Creating a desired lighting pattern with an LED array Ivan Moreno Unidad Academica de Fisica, Universidad Autonoma de Zacatecas, 98060, Zacatecas, Mexico. ABSTRACT An array of spatially distributed LEDs can produce a desired illumination pattern by individually modulating each LED. A target image can be the desired lighting pattern so that the software could find the best solution to match it. Given a desired illuminance distribution on a target surface, the luminous flux of each single LED that most closely matches the desired distribution must be determined. We review a constrained least squares method for this problem. We show how the quality of the rendering depends on the number of LEDs, array-target distance, and the size of the illuminated area. In particular, as we observed, there is an optimum illumination distance, which is proportional to the square root of the target size and varies inversely with a power of the number of LEDs. Key words: light-emitting diode, LED array, illumination distribution, least-square method, normalized-cross correlation. 1. INTRODUCTION The rapid development of LEDs over the last years has opened up new applications in general illumination. An interesting application is to use a cluster of LEDs for producing a given desired illuminance (or irradiance) distribution on a target (see Fig. 1). In general, a high power LED emits light into one hemisphere with some degree of directionality. As a consequence, LED arrays can be designed to easily direct emission into specific lighting patterns without additional optical devices. Such illumination system offers a high flexibility in rendering and changing the target illumination. An adaptive lighting pattern can be adjusted to specific environments and requirements by individually modulating the LEDs. This controllability can provide new benefits in numerous applications as lighting, displays, transportation, agriculture, and medicine. Fig. 1. Cluster of LEDs projecting an arbitrary illumination distribution over a target surface. Lighting applications include illuminated ceilings, interior design, lighting in buildings, and luminous signage.1 The flexibility of an LED array is absent in conventional light sources, because a fluorescent lamp cannot be dimmed, and an incandescent source decreases its efficiency and changes its spectral content when its intensity is dimmed. Moreover, a multicolor LED array can render a color illumination distribution. In other applications like high-contrast liquid crystal displays, spatially distributed LEDs in the backlight system can provide localized illumination effects that increase the image contrast, and reduce the light leakage in image regions that do not need illumination.2-4 An application beyond illumination and lighting involves specialized solid-state micro-projection devices, which contain up to several thousand individually-addressable micro-pixel elements emitting light in the ultraviolet or visible regions of the spectrum.5-8 These 8th International Conference on Solid State Lighting, Ian T. Ferguson, Tsunemasa Taguchi, Ian E. Ashdown, Seong-Ju Park, Editors, Proceedings of SPIE Vol. 7058 (2008) © 2008 SPIE ⋅ ISSN: 0277-786X (print) ⋅ $ 15.00 micro-LED arrays can project controllable illumination distributions into biological cells and tissues, biopolymers, and photoresists. This leads to new possibilities in optical microscopy, bio-sensing and chemical sensing, mask-free lithography and direct writing.5-8 An arbitrary image can be the target light distribution of the LED array. However, the pixels in that image cannot be used directly as intensity weights, since light beams overlap between neighboring LEDs. However, the luminous flux of each single LED that most closely matches the desired light distribution can be determined. In next section, we review a constrained least squares method for this problem. We analyze the effects on the rendering accuracy due to the number of LEDs, array-target distance, and the image size. The accuracy of rendering is evaluated by using normalized cross correlation, which measures the similarity between the target image and the rendered light distribution. As shown in Section 3, there is an optimum target-array spacing, which varies with the square root of the image size and varies inversely with a power (~0.5) of the number of LEDs. 2. CONSTRAINED LEAST SQUARES PROBLEM The problem is stated as follows: given a desired illumination distribution over a flat surface, determine the LED intensities that will most closely match it. In other words, given a desired image matrix IMAGE, determine the LED intensity levels that most closely approximate IMAGE. The general problem involves determining how many LEDs to use, where the LEDs must be placed, as well as the radiation pattern of LEDs. This general case is a non-linear optimization problem. However, if all LEDs are previously positioned, and their radiation patterns fixed, the calculation of the LED intensities is a linear optimization process. The LED radiation pattern is characterized by the angular intensity distribution. Radiant (or luminous) intensity is the radiant (or luminous) flux per solid angle [W/sr or lm/sr] in a given direction from the source. It is the radiometric standard used to rate spatial variation of LEDs. However, the power distribution that illuminates the target is characterized by the irradiance. Irradiance (or illuminance) is the radiant (or luminous) flux incident on a surface per unit area [W/m2 or lm/m2]. Consider Ei is the normalized irradiance (or illuminance) spatial distribution (in arbitrary units) of the i-th LED illuminating the target plane, where i=1,2,…N. Therefore [ E1, E2,.. EN ] is the set of functions resulting from an array of N LEDs illuminating the target (see Fig. 1). These functions can be easily modeled with Gaussians for any kind of LED of practical interest.9 The problem can be stated in terms of finding non-negative weights wi such that the irradiance produced by the LED array N Earray ( x, y ) = ∑ wi Ei ( x, y ) , (1) i =1 approximates a target image, IMAGE, we wish to approximate. The problem is one of least squares (LS), in which |IMAGE−Earray|2 must be minimized. The LS minimization process leads to solve a system of equations given by: N H V H V ∑∑∑ wi Ein,m E nj,m = ∑∑ IMAGEn,m E nj,m , i =1 n =1 m =1 (2) n =1 m =1 where j=1,2,…N., and the target image size in pixels is NHV=H×V. This set of linear equations can be expressed as a matrix product: ⎡∑ H ∑V E1n ,m E1n ,m L ∑ H ∑V E1n ,m E Nn ,m ⎤ ⎡ w1 ⎤ ⎡∑ H ∑V IMAGEn ,m E1n ,m ⎤ n =1 m =1 ⎥ ⎥ ⎢ ⎥ ⎢ n =1 m =1 ⎢ n =1 m =1 M O M M ⎥ , ⎥⎢ M ⎥ = ⎢ ⎢ V H V H V n,m n,m n ,m n ,m ⎥ n,m ⎥ ⎢ ⎢ H L ∑n =1 ∑m =1 E N E N ⎢⎣ wN ⎥⎦ ∑n =1 ∑m =1 IMAGEn ,m E N E E1 n =1 ∑m =1 N ⎣∑ 3 ⎣14444 1 4444444442444444444 3⎦ 12 4244444 3⎦ W EE IE (3) where EE is a symmetric matrix, i.e. EEi,j=EEj,i. Taking into account the symmetric nature of EE, its calculation time can be reduced by nearly half. Vector W gives the irradiance weights wi of LEDs in the array. The problem is to solve Eq. (3), which contains N linear equations in N unknowns. The problem of solving W cannot be based on a direct numerical method because the solution is not stable. This is due to the large number of unknowns and the difficulty of introducing physical constrains (in our case wi≥0). Therefore, an iterative numerical method must be used. We compared two simple methods: the Gauss-Seidel and successive over-relaxation (SOR). The Gauss-Seidel iteration is IEi − ∑ j =1 EEi , j wkj +1 − ∑ j =i +1 EEi , j wkj i −1 wik +1 = N , (4) + (1 − α )wik , (5) EEi ,i and the SOR iteration is IEi − ∑ j =1 EEi , j wkj +1 − ∑ j =i +1 EEi , j wkj i −1 k +1 i w = N EEi ,i α where α is a weighting factor for rapid convergence (0<α<1). Because both methods rendered target images with equal accuracy, we choose using Gauss-Seidel iteration in most simulations. However, for arrays with a few LEDs, the SOR iteration performed better than the Gauss-Seidel. In these cases, we used SOR with α factors in a range of 0.1<α<0.5. The Gauss Seidel iteration is a particular case of the SOR iteration for α=1. We constricted the iterative algorithm so that negative values of wi are clipped to zero during each iteration.10 The k+1 estimate of wi, given by Eq. (5), is modified by10 ( ) wik +1 = max wik +1 ,0 . (6) This constriction does not influence other W elements, because each zero value is ignored during the subsequent iterations.10 Since each iteration reduces the previous wi value, a large number provides a good initial guess wi0. The normalized irradiance Ei of the i-th LED that illuminates the target is a critical parameter for rendering IMAGE. The irradiance produced by an LED over a flat surface (xy plane) at distance z from the LED can be modeled as:9 E ( x, y , z ) = (x z 2 + y + z2 2 ) 3 2 I [θ (x, y, z ), φ ( x, y )] , (7) Here I(θ,φ) is the LED angular intensity distribution, and ⎡ x2 + y2 ⎤ ⎥ , z ⎣⎢ ⎦⎥ θ (x, y, z )= arctan ⎢ ⎛ ⎞ φ (x, y )= arctan⎜ ⎟ . y x ⎝ ⎠ (8) (9) Intensity I(θ,φ) can be easily modeled with Gaussians for any kind of LED of practical interest.9 E(x,y,z) is the irradiance spatial distribution over a target plane x-y of area A. We can scale the image to the target plane, or scale both the target and LED array to the image plane. For simplicity we worked in the image space of area NHV=H×V. Therefore, we evaluated Eq. (7) within the image area, which is equivalent to take a picture of the target plane when it is only illuminated by the i-th LED. We created a vector E with nested irradiance matrices Ei for every LED of the array: Ein ,m = E (n − ni , m − mi , s ) E (0,0, s ) , (10) where s is the target-array distance (see Fig. 1), and n=1,2,…H and m=1,2,…V. Parameters ni and mi are the pixel position of the i-th LED. We analyzed a square LED array with pixel separation between each pair of LEDs ΔH and ΔV, in the horizontal and vertical directions, respectively (see Fig. 2). Edge LEDs are located near the image boundary as seen in Fig. 2. Because the pixel number along the vertical and horizontal directions of the image are integer numbers, distances ΔH, ΔV, ΔH1, ΔV1, ΔH2, and ΔV2 can not take values arbitrarily. In the next section we provide the values we used for each simulation. Calculating matrices Ei from the beginning saves considerable computation time. Additionally, simulating the irradiance pattern produced by the LED array is more rapid when adding N matrices Ei: N E array = ∑ wi Ei , (11) i =1 In order to display an image of Earray, the grey level range must fit that of the target image. Therefore, we escalated Eq.(11) to the pixel range of the target image: E array = max( IMAGE ) E array , max( E array ) (12) Changing any physical parameter of the LED array requires re-computing matrix EE and vector IE. In particular, the EE computation is the one that takes more time. Therefore, analyzing the effects of changing LED radiation pattern and number of LEDs takes a longer time. The computation is considerably faster if only the target image is changed because only the IE vector must be recomputed. Fig. 2. Cluster of LEDs in the image space of area NHV=H×V pixels. We have discussed an iterative method that directly solves the constrained LS problem. Such numerical methods always give the best answer but the price is computation time. However, for video displays, where time is a critical issue, the convolution method is more convenient.2,3 In this paper, we use the constricted LS method because it is more precise than the convolution one for image rendering. In what follows, we need precision to analyze the system performance for creating a desired lighting pattern. 3. SOME RENDERING PROPERTIES The illumination rendering performance of a LED array depends on several parameters as the target-array distance, number of LEDs, illumination area (or escalated to target image size), and the radiation pattern of LEDs. Changing the target-array distance is equivalent to change the packaging density of LEDs. Therefore, in what follows, without loss of generality, we render the target image in function of the target distance. 3.1 Reconstruction accuracy To evaluate the similarity between the rendered irradiance distribution and the target image, we compute the normalized cross correlation (NCC). The NCC is a merit function that is frequently used for image recognition. Therefore, the rendering accuracy of the rendered irradiance pattern Earray with respect to the target image IMAGE is NCC = ∑ ∑ (IMAGE − IMAGE )(E − E ) ∑ ∑ (IMAGE − IMAGE ) ∑ ∑ (E − E ) H V n =1 m =1 H V n =1 m =1 n,m ave 2 n,m ave n,m array H V n =1 m =1 ave n ,m array 2 , (13) ave IMAGEave and Eave are the mean value of IMAGE and Earray across the entire range of pixels, respectively. The degree of accuracy depends on the particular application. A NCC equal to 100% means that IMAGE and Earray are exactly the same. Because the number of LEDs N is generally smaller than the number of pixels in the image, the NCC always is smaller than 100%. Fig. 3. Simulation of the light distribution on the target plane using a 10x15 LED array. The radiation pattern of each LED is perfect Lambertian. (a) Target image is a symbol of batman, whose pixels values are matrix IMAGE. (b)-(c) Illumination patterns for several working distances s, obtained by weighting the light of each LED. (b) Irradiance weights wi computed by the constricted LS method. (c) Irradiance weights wi assigned by sampling the target image in each pixel in front each LED. (d) Target image is a Mona Lisa paint. (e)-(f) Illumination patterns for the same working distances as (b) and (c). (e) Weights wi computed by the constricted LS method. (f) Weights wi assigned by sampling the target image. 3.2 An Example For beginning the analysis, consider the target images shown in Figs. 3(a) and 3(d), whose pixels represent the target matrix IMAGE. Using the constricted LS methodology described in section 2, we calculate the irradiance weights wi and simulate the illumination pattern produced by a LED array with 150 LEDs. For Mona Lisa image, parameters are: image size NHV=100x150 pixels, ΔH=ΔV=10 pixels, ΔH1=ΔV1=6 pixels, and ΔH2=ΔV2=4 pixels. For the Batman image, parameters are: image size NHV =122x83 pixels, ΔH=ΔV=8 pixels, ΔH1=ΔV1=4 pixels, ΔH2=6 pixels, and ΔV2=7 pixels. Figure 3 shows the simulated illumination distributions for several target-array distances s. In this simulation the radiation pattern of each LED is perfect Lambertian,9 i.e. the half width half maximum angle θ1/2=60°. Fig. 3(b) displays the irradiance distribution with weights wi computed by the constricted LS method. For the purpose of comparison we show in Figs. 3(c) and 3(f) the illumination distributions obtained by assigning weights wi by sampling (grey levels right in front of each LED of the array) the target image at each pixel. Fig. 4 shows the NCC of rendered illumination distributions in function of s for both the sampling and LS method. As noted in Figs. 3 and 4, a light distribution rendered by sampling LED levels from IMAGE has lower quality than a pattern rendered with the LS method. This difference becomes smaller for shorter working distances, and for smoother target images as noted by comparing the Batman and Mona Lisa images. However, the LS method considerably gives higher accuracy than the sampling method if array-target distance increases. Fig. 4 shows that there is an optimum illumination distance sopt that maximizes NCC. In addition, it can be noted that the NCC peak of Fig. 4 (a) is near to that one of Fig 4 (b). This lead us to speculate that the ideal target-array distance could not depend significantly on the image content; and it could depend strongly on the target image size, number of LEDs, and the light intensity distribution of LEDs. Fig. 4. Rendering accuracy of illumination pattern with respect to the target image in function of the target-array distance s. The simulation parameters are those used in Fig. 3. The Normalized cross correlation (NCC) is used to measure the accuracy in rendering the target image. This figure shows the NCC obtained by using weights wi computed with the constricted LS method, and obtained from sampling the target image (see Fig. 3). (a) shows the NCC for the Batman image, and (b) shows the NCC for the Monalisa image. 3.3 Optimal illumination distance in function of parameters of influence In order to find the dependence of the NCC peak in function of the parameters of influence, we used several target images with square shape. In these simulations the LED array fits the borders of the image, i.e. ΔH1=ΔV1=ΔH2=ΔV2=0 pixels. For simplicity, the radiation pattern of each LED is perfect Lambertian, i.e. θ1/2=60°. We calculated the NCC of rendered illumination distributions in function of s for several LED array sizes, in particular for arrays with 4x4, 6x6, 7x7, 9x9, 10x10, 11x11, 13x13, 16x16, 21x21, and 25x25 LEDs. In Fig. 5(a) the image size is NHV =120x120 pixels, and the array sizes are: 7x7, 11x11, and 25x25 LEDs. For Fig. 5(b) the image size is NHV=45x45 pixels, and the array sizes are: 6x6, 10x10, and 16x16 LEDs. Fig. 6 displays the irradiance distributions for the optimum working distances of some of the LED arrays. As noted in Fig. 5, the NCC peak shifts to the left for larger LED arrays. Fig. 7 shows the optimum working distance sopt in function of N the number of LEDs of the array. From direct observation, we note that the relationship between sopt and N is a simple power. Moreover, we empirically found that the relation with the target image size NHV is a square root. Then an empiric equation for the optimal target-array distance is sopt = N HV N − f . (14) The f factor could depend on the image content, and other factors as the array geometry and the LED radiation pattern. Units of sopt are pixels. However, Eq. (14) can be scaled to the working space, in terms of the LED array or the illumination area A [m2], it is sopt = A N − f , [m] (15) In particular, for the Batman image of 120x120 pixels, Eq. (14) is s opt = 120 N −0.5328 (16) And for the Playboy image of 45x45 pixels, it is s opt = 45 N −0.5613 (17) Additional simulations with the same Playboy image but with size 120x120 pixels show that Eq. (17) only changes the 45 factor by 120. Fig. 7 includes the plot of Eqs. (16) and (17). Although, illuminating at sopt distance the NCC is maximized, the rendered distribution shows a hot-spot distribution. In LCD TV applications a uniform background is desired.11 In order to ensure uniform illumination, a slightly larger targetarray distance could blur the discrete spots into a smoothly varying field. However, the trade-off between illumination uniformity and rendering accuracy is small because when increasing the working distance the NCC decreases slowly after the peak value. Fig. 5. Rendering accuracy of illumination pattern in function of the target-array distance s. (a) Shows the NCC for a batman image of 120x120 pixels, for LED arrays with 7x7, 11x11 and 25x25 LEDs. (b) Shows the NCC for a playboy image of 45x45 pixels, for LED arrays with 6x6, 10x10 and 16x16 LEDs. Fig. 6. Simulation of the light distribution on the target plane. The target-array distance is that one that maximizes the similarity between the target image and the LED array lighting. This figure shows the illumination distributions using a Batman symbol and a Playboy logo as target images. For the Batman image the LED array has 21x21 and 25x25 LEDs. For the Playboy image LED array has 10x10 and 16x16 LEDs. Fig. 7. Target-array distance for optimal rendering accuracy. (a) Is for the Batman image of 120x120 pixels, for LED arrays with 7x7, 9x9, 11x11, 21x21, and 25x25 LEDs. (b) Is for the Playboy image of 45x45 pixels, for LED arrays with 4x4, 6x6, 10x10 and 16x16 LEDs. 4. CONCLUSION By individually modulating the LEDs of an array of spatially distributed LEDs a desired illumination distribution can be created. An arbitrary image can be the desired lighting pattern so that some software can find the best solution to match it. We reviewed a constrained least squares method for solving this problem. We showed how the rendered illumination pattern depends on the number of LEDs, array-target distance, and the illuminated area. As shown in section 3, there is an optimum illumination distance, which is proportional to the square root of the target size and varies inversely with a power (around 0.5 for LEDs with Lambertian radiation pattern) of the number of LEDs. 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