Fully smoothed L1-TV energies : properties of the optimal solutions, fast minimization and applications. Mila Nikolova

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1 Fully smoothed L1-TV energies : properties of the optimal solutions, fast minimization and applications Mila Nikolova CMLA, ENS Cachan, CNRS 61 Av. President Wilson, F Cachan, FRANCE nikolova@cmla.ens-cachan.fr PGMO group MAthematics and Optimization for Imaging (MAORI) ENSTA Paristech, October 4th 2013

2 A fully smoothed l 1 -TV model [Nikolova, Wen and Chan, JMIV 2012] J(u, f) := Ψ(u, f) + βφ(u), β > 0 Ψ(u, f) := i I n ψ(u[i] f[i]) and Φ(u) := i I r φ(g i u) {g i R 1 n : i I r } forward discretization, options: N4 Only vertical and horizontal differences; N8 Diagonal differences are added. G := [ g T 1,, g T r ] T R r n. ψ( ) := ψ(, α 1 ) : R R and φ( ) := φ(, α 2 ) : R R where (α 1, α 2 ) > 0. They belong to the family of functions θ(, α) : R R satisfying (we set θ (t, α) := d θ(t, α)): dt H 1 For any α > 0 fixed, θ(, α) is C s 2 -continuous, even and θ (t, α) > 0, t R. H 2 For any α > 0 fixed, θ (t, α) < 1 and for t > 0 fixed, it is strictly decreasing in α > 0 with α > 0 lim t θ (t, α) = 1, t R lim α 0 θ (t, α) = 1 and lim α θ (t, α) = 0. 2

3 f1 f2 f3 θ θ θ t t2 + α t2 + α ( α log cosh t α log ( )) t α ( 1 + t α ) ( ) t 1 tanh α α t α + t ( 1 α ( t2 + α ) 3 ( ( )) ) 2 t tanh α α (α + t ) 2 Table 1: Relevant choices for θ(, α) obeying H1 and H θ(t) = t 2 + α θ (t) = t t 2 +α ξ(y) = (θ ) 1 (y) = y α 1 y 2 Assumptions H1 and H2. Function f1: plots for α = 0.05 ( ) and for α = 0.5 ( ). 3

4 Rationale for the choice of J: û tends to decrease the quantization noise Real-valued original f quantized on {0,, 15} Restored f In general fully smoothed L1 - TV are slow to minimize with high precision... 4

5 3. Preliminary facts on J The properties below, derived in [Nikolova, Wen, Chan 12] and [Bauss, Nikolova, Steidl, JIMV 2013] using H1 and H2, underly the proposed algorithm. For any β > 0 and any f, J(, f) has a unique minimizer û. For any β > 0 and any f K n, where K n is a dense open subset of R n, we have û[i] û[j] i, j I n, i j and û[i] f[i] i I n. (1) For any f R n, û satisfies (1) with a very high probability. There is an inverse function ξ(, α 1, ) := (ψ ) 1 (, α 1 ) : ( 1, 1) R which is odd and C s 1. α 1 ξ(y, α 1 ) is strictly increasing on (0, + ), for any y (0, 1). Set η := G 1. If βη < 1 then û f ξ ( ) βη, α 1. This bound is independent of f. And, û f ξ ( ) βη, α 1 as α2 0. 5

6 Semi-Explicit formula for the minimizer J(û, f) = 0 for any i I n ψ (û[i] f[i]) = β φ (g j û)g j [i] û[i] = f[i] + ξ β φ (g j û)g j [i]. j I r j Ir f1 f2 f3 ξ ξ α α y 1 y 2 ( 1 y 2 ) 3 α 2 ln 1 + y α 1 y 1 y 2 αy 1 y α 1 y Table 2: The inverse function ξ(y, α) = (θ ) 1 (y, α) and its derivative ξ with respect to y for all functions in Table 1. 6

7 4. A fast fixed point (FP) algorithm [Nikolova 13] Initialization: u 0 = f for k = 1, 2, u k+1 = X (u k ), X (u) := f ξ ( β Φ(u) ), with ξ applied componentwise. Theorem 1 Let (α 1, α 2 ) > and β > 0 be chosen so that βη < 1 and Then the iteration X converges. β ξ (βη) φ (0, α 2 ) G T G < 1. (2) Corollary 1 (Nikolova, Steidl 13) Let (α 1, α 2 ) > 0 and β > 0 be such that βη < 1. Then u k f ξ(βη, α 1 ) All iterates live in the same ball 7

8 Some values: If G corresponds only to vertical and horizontal differences, then η = G 1 = 4 and G T G = 8. For ψ and φ given by f1 in Table 1 and α 1 = 0.05, α 2 = 0.3 and β = 0.1 X (u) and û f If G also includes diagonal differences, η = G 1 = 8 and G T G = 16. For ψ and φ given by f1 and α 1 = 0.02, α 2 = 0.4 and β = 0.07 we have X (u) and û f FP algorithms are not reputated for their speed... Here we take advantage of two facts: We know the right initialization; All expressions in the algorithm are explicitly known. 8

9 4. Application: Histogram Specification (HS) f input digital gray value M N image / stored as an n := MN vector f[i] {0,, L 1} i I n := {1,, n} 8-bit image L = 256 Histogram H f [k] := 1 n {f[i] = k : i I n} k I L Target histogram ζ = (ζ[1],, ζ[l]) Goal of HS: convert f into f so that H f = ζ order the pixels in f: i j if f[i] < f[j] i 1 i 2 i ζ[1] }{{} ζ[1] i n ζ[l]+1 i n }{{} ζ[l] Ill-posed problem for digital (quantized) images since n L An issue: obtain a meaningful total strict ordering of all pixels in f Histogram equalization (HE) is a particular case of HS where ζ[k] = n/l k I L 9

10 Histogram Equalization original image HE by histeq HE by sort HE ours

11 Modern sorting algorithms (and a smoothed L1-TV) For any pixel f[i], extract K auxiliary information, a k [i], k I K, from f. Set a 0 := f. Then i j if f[i] f[j] and a k [i] < a k [j] for some k {0,, K}. Local Mean Algorithm (LM) [Coltuc, Bolon, Chassery, IEEE TIP, 2006] Idea: use the local mean intensity of a pixel s neighborhood. If two pixels have the same intensity and their local mean is the same, take a larger neighborhood. Repeat until all pixels are ordered. Edges are smoothed. Authors: 6 iterations are enough. Wavelet Approach (WA) [Wan and Shi IEEE TIP, 2007] Idea: use wavelet coefficients from different subbands to order the pixels. If two pixels have the same intensity and the wavelet coefficients in the finest subband are the same, go to next coarser subband. If wavelet coefficients are different, order according to their absolute value. Repeat until all pixels are ordered. Unstable. Authors: 9 iterations are enough. Variational Approach (VA)- fully smoothed L1-TV [Nikolova, Chan, Steidl, Wen 12, 13] Choose α 1, α 2, β so that f û 0.1 and FP converges. Compute û the minimizer of J Sort the pixels of û. No special storage requirements! LM and WA can not cope with large images, e.g (Canon EOS 5D MkII) storage. 11

12 Comparison with the state-of-the-art sorting algorithms The numerical results in [Nikolova, Wen, Chan 12] have shown that SVA clearly outperforms its main competitors LM and WA in terms of quality and memory needs but not in speed. VA G N4, ψ(t)= t 2 +α 1, φ(t)= t 2 +α 2, α 1 =α 2 =0.05, β =0.1., then f û [Nikolova 13]: for FP algor theithm - stopping rule J(u k, f) [Nikolova, Wen, Chan 12]: Polak-Ribière (PR) CG minimization, stopping rule J(u k, f) 10 6 and maximum iterations 35; LM algorithm [Coltuc, Bolon, Chassery 06] for K = 6 WA algorithm [Wan, Shi, 07] for Haar wavelet and K = 9. A much faster FP algorithm for sorting - [Nikolova, Steidll 13 (report)] 12

13 airplane(512 2 ) baril (512 2 ) car (512 2 ) clock (512 2 ) couple (256 2 ) F-16 (512 2 ) house (512 2 ) Lena (512 2 ) man ( ) raffia (512 2 ) sand (512 2 ) stream (512 2 ) Digital images with gray values in {0,, 255} used to compare the algorithms. 13

14 Fail % CPU SVA LM WA SVA LM WA SVA SVA J 10 7 Image PR FP PR FP airplane baril car clock couple F houseb Lena man raffia sand stream means Fail denotes the percentage of pixels that could not be sorted in a strict way. CPU is in seconds. 14

15 Faithful total strict ordering: VA outperforms by far LM and WA. Memory requirements: VA needs only 1 image to sort the pixels; LM - needs 6 and WA - 9. Extensive tests (contrast compression restoration & HE inversion) [Nikolova, Wen, Chan 12] The proposed FP minimization much faster and precise than PR - CG. LM WA Ours Man - zoom of the residuals in HE Inversion. 15

16 Enhancement of color images [Nikolova 13] Strict ordering for the intensity channel HS for the intensity channel Hue and Range-preserving color assignment Fast algorithm in RGB 16

17 Input image ( ) HS by [Han, Yang, Lee 11] ACE (IPOL) HS - ours 17

18 Input image ( ) HS by [Han, Yang, Lee 11] ACE (IPOL) HS - ours 18