for three times. It can be seen how the cosine waves sum to give the resulting amplitude after scaling. It is then possible to see how the phase offsets tk change the interference pattern and reduce the differences between the peaks and troughs of the sampled pixel values. Suitable phase offsets can be determined by inspection, by using a search algorithm or using machine learning. Figure 5 shows the effect of applying pure phase offsets to the cosine waves in Figure 4. The phase-offset cosine waves 502, 504, 506, 508, sum to amplitude indicated by line 510 after scaling, with the pixel sampling positions denoted by dots 512. Figure 6 then shows the complex weight (in this case, a pure phase offset) applied to each spike in the Fourier plane and Figure 7 shows the resulting amplitude variation in R / A across the final image, with a pixel dimension of 702. As with Figure 3, a lighter region in Figure 7 denotes a higher amplitude, the pattern is now simpler and has a smaller difference between peaks and troughs of amplitude. The amplitude variation of Figure 7 is less noticeable than that of Figure 3 on the final hologram. The complex weights (real weights and / or phase offsets) can be predetermined so do not add any significant computational complexity to the method. There are four cosine waves in Figure 5 because the phase offsets applied to the central column of spikes are not identical. The top spike has a different phase offset from the central and bottom spikes. The central and bottom spikes have the same phase offset and so appear as an identical cosine wave, which is accounted for twice in the sum. Thus, in this example, determining an initialising complex array by the sum of a plurality of affine plane phase distributions (five in this case), allows a viewer to receive more than one spike at once in a plane at or near the pupil, such that more than one spike is “passing through” the viewer’s pupil. This gives the viewer improved defocus performance and / or a larger viewing eyebox compared to a prior art “flat phase” approach. While the multiple spikes do create an amplitude variation pattern in R / A, this may not be noticeable in the final image. If it is noticeable, the amplitude variation pattern can be reduced by weighting and / or phase-offsetting the spikes appropriately. Although the examples so far have discussed pure phase-offsetting, it will be appreciated that other examples may be used which combine real weights and phase offsetting, or which use pure real weights without phase offsets. Some of these will be described in more detail below. Figure 8 shows an overview of the entire method this example, from determining the initialising complex array used to initialise a source image through to display. In practice, a single device may not implement the whole of this method. For example, the initialising complex array to initialise the source image is likely to be predetermined and a display device will not determine the values each time an image is displayed. A source image 802 is received for display as a hologram. In this example the method is explained using a monochrome, flat image for display at a particular depth. The source image 802 therefore comprises an array of amplitude values for display. The skilled person is aware that the method may be expanded to non-flat holograms by, for example, using layer-based techniques. Likewise, the method can be expanded to colour images by simultaneously or sequentially displaying images corresponding to different image fields. The source image 802 is then pointwise multiplied at 804 by an initialising complex array 806 having values determined by a weighted sum of affine phase gradients. Each affine phase gradient is associated with a spike in the Fourier space 808 when a Fourier transform is taken of the initialising complex array. The initialising complex array 806 is predetermined based on the desired positions of the spikes. The process by which the initialising complex array is predetermined is shown generally at block 810. First, at 812a, 812b, ... 812n each of the n desired spike positions in the Fourier space is determined and mapped to coordinates in Fourier space. While more spikes will improve defocus effects and potentially the eyebox size, it can be useful to work with a relatively low number of spikes so that the interference effects between the phase profile associated with each spike are less complex. For example, there may be 10 or fewer spikes, or 5 or fewer spikes. There may also be at least 4 spikes to provide improved defocus. Taking the inverse Fourier transform from the spike locations gives a series of phase gradients 814a, 814b, ... 814n in an image space. These are then weighted by a real weight and / or phase-offset to reduce amplitude variation in R / A, i.e. in the image domain at 816 giving a plurality of weighted affine phase gradients 818a, 818b, ... 818n. The weighted affine phase gradients are then summed at 820 to give the initialising complex array 806. Returning to the main process, the pointwise multiplication at 804 results in an initialised image 820 from which a modulation pattern 822 is generated for display. Any suitable method may be used to generate the CGH from the initialised image. In the example depicted overall at 824, the initialised image is Fourier transformed and multiplied by a spherical phase array, such as one represented by a Zernicke polynomial, to apply a desired depth from the viewer. An inverse Fourier transform is then applied to give modulation pattern 822. Next, the modulation pattern is quantised according to a scheme appropriate for the display system. It will be appreciated that other methods of generating the modulation pattern can be used. With the quantised modulation pattern 822 displayed on the SLM, the optical system will then generate a Fourier plane 824 and a corresponding real image 826 at various points in the display system (including on the viewer’s retina where a viewer is involved) as discussed with reference to Figure 1 above. Time multiplexing to reduce amplitude variations in the displayed hologram Should it be desired to further reduce the impact of amplitude variation, the pattern of Figure 7 is well-suited for using time-multiplexing techniques. Timemultiplexing techniques are known for reduction of speckle noise from random phase patterns. Multiple holograms with different random phase patterns at the initialisation stage are determined and displayed in rapid succession such that the viewer’s eye averages over several displayed holograms to reduce speckle noise. However, timemultiplexing is more effective using the methods of this disclosure because the amplitude modulation pattern is deterministic. For example, simply by adjusting the complex weights (real weight and / or phase offset) of the spikes, different amplitude variation patterns can be produced. Taking the example of the five-spike distribution of Figure 2, four time-multiplexed holograms are sufficient to reduce amplitude variation to substantially zero. In other examples, a five-spike distribution may only require two time-multiplexed holograms. This is depicted in Figure 9, which shows the required complex weightings 902a, 902b, 902c, 902d, (pure phase offsets in this case) of each spike, the resulting amplitude variation patterns 904a, 904b, 904c, 904d and the sum 906 of all four amplitude variation patterns. Far fewer successive frames are required than when using time-multiplexing to reduce speckle noise from random phase. With this example, a colour-sequential hologram can be displayed at 30 frames per second with an SLM capable of 360 Hz operation (30 frames / second x 3 colours x 4 frames per colour). However, as the amplitude variation pattern introduced by the initialising complex array is independent of the image to be displayed, the sequence of initialising complex arrays can be applied to image fields for display in sequence, without requiring the same image field to be used for each of the display patterns. In that case, the method can be applied without requiring any faster SLM operation. Put another way, a colour-sequential hologram could be displayed at 90 Hz (30 frames / second x 3 colours), and the sequence of phase patterns applied to the image fields in the order that they are displayed. Thus, in this example, amplitude variation introduced by the initialising complex array may be reduced by time-multiplexing much more effectively than for random phase, potentially removing perceptible amplitude variation completely with relatively few frames. The method of this example is depicted schematically in Figure 10. The method is an extension of Figure 8, applied to sequential image sources for display, as such Figure 10 only depicts certain elements of Figure 8 to aid clarity. A first source image 1002 is processed according to the method of Figure 8 to give a modulation pattern 1022 which is displayed on the SLM at 1028. The display of this image will have amplitude variation, such as that depicted in Figure 7. The next source image 1002’ for display (which may the same as the first source image 1002 or different) is processed similarly but the complex weights used to determine the initialising complex array are different, giving in a different amplitude variation pattern in the resulting modulation pattern 1022’. This is then displayed on the SLM at 1028’. Taking the sequence of Figure 9, a further two modulation patterns are determined and displayed until the initialising complex array returns to the one used for the first modulation pattern (not shown). The amplitude variation is then averaged by the viewer’s eye so that less amplitude variation is perceived by the viewer. Composite spike patterns with sequential display The speed at which a sequence of images (fields or frames) can be displayed on the SLM can present a limitation on expanding the methods to include more spikes in Fourier space when it is desired to use sequential display to reduce perceived amplitude variation. This is because the full sequence should be delivered over a relatively short period to allow the eye to average out the amplitude variations between fields, such as displaying the full sequence in less than 1 / 10th of a second and ideally over a shorter period. In this example, the same physical display system arrangement is used as described above for Figure 1 with an SLM capable of a rapid refresh rate. The SLM is used to populate the Fourier space with nine spikes using time multiplexing by displaying a five-spike and a four-spike pattern sequentially. A suitable commercially available SLM is Sony’s SXRD-241 which runs natively at up to 600 Hz. The complexity of amplitude variation in R / A generally increases with the number of spikes in the Fourier domain, and it becomes harder to determine complex weights (weights and / or phase offsets) that generate a short cycle to reduce amplitude variations, such as the sequence of four in Figure 9. The complexity, and thus the number of successive initialising complex arrays required, can be reduced by interlacing or alternating sequential cycles of at least two subsets of initialising complex arrays. In this example, the set of four five-spike initialising complex arrays from Figure 9 above is used as one of these subsets. That subset is complemented by a further subset consisting of four four-spike initialising complex arrays with spikes at the following positions in Fourier space. r / 1 1) / 3 1X / 1 3\ / 3 1A 4 ’ 4 M 4 7 4 A \ 4 t 4 A V 4 M H (Equation 8) To achieve a secondary cycle of on this subset, and one which leads to patterns in keeping with the cycle of Figure 9, the amplitude of the complex weights also needs to differ between spikes. In this example, two spikes are initialised with a 42-increased magnitude (in the Fourier transform space). This bias is also time-multiplexed, so that across multiple frames the magnitudes given to the spikes is uniform. The ordering of the cycles may be chosen so that the brighter patches are ‘conjugated’ between even and odd frames and are never in the same place between adjacent frames. Figure 11 depicts a schematic diagram of this approach, adding an additional subset of four initialising complex arrays 1002a, 1002b, 1002c and 1002d each consisting of four spikes in the Fourier transform space with different real weighting and phase offsets applied. These have corresponding amplitude variation patterns 1004a, 1004b, 1004c and 1004d. The full sequence of eight initialising complex arrays consists of two interlaced / alternating subsets of initialising complex arrays. Across all eight initialising complex arrays in the set, and across each subset, the amplitude variation sums to a substantially constant amplitude 1006. In the Fourier space, the four-spike and five-spike pattern sum to a nine-spike pattern 1008. It will be appreciated that any other suitable pattern of spikes can be used, with different numbers of spikes and more subsets of spikes than two, such as three, four or more sets. Expanding a visible eyebox with spatial multiplexing All the examples discussed so far have used a same single region of the Fourier plane for all the applied initialising complex arrays. In this example, different arrays target different regions of the Fourier plane. This allows the hologram to be viewed over a larger area, for example by using the principles discussed in WO 2023 / 002175. The same sequence of spike positions are multiplexed across 4 regions or pupils arranged in a diamond shape (‘pupil’ here means a single filtered Fourier plane - its spatial dimensions when re-imaged at the user’s pupil is allowed to be smaller than the user’s physical pupil size). Regions 1202 are used in order to achieve filtering similar to that in previous examples. When the spatial filtering comprises shuttering different physical regions of the Fourier plane, the system operates such that exactly one shutter is open at any given time, and is synchronised with the SLM so that a single hologram is shown within a shutter opening. An area 1204 is also always filtered out. Within a given region, multiplexing may occur across multiple frames. Alternatively (or additionally) a larger area of the Fourier plane can be covered (i.e. the eyebox can be expanded) by using multiple light sources, each illuminating the SLM from different angles. The change in angle means that the Fourier plane of each light source is located at a different position (See for example Figure 8 of WO 2023 / 002175). In some examples, a different initialising complex array is used for each region 1202. For example, when the regions 1202 cover a relatively small area, so that a viewer’s pupil can receive images from all regions, the four different initialising complex arrays from Figure 9 may be applied to each region in turn. In other words, a particular initialising complex array is applied to a one region and the full set of four is received by the viewer receiving light from all four regions. In another example, the same initialising complex array is used for all the regions 1202. This may be appropriate when a viewer’s pupil only receives light from a single aperture location, so the same initialising complex array may be used for all four apertures, before moving to the next initialising complex array and displaying that for all four apertures, and so on. Thus, this example could include one source and multiple shutters; multiple sources and multiple shutters; multiple sources and one shutter i.e. a fixed aperture. The eyebox within which the CGH can be viewed is expanded whilst preserving the benefits of multi-spike within any given pupil. The above embodiments are to be understood as illustrative examples of the invention. Further embodiments of the invention are envisaged. For example, many other spike patterns in the Fourier space can be used, the disclosure is not limited to the five-spike, four-spike and nine-spike patterns of this example. Furthermore, although the example display system used an amplitude-modulating, reflective SLM, the principles described herein apply equally to other SLMs, including transmissive SLMs. It is to be understood that any feature described in relation to any one embodiment may be used alone, or in combination with other features described, and may also be used in combination with one or more features of any other of the embodiments, or any combination of any other of the embodiments. Furthermore, equivalents and modifications not described above may also be employed without departing from the scope of the invention, which is defined in the accompanying claims.

Claims

1. A method of displaying a computer generated hologram, CGH, the method comprising:initialising an image array for display as a CGH using an initialising complex array to give an initialised array, wherein the initialising complex array comprises a sum of a plurality of weighted affine phase profiles, wherein each weighted affine phase profile comprises a real weight, a linear phase gradient and a phase offset, and corresponds to a spike in a Fourier domain;determining a modulation pattern, H, for display by a spatial light modulator using the initialised array, wherein the Fourier Transform of H, F(H), comprises multiple spikes; anddisplaying the modulation pattern using a spatial light modulator and an optical system.

2. The method of claim 1, wherein at least one of the plurality of weighted affine phase profiles has a non-zero phase offset.

3. The method of claim 2, wherein the non-zero phase offset is predetermined to reduce visible amplitude variation compared to the case where all of the plurality of weighted affine phase profiles have a zero phase offset.

4. The method of any preceding claim, wherein at least two of the plurality of weighted affine phase profiles have different real weights from each other.

5. The method of claim 4, wherein the different real weights are predetermined to reduce visible amplitude variation compared to the case where all of the plurality of weighted affine phase profiles have a same weight.

6. The method of any preceding claim, comprising determining a set of initialised arrays for display sequentially in time, each initialised array determined using a respective initialising complex array, wherein the respective initialising complex arraysare predetermined such that a time-averaged visible amplitude variation is reduced compared to a visible amplitude variation when a single initialising complex array is used.

7. The method of claim 6, wherein each respective initialising complex array corresponds to a same positioning of spikes in the Fourier domain, with respective real weights and / or phase offsets applied.

8. The method of claim 6, wherein the respective initialising complex arrays form a set of initialising complex arrays, and the set of initialising complex arrays comprises at least two subsets, wherein each subset comprises initialising complex arrays whose corresponding pluralities of spikes in the Fourier domain share geometric similarity.

9. The method of any preceding claim, comprising determining a set of initialised arrays for display sequentially in time, each initialised array determined using a respective initialising complex array, each respective initialising complex array (i) corresponding to a respective region of a Fourier plane and (ii) corresponding to a plurality of higher intensity spots in the respective region of the Fourier plane; andwherein the displaying comprises spatially filtering in the Fourier plane corresponding to the different respective regions.

10. The method of any preceding claim, wherein the optical system is configured such that a viewer’s pupil receives a plurality of spikes in the Fourier domain when viewing the image.

11. The method of claim 10, wherein the displaying comprises spatially filtering the output of the spatial light modulator using a spatial filter positioned in a Fourier plane of the optical system, wherein the spatial filter defines at least one simply connected region containing at least two higher intensity spots in the Fourier plane.

12. The method of any preceding claim, wherein the plurality of weighted affine phase profiles correspond to a plurality of spikes which are distributed substantially uniformly in the Fourier domain.

13. The method of any preceding claim, wherein the initialising complex array comprises at least four weighted affine phase profiles.

14. The method of any preceding claim, wherein the determining a modulation pattern comprises multiplying the initialised array by at least one non-affine phase profile.

15. The method of any preceding claim, wherein the image array and the initialising complex array have the same dimensions and the initialising comprises a pointwise multiplication of the image array by the complex array.

16. A computer generated hologram display system comprising:an at least partially coherent illumination source;a spatial light modulator arranged to be illuminated by the at least partially coherent illumination source;an optical system for relaying an image formed by the spatial light modulator to a viewer; anda processor configured to control the spatial light modulator to display a computer generated hologram according to the method of any preceding claim.

17. A head-mounted display comprising the computer generated hologram display of claim 16.

18. A vehicle comprising a heads-up display comprising the computer generated hologram display of claim 16.

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