2 Poisson Image Editing DC DC 2 Poisson Image Editing Agarwala 3 4 Agarwala Poisson Image Editing Poisson Image Editing f(u) u 2 u = (x

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1 Poisson Image Editing Poisson Image Editing Stabilization of Poisson Equation for Gradient-Based Image Composing Ryo Kamio Masayuki Tanaka Masatoshi Okutomi Poisson Image Editing is the image composing processing that uses the image gradient information. However, Poisson Image Editing synthesizes an unnatural image for some input images. We propose an image composing algorithm to generate natural images by optimizing over the entire image. Agarwala et al proposed the algorithm by optimizing over the entire image. However, their algorithm is unstable, especially DC and low-frequency components. In addition, they did not mention about an implementation in a discrete system. First, we clarify issues for the implementation of the Poisson equation in the discrete system. Then, we also provide an solution for the implementation issues. 1 Tokyo Institute of Technology 1 1. 1)2) 2 3) Poisson Image Editing 1)2) Poisson Image Editing Poisson Image Editing Agarwala 2) Agarwala DC 1 c 2010 Information Processing Society of Japan

2 Poisson Image Editing DC DC 2 Poisson Image Editing Agarwala 3 4 Agarwala 5 2. 2.1 Poisson Image Editing 2.1.1 Poisson Image Editing f(u) u 2 u = (x, y) = (, ) f(u) x y f(u) Poisson Image Editing ˆf(u) g in(u) 1 ˆf(u) = arg min f(u) 2 f(u) g in(u) 2 du (1) Ω Ω f(u) = g in (u) with f(u) Ω = g bk (u) Ω (2) 3 = 2 x 2 + 2 y 2 (3) Ω Ω g bk (u) (2) Poisson Image Editing Poisson image Editing Ω 2.1.2 Poisson Image Editing (2) Ω 2 f lp (i, j) (4) f lp (i, j) = f(i, j + 1) + f(i, j 1) + f(i + 1, j) + f(i 1, j) 4f(i, j) (4) f(i, j) (i, j) f D f lp f lp = Df (5) g in (i, j) g bk (i, j) g in(i, j) 2 2 c 2010 Information Processing Society of Japan

4 Agarwala Agarwala Ω 0 g bk (u) g in (u) v(u) v(u) (9) { g in(u) u Ω v(u) = (9) g bk (u) otherwise g in lp (i, j) Poisson Image Editing Ω g in lp (i, j) g bk (i, j) Ω Ω (2) (2) Ω (i, j) f lp (i, j) = g in lp (i, j) (6) (4) Ω f(i, j) (7) f(i, j) = (f(i, j + 1) + f(i, j 1) + f(i + 1, j) + f(i 1, j) g in lp (i, j))/4 (7) Ω (7) n Ω f n (i, j) f n (i, j) = (f n 1 (i, j + 1) + f n 1 (i, j 1) + f n 1 (i + 1, j) + f n 1 (i 1, j) g in lp (i, j))/4 (8) 3 Ω (8) Poisson Image Editing Ω 2 2.2 Agarwala Agarwala Ω 0 ˆf(u) 1 ˆf(u) = arg min f(u) Ω 0 2 f(u) v(u) 2 du (10) Poisson Image Editing f(u) = div v(u) with f(u) Ω0 = 0 (11) Ω 0 0 2) (9) DC DC DC 3 3. 3.1 Agarwala Agarwala (11) (11) 2) g(u) g(u) = v(u) divv(u) divv(u) = g(u) (12) Agarwala v(u) (9) 3 c 2010 Information Processing Society of Japan

divv(u) = { g in(u) g bk (u) u Ω otherwise (13) (11) Df = q (14) q divv(u) (14) (15) E(f) = 1 2 Df q 2 2 (15) q v(u) 4 f E f = (DT q D T Df) (16) E f = 0 f Agarwala (11) DCT DCT 0 f Ω0 = 0 4) D 5 DCT W 5)6) D = W 1 diag(ξ)w (17) diag(ξ) ξ ξ D width height 2width 2height DFT 5)6) DCT DCT (16) (15) D T D T D D T = W 1 diag(ξ)w (18) D T D = W 1 diag(ξ 2 )W (19) E f = 0 f 5 DCT f = W 1 (diag(ξ)) 1 Wq (20) = W 1 [Wq ξ] (21) f q DCT Wq ξ IDCT DCT 5 5 (0,0) DC Agarwala DC 0 Agarwala DC 3.2 DC Agarwala h(u) { g in (u) u Ω h(u) = (22) g bk (u) otherwise Agarwala (22) E(f) = 1 2 Df q 2 2 + 1 2 ε f h 2 2 (23) ε q (13) divv(u) h (20) h(u) (21) 4 c 2010 Information Processing Society of Japan

6 7 ε 3.1 DCT f = W 1 (diag(ξ 2 ) + εdiag(1)) 1 (diag(ξ)wq + εdiag(1)wh) (24) = W 1 [(diag(ξ)wq + εdiag(1)) (ξ 2 + ε1)] (25) 1 1 (20) 0 4. Agarwala (13) Agarwala 4.1 Agarwala Agarwala 6 6(a),(b) 6(c) 6(d) f fwd (i) f lp (i) f fwd (i) = f(i + 1) f(i) (26) f lp (i) = f(i + 1) + f(i 1) 2f(i) (27) (26)(27) f lp (i) f fwd (i) f lp (i) = f fwd (i) f fwd (i 1) (28) x y 7 (20) 3.2 (20) DC DC 7(a) 4.2 Agarwala 4.1 Agarwala Agarwala 5 c 2010 Information Processing Society of Japan

8 10 9 2 x y x y 7(c) 7(b) g bk lp (i, j) g in lp (i, j) 7(b) (i) f fwd (i) f bk (i) 8 (i) (i + 1) (i) (i 1) 9 7(b) 4.3 4.2 2 2 2 f fwd (i) = f bk (i + 1) = (f fwd (i) + f bk (i + 1))/2 (29) 2 4.2 5. x x y y 5 3 4 10 (1) 5 (2) (3) (4) 4 (5) 6. Poisson Image Editing Agarwala Agarwala DC DC Agarwala 6 c 2010 Information Processing Society of Japan

11 1 12 2 ε 10 8 10 Poisson Image Editing Agarwala 7. Agarwala 2 Agarwala 1 DC Agarwala DC 2 Agarwala 1) Patrick Peretz Michel Gangnet Andrew Blake Poisson Image Editing 2002. 2) Aseem Agarwala Mira Dontcheva Maneesh Agrawala Steven Drucker Alex Colburn Brian Curless David Salesin Michael Cohen Interactive Digital Photomontage 2004 3) Carsten Rother Vladimir Kolmogorov Andrew Blake GrabCut Interactive Foreground Extraction using Iterated Graph Cuts 2004 4) Amit Agrawal, Rama Chellapa, Ramesh Rasker An Algebraic Approach to Surface Reconstruction from Gradient Fields 2005 5) B Chitprasert K R Rao DISCRETE COSINE TRANSFORM FIL- TERING 1990 6) 1995 7) Vladimir Britanak Patrick C Yip K R Rao Discrete Cosine and Sine Transforms: General Properties Fast Algorithms and Integer Approximations 8) K R Rao Patrick Yip Vladimir Britanak Discrete Cosine Transform: Algorithms Advantages Applications 9) 2008 7 c 2010 Information Processing Society of Japan

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