AUTOSTEREOSCOPIC DISPLAY DEVICE

Abstract

An autostereoscopic display comprises a pixelated display panel comprising an array of single color pixels or an array of sub-pixels of different colors and a view forming arrangement comprising an array of lens elements. The pixels form a hexagonal grid, and the lenses also repeat in a hexagonal grid. A vector p is defined which relates to a mapping between the pixel grid and the lens grid. Regions in the two dimensional space for this vector p are identified which give good or poor banding performance, and the better banding performance regions are selected.

Claims

1. An autostereoscopic display, comprising: a pixelated display panel, pixelated display panel comprising an array of single color pixels or an array of sub-pixels of different colors, wherein each group sub-pixels define full color pixels; and a view forming arrangement, wherein the view forming arrangement is positioned over the display panel, wherein the view forming arrangement is arranged to direct light the light from different pixels or sub-pixels to different spatial locations, thereby enabling different views of a three dimensional scene to be displayed in different spatial locations, wherein the pixels of the display panel form a hexagonal grid, with a maximum internal angle deviation from 120 degrees of 20 degrees or less, wherein the hexagonal grid repeats with translation vectors a and b, and the lengths of the translation vectors a and b have an aspect ratio of the shorter to the longer between 0.66 and 1; wherein the view forming arrangement comprises a two dimensional array of lenses which repeat in a hexagonal grid with translation vectors p′ and q′; wherein defining a dimensionless vector p as (p.sub.a,p.sub.b), which satisfies:
p′=p.sub.aa+p.sub.bb, and defining circular regions in the space of components p.sub.b and p.sub.a for integer n as: $\begin{array}{cc}{E}_n=\left\{pN\left(p-v\right)<r_n^2.Math.\forall v\in {\Gamma }_n\right\}& \\ \mathrm{where}& \\ {\Gamma }_n=\left\{i+\frac{j}{n}i,j\in 2^{}N.Math.\left(j\right)=n\right\}& \end{array}$ where n is an integer number, with r.sub.n=r.sub.0n.sup.−γ defining the radius of each circle, Γ.sub.n defining the circle centers, and with N comprising a vector function for two coordinate vectors defined as: $N\left(\left[\begin{array}{c}a\\ b\end{array}\right]\right)=a^2-\mathrm{ab}+b^2,$ the translation vectors a, b, p′ and q′ are selected with values such that p falls in the vector space which excludes the sets E.sub.1, E.sub.3 or E.sub.4 with r.sub.0−0.1 and γ−0.75.

2. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p falls in the vector space which excludes the set E.sub.1 with r.sub.0=0.25 and γ=0.75.

3. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p falls in the vector space which excludes the set E.sub.3 with r.sub.0=0.25 and γ=0.75.

4. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p falls in the vector space which excludes the set E.sub.4 with r.sub.0=0.25 and γ=0.75.

5. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p is not in the defined set or sets with r.sub.0=0.35.

6. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p is in the set E.sub.7 with r.sub.0=0.35 and γ=0.75.

7. The display as claimed in claim 1, wherein the translation vectors a, b, p′ and q′ have values such that p is in the set E.sub.9 with r.sub.0=0.35 and γ=0.75.

8. The display as claimed claim 1, wherein the pixel hexagonal grid translation vectors a and b have an aspect ratio of the lengths of the shorter to the longer vector between 0.83 and 1.

9. The display as claimed in claim 1, wherein the pixel hexagonal grid has a maximum internal angle deviation from 120 degrees of 5 degrees or less.

10. The portable device comprising a display as claimed in claim 1, wherein the portable device is configurable to operate in a portrait display mode and a landscape display mode.

11. A portable device as claimed in claim 10, comprising a mobile telephone.

12. A portable device as claimed in claim 10, comprising a tablet.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0049] Embodiments of the invention will now be described, purely by way of example, with reference to the accompanying drawings, in which:

[0050] FIG. 1 is a schematic perspective view of a known autostereoscopic display device;

[0051] FIG. 2 is a schematic cross sectional view of the display device shown in FIG. 1;

[0052] FIGS. 3a-e shows various possible pixel grids based on square or near square pixel and lens grids;

[0053] FIG. 4 shows a lens grid overlaid over a square pixel array, with a pitch vector p defining the relationship between them, for the purposes of explaining the analysis used;

[0054] FIG. 5 is a graphical explanation for parameters used to characterize the pixel array and lens grid;

[0055] FIG. 6 shows a plot using moire equations and a visibility function to estimate the amount of visible banding for a given pitch vector p;

[0056] FIG. 7 shows a first possible characterization of regions from the plot of FIG. 6;

[0057] FIG. 8 shows a second possible characterization of regions from the plot of FIG. 6;

[0058] FIGS. 9a-d shows ray trace rendering simulations of the 3D pixel structure for the 2D pixel layout of FIG. 3(c) for different lens designs;

[0059] FIG. 10a-d is a plot of the lightness (L*) as a function of the lens phases in two dimensions, for the same examples as in FIGS. 9a-d;

[0060] FIGS. 11a-d shows a plot of the color deviation for the same examples as in FIG. 9a-d;

[0061] FIGS. 12a-d shows various possible pixel grids based on hexagonal pixel and lens grids;

[0062] FIG. 13 shows a pixel grid based on hexagonal sub-pixels but which in fact form a rectangular grid;

[0063] FIG. 14 shows a hexagonal lens grid overlaid over a hexagonal pixel array, with a pitch vector p defining the relationship between them;

[0064] FIG. 15 is a first graphical explanation for parameters used to characterize the pixel array and lens grid;

[0065] FIG. 16 is a second graphical explanation for parameters used to characterize the pixel array and lens grid which corresponds to the representation in FIG. 5;

[0066] FIG. 17 shows a plot using moire equations and a visibility function to estimate the amount of visible banding for a given pitch vector p.

[0067] FIG. 18 shows a first possible characterization of regions from the plot of FIG. 17; and

[0068] FIG. 19 shows a second possible characterization of regions from the plot of FIG. 17.

[0069] Note that FIGS. 3a-e and 4 are intended to show square pixel and lens grids, FIGS. 12a-d to 14 are intended to show regular hexagonal pixel and lens grids, and FIGS. 5 to 8 and 15 to 19 are intended to show circular regions. Any distortions from square, regular hexagonal and circular representations are the result of inaccurate image reproduction.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0070] The invention provides an autostereoscopic display, comprising a pixelated display panel comprising an array of single color pixels or an array of sub-pixels of different colors and a view forming arrangement comprising an array of lens elements. The pixels form a hexagonal grid, and the lenses also repeat in a hexagonal grid. A vector p is defined which relates to a mapping between the pixel grid and the lens grid. Regions in the two dimensional space for this vector p are identified which give good or poor banding performance, and the better banding performance regions are selected.

[0071] The invention is based on an analysis of the effect of the relationship between the pixel grid and the lens grid on the banding performance. The banding analysis can be applied to different pixel and lens designs. Note that the term “pixel grid” is used to indicate the grid of pixels (if each pixel has only one addressable element), or the grid of sub-pixels (if each pixel has multiple independently addressable sub-pixels).

[0072] To illustrate the analytical approach, a first example will be presented based on square (or near square) pixel grids and lens grids. This invention relates specifically to hexagonal pixel and lens grids, for which an analysis is provided as a second example.

[0073] For the first example of a square pixel grid and lens grid, display panel designs are discussed with pixels on a regular 4-fold symmetric essentially square grid, on top of which there is a light modulator that also has elements in a regular 4-fold symmetric grid. For the purposes of explanation, some definitions are needed. In particular, a coordinate system of the panel (i.e. the pixel grid) needs to be defined, and a coordinate system of the view forming arrangement needs to be defined in terms of geometric (physical) coordinates and logical coordinates that are relative to the coordinate system of the panel.

[0074] FIG. 3 shows various possible pixel grids. Each example shows the smallest unit cell 30 (i.e. the smallest set of sub-pixels 31 which repeat to form the sub-pixel pattern, as defined above) and a pixel 32 using the definition employed in this description. A pixel 32 is the smallest square arrangement of all of the primary colors so that the pixel size and shape is the same in the two orthogonal orientations.

[0075] The sub-pixels are shown as squares. However, the actual sub-pixel shape may be different. For example the actual pixel aperture will typically be an irregular shape as it may for example depend on the size and position of pixel circuit elements, such as the switching transistor in the case of an active matrix display panel. It is the pixel grid shape that is important rather than the precise shape of individual pixels or sub-pixels. The same reasoning applies to the hexagonal pixel grid discussed further below.

[0076] Pixel pitch vectors x and y are also shown. These are translation vectors between adjacent pixel centers in the row direction and the column direction, respectively. The letters in the smallest unit cell 30 indicate the primary colors: R=red, G=green, B=blue, W=white.

[0077] FIG. 3(a) shows an RGGB unit cell and an RGGB pixel, FIG. 3(b) shows an RGBGBGRG unit cell and an RGBG pixel, FIG. 3(c) shows an RGBW unit cell and an RGBW pixel, FIG. 3(d) shows an RGBWBWRG unit cell and an RGBW pixel, and FIG. 3(d) shows a W unit cell and a W pixel.

[0078] A pixel grid is defined based on the two vectors x and y, hereafter referred to as pixel pitch vectors. The vectors form a lattice matrix X=[x y] with length units (e.g. meters). There are multiple possible definitions of a pixel including the smallest unit cell, however for this description, the pixel is approximately square. Therefore X should be chosen to form an approximately square region of sub-pixels. As shown in FIGS. 3(a) to (d), for color displays, the pixel definition most simply results in a region with 2×2 sub-pixels. When the unit cell is larger, as in FIGS. 3(b) and (d), the pixel group appears rotated or mirrored to form the larger unit cell, but also in these cases X remains a 2×2 region. For monochrome displays the pixel is the region of a single sub-pixel.

[0079] The pixels do not need to be perfectly square. They may be approximately square, which is taken to mean that a rotation over any angle, a limited sheer or limited elongation is within scope. The aspect ratio is defined as:

[00003]$a=\frac{.Math.x.Math.}{.Math.y.Math.}$

[0080] and the angle of the grid is:

[00004]$\theta ={\cos }^{-1}.Math.\frac{〈x,y〉}{\sqrt{〈x,x〉.Math.〈y,y〉}}.$

[0081] The sheer is then expressed as |θ−90°|. Hence for an approximately square grid it holds that a≈1 and |θ−90°|≈0°.

[0082] For example, a is preferably between 0.9 and 1.1 and θ is between 80 and 100 degrees (of course, if one pair of corner angles is at 80 degrees, then the other pair will be at 100 degrees).

[0083] To define the lens grid, lens pitch vectors can be defined.

[0084] FIG. 4 shows a lens grid 42 overlaid over a square pixel array 40 with 2×2 sub-pixels 31 per pixel 32 (such as in FIGS. 3(a) and (c). One out of each pixel group of four sub-pixels 31 is highlighted (i.e. shown white). The vectors x and y are the pixel pitch vectors of that grid as explained above. The lens grid 42 comprises a microlens array with spherical lenses 44 organized on a square grid. The vectors p′ and q′ are the pitch vectors of that grid. They are formed by a linear combination of the pixel pitch vectors.

[0085] Instead of physical lens pitch vectors in units of meters, logical and dimensionless lens pitch vectors can be defined as:

p=(p.sub.x,p.sub.y) and

q=(−p.sub.y,p.sub.x)

[0086] for chosen p.sub.x and p.sub.y.

[0087] The geometric (physical) pitch vectors p′ and q′ (e.g. in meters) are defined in terms of the logical lens pitch vectors as:

p′=Xp=p.sub.xx+p.sub.yy,

q′=Xq=−p.sub.yx+p.sub.xY.

[0088] Deformations in the pixel grid should be reflected in equal deformations of the lens grid. Notice that p,q=0 but not necessarily p′,q′=0 as we do not require x,y=0. Similarly |p|=|q| but not necessarily |p′|=|q′|.

[0089] For the purposes of this description, regions are defined P.sub.n,m for integer values n and m. These regions consist of multiple circles, themselves organized on a grid of circles.

[0090] Such a region is defined by:

[00005]$P_{n,m}=\left\{p|.Math.p-v.Math.<r_{n,m}.Math.\forall v\in {ℒ}_{n,m}\right\}.Math..Math.\mathrm{where}$${ℒ}_{n,m}=\left\{i+\frac{j}{n}|i,j\in {\mathbb{Z}}^2〈j,j〉=m\right\}.$

[0091] The p-v term specifies the length of the vector from v to p and thus the inequality defines a set of circles centered with a center defined by v. v is itself a set of vectors defined by the set of L terms. This has a discrete number of members as a result of the conditions placed on the integer values which make up the two dimensional vectors i and j.

[0092] Here r.sub.n,m=r.sub.0n.sup.−γ is the radius of each circle. This radius thus decreases with increasing n. .sub.n,m is the set of centers, and i,i denotes the inner product, such that when i=[i j].sup.T then i,i=i.sup.2+j.sup.2. A shorthand P.sub.n=P.sub.n,n is used in this description. Note that there are integers k for which there are no possible combinations of integers i and j for which j,j=k holds. As a consequence, the P.sub.3, P.sub.6 and P.sub.7 sets are empty.

[0093] As an example, the set P.sub.5 can be explored starting with .sub.5,5.

[0094] With i ∈.sup.2 we indicate all i=[i j].sup.T where i and j are integers (negative, zero or positive). The set of solutions to j ∈.sup.2j,j=5 is:

[00006]$j\in \left\{\left[\begin{array}{c}-2\\ -1\end{array}\right],\left[\begin{array}{c}-2\\ 1\end{array}\right],\left[\begin{array}{c}-1\\ -2\end{array}\right],\left[\begin{array}{c}-1\\ 2\end{array}\right],\left[\begin{array}{c}1\\ -2\end{array}\right],\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ -1\end{array}\right],\left[\begin{array}{c}2\\ 1\end{array}\right]\right\}.$

[0095] There is a graphical explanation of j and j/n as Gaussian integers and the reciprocal lattice thereof respectively shown in FIG. 5.

[0096] Each point in FIG. 5(a) is marked with the coordinate of the Gaussian integer g=a+b where .sup.2=−1 and the norm N(g)=a.sup.2+b.sup.2. FIG. 5(b) consists of the same points but the coordinates of the points are divided by their norm, thus corresponding to j/n instead of j.

[0097] Any combination

[00007]$i+\frac{j}{n}$

from the set of solutions for j shown above is in .sub.5,5. Two examples are

[00008]${\left[\begin{array}{cc}3.Math.\frac{2}{5}& 2.Math.\frac{1}{5}\end{array}\right]}^T.Math..Math.{\mathrm{and}.Math.\left[\begin{array}{cc}1.Math.\frac{4}{5}& \frac{2}{5}\end{array}\right]}^T.$

The region P.sub.5 then consists of circular regions with those centers and radius r.sub.5=r.sub.05.sup.−γ. Note that there are eight P.sub.5 circles around each P.sub.1 circle because there are eight solutions to j ∈.sup.2j,j=5.

[0098] In order to minimize the problems of banding for rotatable displays with pixels on an approximately square grid a display design is presented in which an array of view forming arrangements (typically a micro-lens array) forms a square grid that can be described by the direction p in terms of pixel coordinates where p is chosen outside of regions P.sub.n that give rise to banding.

[0099] To analyze the banding problem, two models have been used. The first model is based on an analysis of the spatial frequencies in both the pixel structure and the lens structure and the second one is based on ray tracing.

[0100] The first model uses moire equations and a visibility function to estimate the amount of visible banding for a given pitch vector p.

[0101] This model results in a map such as FIG. 6 where brighter areas indicate more banding (on a log scale). FIG. 6 plots the p.sub.y versus p.sub.x. It should be understood that the actual map depends on parameters such as the visual angle of the microlenses and the pixel structure. The map in FIG. 6 is generated for the case of a pixel with a single emitting area with aperture ⅛ of the whole pixel surface, a Gaussian lens point spread function (PSF) that scales with the lens aperture, and a constant lens visual angle of 20 arcsec.

[0102] As a consequence of the PSF scaling more banding components are visible for smaller |p| (in the top left part of FIG. 6) because of the more accurate focus. It has been observed that the strength of various banding “blobs” depends on the actual pixel structure (see FIG. 3) but the position of the blobs is always the same.

[0103] The analysis is based in part of the recognition that most of the structure in this banding map can be explained using the P.sub.n areas where P.sub.n with higher n correspond to smaller areas. Most of the areas with significant banding are explained by P.sub.1 . . . P.sub.8.

[0104] By fitting a radius r.sub.0=0.35 and γ=0.75 to this map, the image shown in FIG. 7 results. In other situations there might be less banding and as a consequence r.sub.0=0.25 is sufficiently stringent. FIG. 8 shows the results of fitting a radius r.sub.0=0.25 to the map of FIG. 5.

[0105] In FIGS. 7 and 8, preferred regions are also plotted for the square grid examples, namely P.sub.9,18 and P.sub.14,26. These regions are best described by r.sub.0=0.35.

[0106] The approach of this invention is based on avoiding the zones that give rise to banding, namely avoiding certain ranges of values of the vector p=(p.sub.x,p.sub.y).

[0107] The first zones to avoid are the regions P.sub.1 (i.e. P.sub.1,1) which give rise to the greatest banding. In FIG. 8, with smaller radius values, the excluded zone is smaller. Thus, a first zone to exclude is based on r.sub.0=0.25.

[0108] The zones to exclude when designing the relationship between the pixel grid and the lens grid for this square example are:

[0109] 1. p .Math. P.sub.1 with radius r.sub.0=0.25 and γ=0.75,

[0110] 2. As directly above and also p .Math. P.sub.2,

[0111] 3. As directly above and also p .Math. P.sub.4,

[0112] 4. As directly above and also p .Math. P.sub.5,

[0113] 5. As directly above and also p .Math. P.sub.8,

[0114] 6. Any of the above but with radius r.sub.0=0.35.

[0115] Within the space that is left by excluding the regions, there are some regions that are of particular interest because banding is especially low for a wide range of parameters. These regions are:

[0116] 1. p ∈ P.sub.9,18 with radius r.sub.0=0.35,

[0117] 2. p ∈ P.sub.14,26 with radius r.sub.0=0.35.

[0118] Preferably, for the square grid example, the sub-pixels are on a perfectly square grid but small variations are possible. The aspect ratio is preferably limited to

[00009]$\frac{2}{3}\le a\le \frac{3}{2},$

or more preferably to

[00010]$\frac{5}{6}\le a\le \frac{6}{5}.$

The sneer of me grid from a square/rectangle to a rhombus/parallelogram is preferably to |θ−90°|≦20°, or even to |θ−90°|≦5°.

[0119] An alternative for moiré equations to illustrate the approach is to ray trace a model of a display with a lens that displays a fully white image.

[0120] FIG. 9 shows such rendering for the 2D pixel layout as of FIG. 3(c). Any rendering of a banding-free design would appear to be on average white, while for a design with banding, the intensity and/or color depend on the viewer position (i.e. the lens phase).

[0121] FIG. 9(a) shows renderings for a lens design in a P.sub.1 region for a lens phase. Although not shown in the rendition of FIG. 9(a), the white and most of the blue primary is missing. FIG. 9(b) shows renderings for a lens design in a P.sub.2 region for a lens phase where more than average amount of black matrix is visible. FIG. 9(c) shows renderings for a lens design in a P.sub.4 region for a lens phase where almost no black matrix is visible. FIG. 9(d) shows renderings for a lens design at a P.sub.14,26 center with (virtually) equal distribution of primaries within this patch for this and all other phases.

[0122] A patch such as shown in FIG. 9 can be rendered for various lens phases, since different lens phases (by which is meant lens position which is responsible for generating the view to a particular viewing location) give rise to different distributions of sub-pixels. More effective is to compute the mean CIE 1931 XYZ color value is computed for each such patch. From that mean, the CIE L*a*b* color value can be computed which gives quantitative means of comparing perceptual banding effects.

[0123] In this perceptual color space the L.sub.2 distance between two color values (denoted ΔE below) is indicative of the perceived difference between those colors.

[0124] The target is white corresponding to (L*, a*, b*)=(100, 0, 0).

[0125] In FIG. 10 the brightness (L*) is plotted as a function of the lens phases in two dimensions, corresponding to different views projected by the lenses to different viewer positions, for the same examples as in FIG. 9. The dimensionless lens phase variable has values in the range of (0,1). Due to the periodicity of the pixel grid and the lens grid, lens phases 0 and 1 correspond to the same generated views. Because the display uses a 2D microlens array, the lens phase itself is also 2D.

[0126] In FIG. 11, the color error (ΔE) is plotted again for the same examples.

[0127] Depending on the situation ΔE≈1 is just visible. The banding-free example in FIGS. 10(d) and 11(d) appears as uniform L*=100 and ΔE=0 respectively, while the other examples clearly have banding as the color varies with the viewer position (i.e. lens phase).

[0128] Because the display uses a 2D microlens array, the lens phase itself is also 2D.

[0129] The plots can be summarized by taking the root-mean-square (RMS) value of ΔE over the entire phase space.

[0130] In the table below, this has been done for a list of points that correspond to regions that according to the banding model explained above should be excluded or included.

TABLE-US-00001 Region p.sub.x p.sub.y ΔE.sub.RMS P.sub.1 6.000 2.000 111.576 P.sub.2 7.000 3.000 63.375 P.sub.4 6.000 3.000 12.723 P.sub.5 7.200 3.600 3.609 P.sub.5 7.600 3.200 5.738 P.sub.8 6.500 2.500 2.289 P.sub.8 4.500 4.500 1.495 P.sub.9,18 7.333 3.333 0.467 P.sub.9,18 2.600 2.600 1.308 P.sub.9,18 3.350 3.350 0.796 P.sub.9,18 3.400 3.400 0.871 P.sub.14,26 6.143 3.286 0.180 P.sub.14,26 7.286 2.143 0.185 In between two P.sub.14,26 circles 6.000 3.286 0.155 In between two P.sub.5 circles 7.000 3.600 0.611 In between two P.sub.5 circles 5.000 3.400 0.289

[0131] From this table it is clear that the two models are largely consistent in terms of banding prediction. The positive areas have low ΔE.sub.RMS values, and the biggest negative areas (with lowest ordinals) have the highest ΔE.sub.RMS values.

[0132] The first model above provides an overview of the banding effect, while the second model provides more details and visualization.

[0133] An analogous analysis will now be presented for the example of a hexagonal pixel grid.

[0134] This invention relates specifically to panels with pixels (or sub-pixels) on a hexagonal grid (which is preferably a regular hexagonal grid, although it may deviate from a regular grid) on top of which there is a view forming arrangement that also has elements on a hexagonal grid.

[0135] As in the example above, the coordinate system of the panel is defined, then the coordinate system of the view forming arrangement is defined in terms of geometric (physical) coordinates and logical coordinates that are relative to the coordinate system of the panel. Parametric regions in the parameter space are again defined which can be selected to achieve desired performance, for example with respect to banding.

[0136] Pixel pitch vectors are again defined and for this example vectors a and b are defined, analogous to the vectors x and y in the example above.

[0137] Vectors a and b, are the pixel pitch vectors which form a lattice matrix X=[a b] with length units (e.g. meters). There are multiple possible definitions of a pixel including the smallest unit cell, however for this invention the pixel grid is hexagonal, for example at least approximately regular hexagonal. Therefore X should be chosen to form an hexagonal region of sub-pixels.

[0138] Examples are shown in FIG. 12.

[0139] For color displays the pixel area 32 is most likely a triangular region with 3 or maybe 4 sub-pixels 31. Sometimes such a group appears rotated or mirrored to form a larger and possibly elongated unit cell, but also in that case X is a region with 3 or 4 sub-pixels 31. For monochrome displays, the unit cell 30 is the region of a single pixel 32. Important is the grid of pixels 32 rather than the shape or grid of sub-pixels 31.

[0140] FIG. 12(a) shows a hexagonal grid in which each pixel 32 is formed as a triangle of three RGB sub pixels 31. The unit cell 30 is the same.

[0141] FIG. 12(b) shows a hexagonal grid in which each pixel 32 is formed as group of four RGBW sub pixels 31, forming a shape which is essentially a rhombus (but without straight sides). The unit cell 30 is the same.

[0142] FIG. 12(c) shows a hexagonal grid in which each pixel 32 is formed from seven sub pixels 31 (one in the center and six around the outside). However, the outer sub-pixels are shared with adjacent pixels so that on average there are 4 (RGBW) sub-pixels per pixel. The unit cell 30 (the smallest element which can be translated to form the full overall sub-pixel pattern) is larger, because there are two types of pixel.

[0143] FIG. 12(d) shows a hexagonal grid of single color pixels. The unit cell 30 is a single pixel 32.

[0144] The layout of FIG. 13 is a counter example because although the sub-pixels are hexagons and are arranged on a hexagonal grid, the pixel grid is actually rectangular. The pixel grid is defined by vectors which translate from one pixel to the same location within the adjacent pixels.

[0145] As in the example above, the invention does not require perfectly hexagonal grids nor is the angular orientation relevant. A rotation over any angle, a limited sheer or limited elongation is also possible.

[0146] The aspect ratio for the hexagonal pixel grid is defined as

[00011]$\beta =\frac{.Math.a.Math.}{.Math.b.Math.}$

[0147] and the angle of the grid is:

[00012]$\theta ={\cos }^{-1}.Math.\frac{〈a,b〉}{\sqrt{〈a,b〉.Math.〈a,b〉}}.$

[0148] An interior angle of 120 corresponds to a regular hexagonal grid. An amount of sheer can thus be expressed as |θ−120°|. Hence for an approximately regular hexagonal grid it holds that β≈1 and |θ−120°|≈0°.

[0149] As in the example above, lens pitch vectors are also defined. The definition of the logical and dimensionless lens pitch vectors are p=(p.sub.a, p.sub.b) for chosen p.sub.a and p.sub.b

[0150] The vectors relevant to the hexagonal case are shown in FIG. 14, which like FIG. 4 shows the lens grid 42 over the pixel array 40. This is based on the three sub-pixel pixel of FIG. 12(a). The lens grid is formed by the real vectors p′ and q′.

[0151] The vectors p′ and q′ have the same length and the angle between p′ and q′ is 120°. The geometric (physical) pitch vectors p′ and q′ (e.g. in meters) are defined in terms of the logical lens pitch vectors where deformations (e.g. rotation, sheer, scaling) in the pixel grid should be reflected in equal deformations of the lens grid. This can be understood by considering a flexible autostereoscopic display being stretched.

[0152] The dimensionless pitch vector p again defines a mapping between the pixel grid and the lens grid and in this case is defined by:

p′=p.sub.aa+p.sub.bb,

[0153] For this example regions E.sub.n are defined for integers n that consist of multiple circles, themselves organized on a grid of circles. Such regions are defined by:

[00013]$\begin{array}{cc}{E}_n=\left\{pN\left(p-v\right)<r_n^2.Math.\forall v\in {\Gamma }_n\right\}& \\ \mathrm{where}& \\ {\Gamma }_n=\left\{i+\frac{j}{n}i,j\in 2^{}N\left(j\right)=n\right\}.& \end{array}$

[0154] Again r.sub.n=r.sub.0n.sup.−γ is the radius of each circle, Γ.sub.n is the set of centers, and N(j) is the norm akin the Eisenstein integer norm defined as:

[00014]$N\left(\left[\begin{array}{c}a\\ b\end{array}\right]\right)=a^2-\mathrm{ab}+b^2.$

[0155] This defines a hexagonal lattice of centers. As in the example above, the p-v term specifies the vector from v to p and thus the inequality, which is essentially based on the norm of the space (distance squared), This defines a set of circles with a center defined by v. v is itself a set of vectors defined by the set of Γ.sub.n terms. This has a discrete number of members as a result of the conditions placed on the integer values which make up the two dimensional vectors i and j.

[0156] As an example, explore E.sub.4 is considered, starting with Γ.sub.4. The set of solutions to j ∈.sup.2N(j)=4 is:

[00015]$j\in \left\{\left[\begin{array}{c}-2\\ -2\end{array}\right],\left[\begin{array}{c}-2\\ 0\end{array}\right],\left[\begin{array}{c}0\\ -2\end{array}\right],\left[\begin{array}{c}0\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 2\end{array}\right]\right\}.$

[0157] Any combination

[00016]$i+\frac{j}{4}$

is in Γ.sub.4. Two examples are

[00017]${\left[\begin{array}{cc}3.Math.\frac{1}{2}& 2.Math.\frac{1}{2}\end{array}\right]}^T.Math..Math.{\mathrm{and}.Math.\left[\begin{array}{cc}1& 2.Math.\frac{1}{2}\end{array}\right]}^T.$

The region E.sub.4 then consists of circular regions with those centers and radius r.sub.4=r.sub.04.sup.−γ. There is a graphical explanation of j and j/n as Eisenstein integers (that form a hexagonal lattice in the complex plane) and the reciprocal lattice thereof respectively as shown in FIG. 15.

[0158] Each point in the left subfigure is marked with the coordinate of the Eisenstein integer c=a+ωb, and the norm N([a b].sup.T). The right subfigure consists of the same points but divided by their norm, thus corresponding to j/n instead of j.

[0159] Again there are integers k for which there are no j for which N(j)=k holds. As a consequence, the E.sub.2, E.sub.5 and E.sub.6 sets are empty.

[0160] In the example above based on square grids, a Cartesian norm is used, namely j,j=j.sup.Tj and in a graphical explanation Gaussian integers are used that from a square lattice in the complex plane, instead of Eisenstein integers. FIG. 16 shows this approach for comparison with FIG. 5.

[0161] The approach explained above is used to analyze the banding effect of different designs. The resulting map, again based on moire equations and a visibility function to estimate the amount of visible banding for a given pitch vector p, is shown in FIG. 17. This is a plot of p.sub.b versus p.sub.a and again brighter areas indicate more banding.

[0162] It should be understood that the actual map depends on parameters such as the visual angle of the microlenses and the pixel structure. The map in FIG. 17 is generated for the case of a pixel with a single emitting area with aperture ⅙ of the whole pixel surface, a Gaussian lens point spread function (PSF) that scales with the lens aperture, and a constant lens visual angle of 20 arcsec. As a consequence of the PSF scaling more banding components are visible for smaller |p| because of the more accurate focus.

[0163] Most of the structure in this banding map can be explained using the E.sub.n areas where E.sub.n with higher n correspond to smaller areas. Most of the areas with significant banding are explained by E.sub.1 . . . E.sub.4.

[0164] As in the examples above, r.sub.0=0.35 and γ=0.75 are used to generate the image of FIG. 18. In other situations there might be less banding and as a consequence r.sub.0=0.25 is sufficiently stringent. FIG. 19 shows the results of fitting a radius r.sub.0=0.25 to the map of FIG. 17.

[0165] Note that in FIGS. 18 and 19, the regions are labeled P.sub.x for simple comparison with FIGS. 7 and 8. These regions however are the regions E.sub.x. as defined by the equations above.

[0166] In FIGS. 18 and 19, preferred regions are plotted, namely E.sub.7 and E.sub.9 (shown as P.sub.7 and P.sub.9). These regions are best described by r.sub.0=0.35.

[0167] The invention is based on avoiding the zones that give rise to banding, namely the value of the vector p=(p.sub.a,p.sub.b).

[0168] The first zones to avoid are the regions E.sub.1 which give rise to the greatest banding. In FIG. 19, with smaller radius values, the excluded zone is smaller. Thus, a first zone to exclude is based on r.sub.0=0.25.

[0169] The zones to exclude when designing the relationship between the pixel grid and the lens grid are:

[0170] 1. p .Math. E.sub.1 with radius r.sub.0=0.25 and γ=0.75,

[0171] 2. As directly above and also p .Math. E.sub.3,

[0172] 3. As directly above and also p .Math. E.sub.4,

[0173] 4. Any of the above but with radius r.sub.0=0.35.

[0174] Within the space that is left by excluding the regions, there are some regions that are of particular interest because banding is especially low for a wide range of parameters. These regions are:

[0175] 1. p ∈ E.sub.7 with radius r.sub.0=0.35,

[0176] 2. p ∈ E.sub.9 with radius r.sub.0=0.35.

[0177] Preferably, sub-pixels are on a regular hexagonal grid but small variations are within the scope of the invention: The aspect ratio is preferably limited to

[00018]$\frac{2}{3}\le a\le \frac{3}{2},$

or more preferably to

[00019]$\frac{5}{6}\le a\le \frac{6}{5}.$

The sheer of the grid away from a regular hexagon is preferably limited to |θ−120°|≦20°, or even to |θ−120°|≦5°.

[0178] The invention is applicable to the field of autostereoscopic 3D displays, more specifically to full-parallax rotatable multi-view auto-stereoscopic displays.

[0179] The invention relates to the relationship between the pixel grid and the lens grid. It can be applied to any display technology.

[0180] Other variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope.