Zhang Wu Directional LMMSE Image Demosaicking

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2 Pascal Getreuer 2 LMMSE Estimation We review linear minimum mean-square error (LMMSE) estimation for scalar variables. Suppose X and Y are scalar random variables, and we wish to use the (assumed known) value of Y to estimate the (unknown) value of X. The minimum mean-square error (MMSE) estimate is ˆX MMSE := arg min F E[( F (Y ) X ) 2], (1) where F (Y ) is any function of Y. Under mild conditions the MMSE estimator is E[X Y ], the conditional expectation of X given Y. However, in many applications only limited information is available about X and Y. LMMSE estimation is useful in these cases since it only requires second-order statistics. The LMMSE estimator is the best estimate of the form The optimal coefficients a and b minimize the mean-square error Differentiating with respect to a and b yields ˆX LMMSE := a(y EY ) + b. (2) min E[( (a(y EY ) + b) X ) 2]. (3) a,b 0 = a E[( a(y EY ) + b X ) 2] = 2E [( a(y EY ) + b X ) (Y EY ) ] = 2 ( a Var Y Cov(X, Y ) ) a = Cov(X, Y ), (4) Var Y 0 = E[( a(y EY ) + b X ) 2] b = 2 ( ) b = EX. (5) b EX So the LMMSE estimate is and its mean-square error is ˆX LMMSE = Cov(X, Y ) (Y EY ) + EX, (6) Var Y E [ ( ˆX LMMSE X) 2] = a 2 Var Y 2a Cov(X, Y ) + Var X = Var X Cov(X, Y )2. (7) Var Y To avoid division by zero, a small value 0.1 is added (relative to the intensity range [0, 255]) in the denominators to Var Y. 3 LMMSE Denoising Zhang and Wu [4] describe the following simple method based on LMMSE estimation for signal denoising. Suppose that a smooth one-dimensional signal u n is observed with independent zeromean additive noise, f n = u n + ɛ n. (8) 118

3 Zhang Wu Directional LMMSE Image Demosaicking Assuming the noise ɛ has limited low frequency content, an effective estimate is s = f h where h is a lowpass filter. However, lowpass filtering is a balance between noise reduction and blurring the true signal content, which is especially problematic if the noise is spatially varying. Since u is assumed to be smooth, the lowpass filtered signal s should be closer than f to u. Statistics of s and (f s) can be used to estimate statistics of u and ɛ: µ = 1 2M+1 σ 2 u = 1 2M+1 σ 2 ɛ = 1 2M+1 M m= M M m= M M m= M s n+m Eu n (9) (s n+m µ) 2 Var u n (10) (s n+m f n+m ) 2 Var ɛ n, (11) where M is a fixed window radius. From these statistics we obtain the LMMSE denoised estimate σ2 u û n = µ + (f σu 2 + σɛ 2 n µ), (12) and an estimate of the error E[(û n u n ) 2 ] σu 2 (σ2 u) 2. (13) σu 2 + σɛ u f u s u û LMMSE denoising experiment. n The figure shows an LMMSE denoising experiment on a sequence of 100 samples with spatially varying non-gaussian noise. The top shows the exact signal u and the observed signal f. The middle shows the lowpass filtered signal s = f h. The bottom shows the LMMSE denoised signal. 119

4 Pascal Getreuer 4 Algorithm For typical natural images, the color channels are strongly correlated. Most demosaicking methods combine information across color channels to take advantage of this correlation. Let R n, G n, B n denote the color components of the nth pixel of a row of the image and define the primary difference signals Φ g,r (n) = G n R n, Φ g,b (n) = G n B n. (14) The algorithm outline is 1. The primary difference signals are estimated in the horizontal direction and denoised using LMMSE estimation. 2. Similarly, the primary difference signals are estimated and denoised in the vertical direction. 3. The two denoised directional estimates are fused to obtain a full resolution green channel. 4. The primary difference signals are interpolated and then subtracted from green to obtain the red and blue channels. We discuss each step in detail below. 4.1 Directional Estimates We focus on the estimation of a red row; the process is analogous for a blue row. For each pixel in a red row, either G n or R n but not both are known. Zhang and Wu suggest based on empirical evidence that it is accurate to assume that the primary difference signals are smooth. Second-order Laplacian interpolation is used to obtain the missing samples, which provides an estimate of the difference signal, Ĝ n = 1(G 2 n 1 + G n+1 ) 1(R 4 n 2 2R n + R n+2 ), (15) ˆR n = 1(R 2 n 1 + R n+1 ) 1(G 4 n 2 2G n + G n+2 ), (16) ˆΦ h g,r = {Ĝn R n G n ˆR n n is a red pixel, n is a green pixel. (17) The estimate is denoised as described in the section on LMMSE Denoising. The suggested parameters are a window radius M = 4 and an approximately Gaussian lowpass filter h(z) = (z + z 1 ) (z2 + z 2 ) (z3 + z 3 ) (z4 + z 4 ). (18) The LMMSE denoising error estimate is also noted since it will be used in the following step. 4.2 Fusion of the Directional Interpolations The previous steps produced horizontally and vertically directed estimates of the difference signals Φ g,r and Φ g,b. The directional estimates are now fused to yield the final estimate of the difference signals. Since the objective is to interpolate the green channel, fusion is only performed at red and blue pixel locations (where the green component is unknown). LMMSE estimation is applied again here to develop a method for fusion. 120

5 Zhang Wu Directional LMMSE Image Demosaicking Let h denote the horizontally directed estimate, ɛ h the estimation error, and σh 2 the LMMSE denoising error estimate, and similarly let v, ɛ v, and σv 2 denote the analogous quantities for the vertically directed estimate. The fused estimate is w = (1 λ)h + λv (19) where λ is selected to minimize the estimated mean-square error, 0 = λ E[( (1 λ)ɛ h + λɛ v ) 2 ] = 2E [ (1 λ)ɛ 2 h + λɛ 2 v + (1 2λ)ɛ h ɛ v ] 2(1 λ)σ 2 h + 2λσ 2 v, λ = σ2 h σh 2 +. (20) σ2 v In the last line, it is assumed that ɛ h and ɛ v are uncorrelated. Zhang and Wu claim that this is usually true, particularly on edges and texture where accurate fusion is important. To avoid division by zero, a small value 0.2 is added (relative to the intensity range [0, 255]) in the denominator when computing λ. With the fused estimates of Φ g,r and Φ g,b, the green component G n is now available at every pixel. Demosaicking of the green channel is now complete. 4.3 Obtaining the Red and Blue Channels Based on original Bayer samples and the fully interpolated green channel, primary difference signal Φ g,r = G R is known at red locations and Φ g,b = G B is known at blue locations. The primary difference signals are interpolated in two steps. First, Φ g,r is interpolated at blue locations by averaging its four diagonal neighbors, all of which are red locations, Φ g,r (i, j) = 1 4( Φg,r (i 1, j 1) + Φ g,r (i + 1, j 1) + Φ g,r (i 1, j + 1) + Φ g,r (i + 1, j + 1) ). (21) Primary difference signal Φ g,b is similarly interpolated at red locations. Second, Φ g,r and Φ g,b are interpolated at green locations by averaging their four axial neighbors, Φ g,r (i, j) = 1 4( Φg,r (i 1, j) + Φ g,r (i + 1, j) + Φ g,r (i, j 1) + Φ g,r (i, j + 1) ). (22) Finally, the red and blue channels are obtained as R = G Φ g,r and B = G Φ g,b. 5 Examples In the following examples, an exact image is mosaicked over the Bayer pattern and restored with Zhang Wu demosaicking and other demosaicking methods. The difference between the exact and demosaicked images is measured with the peak signal to noise ratio (PSNR) over all three color channels. Consider two color images A and B as vectors in R 3N, where N is the number of pixels and the components with values in [0, 255] represent RGB intensities, then the PSNR is PSNR(A, B) := 10 log 10 1 A. (23) 3N B 2 2 A typical artifact of demosaicking methods is the appearance of zipper patterns, a sequence of adjacent pixels of alternating brightness. It is also common to see false colors and desaturation. The test images here are selected to highlight these problems. 121

6 Pascal Getreuer To avoid effects from the image boundaries, a margin of 5 pixels has been removed before computing PSNR and in the displayed results. All images are enlarged by nearest neighbor interpolation to show individual pixels more clearly. Zhang Wu is especially competitive on images with aliasing. The fence image is increasingly difficult toward the right side. The red and blue channels are sampled well below the Nyquist rate needed to sample the fence slats. Exact Observed Image Bilinear (PSNR 20.42) Hamilton Adams [2] Gunturk et al. [3] Zhang Wu (PSNR 30.19) (PSNR 32.35) (PSNR 38.29) 122

7 Zhang Wu Directional LMMSE Image Demosaicking In the window image, the window blinds violate the Nyquist rate when subsampled by the CFA. While in the previous example Zhang Wu performs well on the aliased pattern of the fence, Zhang Wu is less successful on the abrupt T-junctions formed between the slats and the window frame. Exact Observed Image Bilinear (PSNR 23.98) Hamilton Adams [2] Gunturk et al. [3] Zhang Wu (PSNR 32.38) (PSNR 34.32) (PSNR 36.75) This example is on a mostly gray image of a sail. Demosaicking such an image is useful to compare to what degree the methods create false colors. Exact Observed Image Bilinear (PSNR 29.91) 123

8 Pascal Getreuer Hamilton Adams [2] Gunturk et al. [3] Zhang Wu (PSNR 37.45) (PSNR 40.95) (PSNR 41.65) The DLMMSE Zhang Wu scheme described here is known to work very well on the Kodak dataset. However, as shown by the same authors [5], the choice of dataset significantly affects the outcome when evaluating demosaicking algorithms. The Kodak dataset is somewhat gray and blurred, and Zhang Wu was designed for these kinds of images. Zhang Wu is less effective for example on the more colorful IMAX and McMaster datasets. For example, significant zipper artifacts appear on McMaster image 9: Exact Observed Image Bilinear (PSNR 31.08) Hamilton Adams [2] Gunturk et al. [3] Zhang Wu (PSNR 30.27) (PSNR 27.11) (PSNR 28.34) 124

9 Zhang Wu Directional LMMSE Image Demosaicking As another demonstration, the last example is on the peppers image. This image has significant color discontinuities at the edges, where the channel correlation assumption is false. Exact Observed Image Bilinear (PSNR 29.63) Hamilton Adams [2] Gunturk et al. [3] Zhang Wu (PSNR 29.40) (PSNR 27.02) (PSNR 28.09) Acknowledgments This material is based upon work supported by the National Science Foundation under Award No. DMS Work partially supported by the Office of Naval Research under grant N and by the European Research Council, advanced grant Twelve labours. Image Credits Kodak Image Suite, image 1 ( Kodak Image Suite, image 7 ( Kodak Image Suite, image 9 ( 125

10 Pascal Getreuer Kodak Image Suite, image 19 ( McMaster Dataset, image 9 ( Peppers standard test image References [1] B. E. Bayer, Color imaging array, U.S. Patent , [2] J. F. Hamilton, Jr. and J. E. Adams, Jr., Adaptive color plan interpolation in single sensor color electronic camera, U.S. Patent , [3] B. K. Gunturk, Y. Altunbasak, and R. M. Mersereau, Color plane interpolation using alternating projections, IEEE Transactions on Image Processing, vol. 11, no. 9, pp , http: //dx.doi.org/ /tip [4] Lei Zhang and Xiaolin Wu, Color demosaicking via directional linear minimum mean squareerror estimation, IEEE Transactions on Image Processing, vol. 14, no. 12, pp , [5] Lei Zhang, Xiaolin Wu, Antoni Buades, and Xin Li, Color Demosaicking by Local Directional Interpolation and Non-local Adaptive Thresholding, Journal of Electronic Imaging 20(2), ,

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