Frequency-Based Environment Matting of Transparent Objects Using Kaczmarz Method *

Transcription

1 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 26, (2010) Frequency-Based Environment Matting of Transparent Objects Using Kaczmarz Method * I-CHENG CHANG +, TIAN-LIN YANG AND CHUNG-LIN HUANG + Department of Computer Science and Information Engineering National Dong Hwa University Hualien, 974 Taiwan icchang@mailndhuedutw Department of Electrical Engineering National Tsing Hua University Hsinchu, 300 Taiwan Digital compositing is an important topic in the field of computer graphics and image processing Many works have sought a good method to prevent composited images from being distinguishable from real images The paper proposes a new environment matting algorithm to model the appearance of transparent object under different backgrounds In the proposed approach, the frequency-domain analysis is adopted to compute the relationship between the area of foreground object and background image Furthermore, the Kaczmarz method is applied to compute the caustics from refraction by transparent objects The experimental results show that the algorithm effectively improves the quality of composited image and has a high PSNR In addition, the composited results are also demonstrated by a video sequence involved of the motion, rotation and scaling of the transparent object Keywords: environment matting, transparent object, Kaczmarz method, weight matrix, iterative projection 1 INTRODUCTION Digital compositing is an important topic in the field of computer graphics and image processing Many works have sought a good method to prevent composited images from being distinguishable from real images Based on their material and reflective characteristics, foreground objects can be classified as opaque or transparent It needs quite different approaches to model the two classes of objects Opaque objects are usually modeled using a Bidirectional Reflectance Distribution Function (BRDF) or Image-based Rendering (IBR) This work addresses on the environment matting of a transparent object Previous studies have presented two approaches: one based on spatial-domain analysis and the other based on frequency-domain analysis Both methods use a monitor as a light source to change the position and intensity of the light source conveniently; both record the change in the lighting pattern using a camera, finally constructing a model of the object from the acquired images The main difference between the approaches is in the lighting pattern Spatial-domain analysis examines the variation of the lighting pattern in the spatial domain However, frequency-domain analysis inspects variations of the lighting pattern in both the spatial and the frequency domains When images are acquired, Received May 9, 2008; revised October 27, 2008; accepted December 1, 2008 Communicated by H Y Mark Liao * This work was supported by the Ministry of Economic Affairs, Taiwan, ROC, under Grant No 97-EC-17- A-02-S

2 1414 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG some unexpected noise may be introduced Such noise affects the accuracy in the analysis of the object model The design of frequency-domain analysis is to allocate different frequency information related to different position of the light source plane, so one can separate noises from the original image series Hence, a more accurate compositing image is obtained Zongker et al [1] adopted coarse-to-fine images as backdrops to be displayed on a CRT screen The object is placed in front of the CRT screen, and a camera records the change in the various backdrops They optimized the contribution of the light sources in the background for each pixel in the imaging plane If several light sources of the background contribute to a single pixel, then this algorithm finds only the position of the strongest one Other researchers [2, 3] have utilized stripes of one-dimensional Gaussian as the sweep line The camera captures the variation of color when the stripes shift at a regular step Finally, Levenburg-Marquardt optimization procedure was adopted to determine the matting model in the form of a 2D Gaussian Li [10] analyzed the radiance value instead of the color value, and extract the environment matte of a colorless and transparent object from a single background picture The method can decrease the error caused by the nonlinear response of optical system Wexler et al [4] proposed an algorithm that does not require a priori knowledge of the content of the backdrops They moved the object in front of a fixed background picture, and used geometric registration to determine the characteristics of light refraction and the relative position between the object and the background Peers and Dutre [5] applied the wavelet patterns as the backdrop When the backdrop changes in the acquisition process, the algorithm computes the contribution of the wavelet pattern that corresponds to each level Zhu and Yang [6] adopted the frequency response as a basis to establish a matting model They used the row-based and column-based stripe patterns, in which each stripe corresponded to a particular frequency Then they used DFT to analyze the contributions of those frequencies, and the matting model of the object is obtained from the contributions of the row-based and column-based areas We developed a new algorithm that extended the method of frequency-domain analysis The proposed algorithm solves the derived simultaneous equations that represent the distribution of the light source to yield a more accurate model of the object The experiments show that our approach can improve the quality of the composited image and has a high PSNR The rest of this paper is organized as follows Section 2 describes the proposed environment matting equation Section 3 presents the method for extracting the area of the foreground object, and section 4 shows how to generate the background patterns Section 5 describes the computation of the weight matrices, and the experimental results are demonstrated in section 6 Finally, section 7 draws conclusions 2 ENVIRONMENT MATTING EQUATION In the traditional matting equation [9], the compositing image C is described as the compound of a foreground image F, a background image B, and alpha channel α The relation is described as: C = αf + (1 α)b (1)

3 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1415 The alpha channel plays a dual role: it is used to represent simultaneously both the coverage of a pixel by a foreground element, and the opacity of this foreground element In image compositing, the foreground element can be regarded as a synthesis of two components the emissive component derived from the foreground element itself and the reflection or refraction component from light sources in the environment This work presents a formula for describing how a foreground element reflects and refracts light in the environment The new environment matting equation is as follows C = αf + (1 α RBˆ + αφ, (2) ) g where C represents the color of the recorded image in the imaging plane; F represents the emissive component derived from the foreground element, and α represents whether the background is covered by the foreground element If α = 0, then the corresponding pixel belongs to the background; if α = 1, then it belongs to the foreground B ˆg represents the background image of n m pixels in the camera coordinates R represents the attenuation from the position of the light source to the camera Φ denotes the proportion of the light from the environment that passes through the foreground element The light received by the camera comes from the light sources in the environment Therefore, the total amount Φ p of light received by point p in the imaging plane is given by an integral over the area of all light from the environment that contributes to point p The definition of Φ p is described as follows: Φ P = W( λ) E( λ) dλ (3) where λ denotes the position of the light source and E(λ) is its illumination distribution The weighting function W(λ) depicts the transport of the environment lighting from the position λ through the foreground object to the camera The background B g is divided into S T axis-aligned regions, and then the mean over every axis-aligned region is denoted as B g (s, t) Since Φ is the set of all foreground pixels in the imaging plane, it is expressed as follows: s= S, t= T Φ= W(,) s t B (,) s t (4) s= 1, t= 1 g where W(s, t) is the weight matrix, which represents the attenuation of the axis-aligned region B g (s, t) from the position of the light source to the camera Fig 1 shows the relationship between the image plane and the background, where several background sources would corresponds to one pixel in the image plane Therefore, the environment matting equation for a transparent object is described as: s= S, t= T C = αf + (1 α) RBˆ + α W( s, t) B ( s, t) (5) g s= 1, t= 1 The principal problem here is to determine the attenuation value of the weight matrix W(s, t) g

4 1416 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG Fig 1 Illustration of the environment matting of a transparent object The light from several different positions may be received by a cell p in image plane (a) Projection pattern (n = + 1) with foreground object (b) Projection pattern (n = + 1) without foreground object (c) Projection pattern (n = 1) with foreground object (d) Projection pattern (n = 1) without foreground object (e) Foreground object area Fig 2 Projection of sinusoidal background patterns and foreground object area 3 SEGMENTATION OF THE FOREGROUND OBJECT The image from image plane can be divided into two areas covered area and uncovered area The covered area is the projection region of a transparent foreground object and the uncovered area represents the projection of the backdrop Therefore, only the weight matrices of the elements that belong to the covered area are computed This section describes the segmentation technique that extracts the covered area from the image plane The sinusoidal background patterns ([6]) are adopted to sample the covered area, and the pattern is described as, 2 π( x+ y) π Ci ( x, y, n) = (1 + nsin( + i )) 127, (6) λ 3

5 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1417 where i = 0, 1, 2 for the R, G, B domain and n = 1 or 1 C i (x, y, n) is the intensity of color channel i at pixel location (x, y) To maximize the per-pixel difference between the two background patterns, the phase of the pattern is shifted by 180 o (n = 1 or 1) The user can define the period of the sinusoidal stripes with parameter λ Here, λ = 50 is set and four images are captured For instance, when n = 1, the captured images of Figs 2 (a) and (b) are obtained with and without the object, respectively When n = 1, the images of Figs 2 (c) and (d) are obtained with and without an object, respectively After the image is subtracted and post-processed, the covered area of the foreground object can be obtained Fig 2 (e) presents the results of this process 4 DESIGN OF THE BACKGROUND PATTERNS In the work, the background is segmented into several axis-aligned regions of equal size, and each is assigned a cosine waveform of a specific frequency Illustration of setting of pixel values of background patterns is shown in Figs 3 (a) and (b) first frame B(1, 1) B(2, 1) B(3, 1) B(S, 1) first frame B(1, 1) B(2, 1) B(3, 1) B(T, 1) second frame B(1, 2) B(2, 2) B(3, 2) B(S, 2) second frame B(1, 2) B(2, 2) B(3, 2) B(T, 2) Nth frame B(1, N) B(2, N) B(3, N) B(S, N) Nth frame B(1, N) B(2, N) B(3, N) B(T, N) (a) Column-based background patterns (b) Row-based background patterns Fig 3 Illustration of setting of pixel values of background patterns

6 1418 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG Every column-based and row-based area is assigned a specific frequency, and the pixel values B(f c, n) of every area of background patterns are designed as: B(f c, n) = ρ(1 + cos(2π f c nδt)) (7) where n is the time index, ranging from 1 to N f c represents setting frequency of the area, where the range is [1, S] for column-based area and [1, T] for row-based area Δt depicts the sampling period of cosine function in time domain To prevent aliasing in the results of DFT, Δt should be selected to satisfy the Nyquist theorem The range of values of cosine function varies in [ 1, 1], ρ is set to 1275 such that the range of pixel values be [0, 255] The spectrum of the signal is computed by performing Fourier transform Accordingly, the variation in the color of the column-based and row-based background patterns between the imaging plane and the background is analyzed Therefore, the positions of the light sources that contribute to the brightness of every position of the imaging plane are obtained The light attenuation from the background to the imaging plane can also be obtained When a small area owns many signals of different frequencies, noise with low frequency (under than 5Hz) is present in the analysis of the background pattern Such noise may be caused by breeding effect of CCD cells or a non-defined light source Hence, the setting frequency f c is shifted up by 10Hz to prevent the interference caused by background noise, and the definition of background pattern is modified as follows B(f c, n) = ρ(1 + cos(2π f c nδt)) (8) where f c = f c + 10, so the frequency range of f c is [11, S + 11] for column-based area and [11, T + 11] for row-based area 5 WEIGHT MATRIX COMPUTATION The section describes the application of the iterative procedure to determine the weight matrix The algebraic reconstruction technique (ART), based on the Kaczmarz algorithm in 1937, is an iterative method for solving sparse systems of linear equations, and it is widely used to the problem of image reconstruction from projections in computerized tomography (CT) At each iterative step, the parameters of current state are obtained by projecting the previous results to next hyperplane defined by the linear equations In the work, we formulize the computation of weight matrix as a problem of image reconstruction The Kaczmarz method is more suitable to compute the weights of matrices than other traditional methods Section 51 briefly introduces the concept of the Kaczmarz method Section 52 describes the proposed approach, which consists of two procedures: (1) determination of the initial weight values and (2) iterative projection 51 The Kaczmarz Method The Kaczmarz method [7, 8] is an iterative algorithm and commonly used to solve a system of linear equations If a system of algebraic equation is given by

7 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1419 B11W1 + B12W2 + B13W3 + L+ B1 NWN = R1 B21W1 + B22W2 + B23W3 + L+ B2 NWN = R2 M B W + B W + B W + L+ B W = R M1 1 M2 2 M3 3 MN N M (9) where B i1, B i2, B i3,, B in, R 1, R 2, R 3,, R M is known, coefficients W 1, W 2, W 3,, W N are to be solved Fig 4 presents the determination of unknown parameters using the Kaczmarz method () i The projected result W r r ( i 1) j is obtained by projecting the preceding result W j onto the next hyperplane Accordingly, the unknown parameters can be obtained by a series of projection operations Each projection operation is described as, r r r r r W W r r B (10) ( i 1) ( W ) () i ( i 1) j Bi R i j = j i Bi Bi The above equation can be further modified to the form, where R Q W W B () i ( i 1) i i j = j + N ij 2 Bik k = 1 r r Q W B W B N ( i 1) ( i 1) i = j = k ik k = 1, (12) (11) B 21W 1 + B22W2 + B23W 3 + L+ B2 N WN = p2 W (2) W (1) W (0) B 11W 1 + B12W2 + B13W 3 + L+ B1 N WN = p1 Fig 4 An illustration of the projection operation of the Kaczmarz method for two equations

8 1420 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG W r(1) W (1,1) W (2,1) W (S,1) W c(1) W c(2) W c(s) W r(2) W (1,2) W (2,2) W (S,2) W r(t) W (1,T) W (2,T) W (S,T) (a) (b) (c) Fig 5 (a) The contribution weight of every column position of the background; (b) The contribution weight of every row position of the background; (c) The contribution weight of every axisaligned region of the background 52 Computation Procedures The computation procedures of weight matrix by using the Kaczmarz method contain two steps: Step 1: Determination of Initial Weight Values Figs 5 (a) and (b) show the distribution of the contribution weights that correspond to every column and row of the background Analyzing the spectrum of the Fourier transform of column-based and row-based background patterns, we can get the matrices (W c (1), W c (2),, W c (S)) and (W r (1), W r (2),, W r (T)), corresponding to each position of the imaging plane These matrices describe the contribution of the background to each pixel Next, these matrices are normalized to yield (W c (1), W c (2),, W c (S)) and (W r (1), W r (2),, W r (T)) Then, the normalized values in the weight matrix (Fig 5 (c)) are computed to be W (s, t) = W c (s)w r (t), s = 1, 2,, S, t = 1, 2,, T (13) After executing the de-normalization process, we obtain the initial weight matrix for the axis-aligned region (Eq (14)) Wc () s W(,) s t = W (,) s t, W () s c s = 1, 2,, S, t = 1, 2,, T (14) The distribution of the contributions of light source to every pixel of the imaging plane is now derived Every pixel of the imaging plane can yield a set of weights of all axis-aligned regions Step 2: Iterative Projection Let B(f c, t) denote the pixel value of column and row background patterns, and R n represent the value that corresponds to the nth background pattern in the imaging plane These two items together with the weight matrix W(s, t) are related as follows s= S, t= T BsnWst (, ) (,) = Rn (15) s= 1, t= 1

9 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1421 where B(s, n) represents the pixel value B(f c, t) of column-based and row-based background patterns Then, these equations can be described as two sets of simultaneous equations (Eqs (16) and (17)) B(1,1) W(1,1) + B(2,1) W(2,1) + L+ B( S,1) W( S,1) + B(1,1) W(1,2) + B(2,1) W(2,2) + L+ B( S,1) W( S,2) + L+ B(1,1) W(1, T) + B(2,1) W(2, T) + L+ B( S,1) W( S, T) = R1 B(1,2) W(1,1) + B(2,2) W(2,1) + L+ B( S,2) W( S,1) + B(1,2) W(1,2) + B(2,2) W(2,2) + L+ B( S, 2) W( S,1) + L+ B(1,2) W(1, T) + B(2,2) W(2, T) + L+ B( S,2) W( S, T) = R2 M B(1, NW ) (1,1) + B(2, NW ) (2,1) + L+ BSNW (, ) ( S,1) + B(1, NW ) (1, 2) + B(2, NW ) (2, 2) + L+ BSNW (, ) ( S, 2) + L+ B(1, N) W(1, T) + B(2, N) W(2, T) + L+ B( S, N) W( S, T) = RN B(1,1) W(1,1) + B(1,1) W(2,1) + L+ B(1,1) W( S,1) + B(2,1) W(1,2) + B(2,1) W(2,2) + L+ B(2,1) W( S,2) + L+ BT (,1) W(1, T) + BT (,1) W(2, T) + L+ BT (,1) W( ST, ) = R1 B(1,2) W(1,1) + B(1,2) W(2,1) + L+ B(1,2) W( S,1) + B(2,2) W(1,2) + B(2,2) W(2,2) + L+ B(2, 2) W( S,1) + L+ BT (,2) W(1, T) + BT (,2) W(2, T) + L+ BT (,2) W( ST, ) = R2 M B(1, N) W(1,1) + B(1, N) W(2,1) + L+ B(1, N) W( S,1) + B(2, N) W(1,2) + B(2, N) W(2,2) + L+ B(2, N) W( S,2) + L+ BT (, NW ) (1, T) + BT (, NW ) (2, T) + L+ BT (, NW ) ( ST, ) = RN (16) (17) where B(1, 1), B(1, 2),, B(S, N),, B(T, N) are the results of sampling B(f c, t) of Eq (8); R 1, R 2,, R N are the pixel values at the same locations on different captured images, and W(1, 1), W(1, 2),, W(S, T) are the parameters of the weight matrix The iterative Kaczmarz procedure (Eqs (11) and (12)) is employed to solve the simultaneous equations, yielding the weight matrix 6 EXPERIMENTAL RESULTS This section demonstrates the performance of the proposed algorithm A Canon EOS D60 is used as the acquisition device A P4-24G CPU with 1G byte RAM is used and the operating system is Windows XP An LCD monitor is used to project the background patterns, and the digital camera records the images in the response of each backdrop

10 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG 1422 pattern The size of the backdrop pattern is set to pixels, and the size of the axis-aligned region is set to 4 4 pixels Fig 6 displays the acquisition environment Fig 6 Acquisition environment Fig 7 (a) Real picture with new background and object (b) (i) (c) (j) (d) (k) (e) (l) (f) (m) (g) (n) (h) (o) (b-h) are the results by using the proposed method (i-o) by using the with Zhu s method Fig 7 Illustration of composition results The thresholds are max/2, max/4, max/8, max/16, max/32, max/64, and max/128, respectively

11 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1423 Experiment 1: In the experiment, the composited results of a glass and three background images are presented to evaluate the proposed algorithm After information is obtained from all images, the spectrum of every pixel of the imaging plane is analyzed to compute the weight matrix Meanwhile, a threshold is established by examining the largest magnitude from the spectra of every pixel, and only the magnitudes that exceed the threshold are retained Therefore, only those light source positions with larger contribution need to be calculated, and the values of the other positions are set to zero (a) Real picture with new background and object (b) (i) (c) (j) (d) (k) (e) (l) (f) (m) (g) (n) (h) (o) (b-h) are the results by using the proposed method (i-o) by using the with Zhu s method Fig 8 Illustration of composition results The thresholds are max/2, max/4, max/8, max/16, max/32, max/64, and max/128, respectively

12 1424 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG (a) Real picture with new background and object (b) (i) (c) (j) (d) (k) (e) (l) (f) (m) (g) (n) (h) (o) (b-h) are the results by using the proposed method (i-o) by using the with Zhu s method Fig 9 Illustration of composition results The thresholds are max/2, max/4, max/8, max/16, max/32, max/64, and max/128, respectively The results of the proposed method and Zhu s method are compared, with seven thresholds (max/k, k = 2, 4, 8, 16, 32, 64, 128) are set to analyze and compare their visual results Figs 7, 8 and 9 (b-h) present the composited images obtained using the proposed algorithm and Figs 7, 8 and 9 (i-o) show the results obtained using Zhu s method The visual performance of the proposed algorithm clearly exceeds that of the other The PSNRs obtained from the two algorithms are also compared Fig 10 shows the PSNR vs threshold curves for the two algorithms The proposed algorithm maintains PSNR around 35, but the PSNR of Zhu s algorithm is between 254 and 3262

13 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS 1425 * represents the result of the proposed method, and the corresponding PSNRs are 3435, 3480, 3506, 3515, 3508, 3499, and represents the result of Zhu s method, and the corresponding PSNRs are 2759, 3080, 3210, 3262, 3124, 2893, and 2545 Fig 10 PSNR comparison The PNSR is lower when the threshold is higher (such as max/2), because the contribution of the light source is limited to only a few positions However, selecting a lower threshold, such as max/128 increases the noise in the spectrum The noise from the LCD screen or the digital camera influences PSNR if the threshold is set low This phenomenon is evident in Zhu s method, but the proposed method is robust against such a situation Table 1 presents the times required under different thresholds We select a rectangular area of the imaging plane, and then compare the consuming time for calculating the weight matrices of all pixels inside this rectangle Since the proposed approach solves the simultaneous equations iteratively, it takes more time than Zhu s method at the same threshold However, Zhu s method achieves its highest PSNR(3262) at the threshold of max/16, and it takes 61437s Our method achieves a higher PSNR(3435) at the threshold max/2, and it takes only 55343s Therefore, the proposed approach can use a higher threshold to reduce the computing time while obtaining PSNR that exceeds that of Zhu s method For example, if max/11 is chosen to be the threshold, the computing time can be further reduced to 37641s and the PSNR can maintain 3382 Table 1 The computing time (s) for different thresholds Max/2 Max/4 Max/8 Max/16 Max/32 Max/64 Max/128 Zhu s method Our method Experiment 2: The experiment demonstrates the visual effect of a transparent object with and without foreground lighting Fig 11 (a) presents the new background image The composited image in Fig 11 (b) considers only the light source behind the object in the synthesis However, Fig 11 (c) is associated with not only the light source behind the object but also foreground lighting Comparing the composited image in Fig 11 (b) and

14 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG 1426 the composited image with the foreground lighting in Fig 11 (c) shows that the latter appears rather visually lifelike because it exhibits the specular effect Fig 11 (d) shows a captured image, where a transparent glass cup is placed in front of the LCD screen Because we need to highlight the surrounding of object while getting foreground color, the LCD screen behind the object also reflects light that is recorded by digital camera Consequently, the actual image is brighter than the synthesized image (Fig 11 (c)) Three composited videos are constructed for the transparent model Fig 12 shows the results of object motion, object rotation, and object scaling (a) (b) (c) (d) Fig 11 (a) New background image; (b) The compositing image just takes the light source behind into consideration for synthesizing; (c) The compositing image involves the LCD light source and the foreground lighting; (d) The real captured image (a) Object motion (b) Object rotation (c) Object scaling Fig 12 Images of composited videos

15 FREQUENCY-BASED ENVIRONMENT MATTING OF TRANSPARENT OBJECTS CONCLUSION This work develops a new frequency-domain matting algorithm for improving the composited image by refining the weight matrices The proposed algorithm can generate result images of good quality and with a higher PSNR than earlier methods Additionally, the image quality can be maintained while the threshold is set high to reduce the computational time Results of this study demonstrate that the composited result is favorable even when the threshold is set to a high value REFERENCES 1 D E Zongker, D M Werner, B Curless, and D H Salesin, Environment matting and compositing, in Proceedings of ACM SIGGRAPH, 1999, pp , 2 Y Y Chuang, D E Zongker, J Hindorff, B Curless, D H Salesin, and R Szeliski, Environment matting extensions: Towards higher accuracy and real-time capture, in Proceedings of ACM SIGGRAPH, 2000, pp W Matusik, H Pfister, R Ziegler, A Ngan, and L McMillan, Acquisition and rendering of transparent and refractive objects, in Proceedings of the 13th EURO- GRAPHICS Workshop on Rendering, 2002, pp Y Wexler, A W Fitzgibbon, and A Zisserman, Image-based environment matting, in Proceedings of EUROGRAPHICS Workshop on Rendering, 2002, pp P Peers and P Dutre, Wavelet environment matting, in Proceedings of EURO- GRAPHICS Workshop on Rendering, 2003, pp J Zhu and Y H Yang Frequency-based environment matting, in Proceedings of the 12th Pacific Conference on Computer Graphics and Applications, 2004, pp A C Kak and M Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, K Tanabe, Projection method for solving a singular system of linear equation and applications, Numerical Mathematics, Vol 17, 1971, pp T Porter and T Duff, Compositing digital images, Computer Graphics, Vol 18, 1984, pp J Li, S Xiao, X Yang, and L Ma, Dynamic environment matting using radiance, International Conference on Computational Intelligence for Modelling Control and Automation, 2006, pp I-Cheng Chang ( ) received the BS degree in Nuclear Engineering in 1987, and the MS and PhD degrees in Electrical Engineering in 1991 and 1999, respectively, all from National Tsing Hua University, Hsinchu, Taiwan In 1999, he joined Opto-Electronics and Systems Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, as an engineer and project leader In the autumn of 2003, he

16 1428 I-CHENG CHANG, TIAN-LIN YANG AND CHUNG-LIN HUANG joined the Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, Taiwan, and he is currently an assistant professor His research interests include image/video processing, computer vision and graphics, and multimedia system design Dr Chang received the Annual Best Paper Award from Journal of Information Science and Engineering in 2002, and the Research Awards from Industrial Technology Research Institute in 2002 and 2003 He is a member of the IEEE, and the IPPR of Taiwan, ROC Tian-Lin Yang ( ) was born in Taiwan, ROC, on April 23, 1977 He received the BS degree in Electrical Engineering from National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, in 2002, and MS degree in Electrical Engineering from National Tsing Hua University, Hsinchu, Taiwan, in 2005, respectively Chung-Lin Huang ( ) received his BS degree in Nuclear Engineering from the National Tsing Hua University, Hsinchu, Taiwan, ROC, in 1977, and MS degree in Electrical Engineering from National Taiwan University, Taipei, Taiwan, ROC, in 1979 respectively He obtained his PhD degree in Electrical Engineering from the University of Florida, Gainesville, FL, USA, in 1987 From 1987 to 1988, he worked for the Unisys Co, Orange County, CA, USA, as a project engineer Since August 1988 he has been with the Electrical Engineering Department, National Tsing Hua University, Hsinchu, Taiwan, ROC Currently, he is a professor in the same department In 1993 and 1994, he had received the Distinguish Research Awards from the National Science Council, Taiwan, ROC In Nov 1993, he received the best paper award from the ACCV, Osaka, Japan, and in Aug 1996, he received the best paper award form the CVGIP Society, Taiwan, ROC In Dec 1997, he received the best paper award from IEEE ISMIP Conference held Academia Sinica, Taipei In 2002, he received the best paper annual award from the Journal of Information Science and Engineering, Academia Sinica, Taiwan His research interests are in the area of image processing, computer vision, and visual communication Dr Huang is a senior member of IEEE