Exact discrete minimization for TV+L0 image decomposition models

Transcription

1 Exact discrete minimization for TV+L0 image decomposition models Loïc Denis 1, Florence Tupin 2 and Xavier Rondeau 2 1. Observatory of Lyon (CNRS / Univ. Lyon 1 / ENS de Lyon), France 2. Telecom ParisTech (Institut Telecom / CNRS LTCI), Paris, France this work has been funded by DGA under contract REI IEEE ICIP 2010 Image Enhancement II 28 Sept. 2010

2 Context Synthetic aperture radar (SAR) images denoising: = noisy data noise scene v n u noise / signal separation using a variational approach: recover scene u as the minimizer of E data (u, v) + E reg (u) Radar scene distinctive feature: strong scatterers (very bright dots) Q: How to model such scenes? Q: How to compute the corresponding minimizers? 1 / 10

3 Overview 1. TV+L0 image decomposition models 2. Exact discrete minimization by graph-cuts 3. Results and discussion

4 1. Denoising and image decomposition Total variation denoising: E reg (u) = TV(u) := û = arg min E data (u, v) + E reg (u) u ( x u) 2 + ( y u) 2 dx dy preserves sharp boundaries cartoon-like images (staircasing effect), favors larger regions Image decomposition: e.g., E data (u v) = u v 1 = + image texture geometry (illustration from [Yin, Goldfarb & Osher 2005]) 2 / 10

5 1. Denoising and image decomposition TV denoising vs TV+L0 image decomposition: 3 / 10

6 1. Denoising and image decomposition TV denoising vs TV+L0 image decomposition: 3 / 10

7 1. Denoising and image decomposition TV denoising vs TV+L0 image decomposition: 3 / 10

8 2. Energy minimization problem (u BV, u S ) = arg min (u BV,u S ) E data (v, u BV, u S ) + E reg (u BV, u S ) 4 / 10

9 2. Energy minimization problem (u BV, u S ) = arg min (u BV,u S ) E data (v, u BV, u S ) + E reg (u BV, u S ) 1. Image formation model Assumption: separable likelihood (no blurring, uncorrelated noise) E data = k log p(v k u BVk, u Sk ) Speckle noise Rayleigh distribution: E data = k Positivity constraints: v k 2 (u BVk +u Sk ) 2 +2 log(u BVk +u Sk ) k, u BVk > 0 and u Sk 0 4 / 10

10 2. Energy minimization problem (u BV, u S ) = arg min (u BV,u S ) E data (v, u BV, u S ) + E reg (u BV, u S ) 2. Image decomposition model Prior model: decomposition into sparse and bounded variations components E reg = β S L0(u S ) + β BV TV(u BV ) 4 / 10

11 2. Energy minimization problem (u BV, u S ) = arg min (u BV,u S ) E data (v, u BV, u S ) + E reg (u BV, u S ) Minimization problem E data is non-convex (quasi-convex) E reg is non-convex (due to L0 term) the problem is non-convex Variable coupling: - u BV and u S are coupled - u BV is spatially coupled global minimization is hard... 4 / 10

12 2. Energy minimization problem (u BV, u S ) = arg min (u BV,u S ) E data (v, u BV, u S ) + E reg (u BV, u S ) Minimization problem E data is non-convex (quasi-convex) E reg is non-convex (due to L0 term) the problem is non-convex Variable coupling: - u BV and u S are coupled - u BV is spatially coupled global minimization is hard... 4 / 10

13 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) which is of the form: arg min u BV k f k(u BVk ) + (k,l) g kl(u BVk, u BVl ) 3 Exact discrete minimization is possible with a maximum-flow / minimum s-t cut algorithm, due to the structure of the problem: it is the sum of a separable and a convex term involving only first-order cliques. 5 / 10

14 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) 5 / 10

15 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) 5 / 10

16 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) 5 / 10

17 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) which is of the form: arg min u BV k f k(u BVk ) + (k,l) g kl(u BVk, u BVl ) 3 Exact discrete minimization is possible with a maximum-flow / minimum s-t cut algorithm, due to the structure of the problem: it is the sum of a separable and a convex term involving only first-order cliques. 5 / 10

18 2. Energy minimization problem: reformulation 1 Consider u BV fixed. The restricted problem is spatially separable: u S (u BV) = arg min u S E data (v, u BV, u S ) + β S L0(u S )+β BV TV(u BV ) The problem reduces to a 1D problem per pixel (easy). 2 The original problem can be reformulated with u BV only: arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) which is of the form: arg min u BV k f k(u BVk ) + (k,l) g kl(u BVk, u BVl ) 3 Exact discrete minimization is possible with a maximum-flow / minimum s-t cut algorithm, due to the structure of the problem: it is the sum of a separable and a convex term involving only first-order cliques. 5 / 10

19 2. Energy minimization problem: graph-cuts methodology 1 The pixel grid is mapped to a graph with two terminal nodes: 2 A minimum s-t-cut is computed: 3 The cut is interpreted as a solution of the original problem: 6 / 10

20 2. Energy minimization problem: graph construction arg min u BV E data (v, u BV, u S (u BV)) + β S L0(u S (u BV)) + β BV TV(u BV ) - u BV is decomposed into its level sets - each level is represented by a layer of the graph - vertical arcs going downstream represent E data ( ) + β S L0( ) - horizontal arcs represent β BV TV(u BV) - positivity is naturally enforced 7 / 10

21 3. Results: TV vs TV+L0 decomposition Noisy data c CNES/DGA TV denoising 8 / 10

22 3. Results: TV vs TV+L0 decomposition Strong scatterers u S Homogeneous regions u BV 8 / 10

23 3. Results: TV vs TV+L0 decomposition Image decomposition: suppresses the bias on strong scatterers (i.e., loss of contrast and suppression of point-like objects) better preserves resolution (strong scatterers do not spread) 9 / 10

24 3. Results: TV vs TV+L0 decomposition Image decomposition: suppresses the bias on strong scatterers (i.e., loss of contrast and suppression of point-like objects) better preserves resolution (strong scatterers do not spread) 9 / 10

25 3. Results: TV vs TV+L0 decomposition Image decomposition: suppresses the bias on strong scatterers (i.e., loss of contrast and suppression of point-like objects) better preserves resolution (strong scatterers do not spread) 9 / 10

26 3. Results: TV vs TV+L0 decomposition Image decomposition: suppresses the bias on strong scatterers (i.e., loss of contrast and suppression of point-like objects) better preserves resolution (strong scatterers do not spread) 9 / 10

27 3. Results: TV vs TV+L0 decomposition Image decomposition: suppresses the bias on strong scatterers (i.e., loss of contrast and suppression of point-like objects) better preserves resolution (strong scatterers do not spread) 9 / 10

28 3. Conclusion The prior model benefits from image decomposition Decomposition choice: a component with bounded variations u BV and a sparse component u S Minimization of TV+L0 is challenging but exact discrete minimization is possible with graph-cuts A drawback of this minimization approach is its memory cost: O(number of pixels number of quantization levels) More elaborate speckle noise models (strong scatterer + random phasors) can be applied with the proposed decomposition for SAR image denoising ( Rice distribution, see paper) 10 / 10

29 Questions? the slides can be downloaded from my homepage ( 10 / 10