ARS Workshop Lucas Létocart ARS Workshop Context Markov Random Fields minimization and minimal cuts in Exact total variation minimization image restoration Total variation and regularization TV models October 31, 2008 Minimization Minimal cut (graph cut) as energy minimization Notations General principle Maximum flow / minimal cut Energy representation Results More results Further results for 3D images Conclusion Conclusion Lucas Létocart Perspectives François Malgouyres Nicolas Lermé LIPN and LAGA – Université Paris 13 1
Outline ARS Workshop Lucas Létocart Context 1 Context Exact total variation minimization Total variation and 2 Exact total variation minimization regularization Total variation and regularization TV models TV models Minimization Minimization Minimal cut (graph cut) as energy minimization Notations General principle 3 Minimal cut (graph cut) as energy minimization Maximum flow / minimal cut Notations Energy representation General principle Results Maximum flow / minimal cut More results Energy representation Further results for 3D images Results More results Conclusion Further results for 3D images Conclusion Perspectives 4 Conclusion Conclusion Perspectives 2
ARS Workshop Lucas Létocart Context Exact total variation minimization Total variation and regularization TV models Minimization Minimal cut (graph cut) as Context energy minimization Notations General principle Maximum flow / minimal cut Energy representation Results More results Further results for 3D images Conclusion Conclusion Perspectives 3
Main context ARS Workshop Lucas Létocart Context Exact total variation minimization Total variation and Image degradation regularization TV models v = Hu + η Minimization Minimal cut (graph cut) as v → Observed image energy minimization Notations u → Original image General principle η → Noise Maximum flow / minimal cut H → Linear degradation Energy representation Results More results Goal Further results for 3D images Conclusion Obtain the best estimation ¯ u from u when H = identity . Conclusion Perspectives 4
Energy minimization ARS Workshop Lucas Létocart Context Exact total variation minimization First approach Total variation and regularization TV models Restoration corresponds to find the minimum of Minimization X Minimal cut (graph cut) as E ( u , v ) = F p ( u p , v p ) energy minimization p ∈ Ω Notations General principle with Ω ⊂ R 2 . Maximum flow / minimal cut Energy representation Inverse problem (Hadamard) ⇒ noise amplification (when H � = Id). Results More results Need to regularize the solution. Further results for 3D images X X ∀ β ∈ R + E ( u , v ) = F p ( u p , v p ) + β · G p , q ( u p , u q ) ∗ Conclusion | {z } | {z } p ∈ Ω p , q ∈ Ω Conclusion Data fidelity term { p , q }∈N Regularization Perspectives 5
Energy minimization ARS Workshop Lucas Létocart Context Exact total variation minimization Standard minimization methods Total variation and regularization → Continuous TV models Gradient descent. Minimization Graduated Non Convexity (GCN). Minimal cut (graph cut) as energy minimization Notations → Discrete General principle Maximum flow / minimal cut Dynamic programming (only in 1D). Energy representation Simulated annealing. Results Iterated Conditional Modes. More results Further results for 3D images Conclusion Problems Conclusion Perspectives No or poor convergence guarantees. Solution not ever optimal. 6
ARS Workshop Lucas Létocart Context Exact total variation minimization Total variation and regularization TV models Minimization Minimal cut (graph cut) as Exact total variation minimization energy minimization Notations General principle Maximum flow / minimal cut Energy representation Results More results Further results for 3D images Conclusion Conclusion Perspectives 7
Regularization ARS Workshop Lucas Létocart Regularization (Tikhonov) Context Exact total variation From : Introduce by A. N. Tikhonov in 1963 minimization Goal : Consider restoration as find the minimum of Total variation and regularization Z 1 TV models | u ( x ) | p dx ) E ( u ) = � u − v � 2 L 2 + β · �∇ u � 2 � u � Lp = ( p where L 2 Minimization Ω Minimal cut (graph cut) as energy minimization Notations Problem General principle Maximum flow / minimal cut Energy representation Results More results Further results for 3D images Conclusion Conclusion “Cubes” image σ b = 30 Tikhonov restoration Perspectives Solution Regularize differently. Decrease the weight of big gradients. 8
BV Space ARS Workshop Lucas Létocart Context Definition Exact total variation minimization BV ⇒ Space of functions with bounded variations. Total variation and regularization TV models Z BV (Ω) = { u ∈ L 1 (Ω) | |∇ u | < + ∞} Minimization Ω Minimal cut (graph cut) as energy minimization Exact definition uses duality, because |∇ u | can be a measure. Notations General principle with the semi-norm Maximum flow / minimal cut Energy representation Z Results | u | BV = |∇ u | = TV ( u ) ⇒ Total Variation More results Ω Further results for 3D images Conclusion Advantages Conclusion Perspectives Discontinuities are authorized along curves. Good space for geometric images. Existence and unicity of the solution. 9
Total Variation ARS Workshop Lucas Létocart Context Definition (co-area – continuous ) Exact total variation Let u ∈ BV (Ω) . Total variation of u is minimization Total variation and regularization Z Z Z TV ( u ) = |∇ u | = ds d λ, TV models Ω d { u ≤ λ } R Minimization Minimal cut (graph cut) as where { u ≤ λ } is equivalent to { u ( x ) ∈ Ω | u ( x ) ≤ λ } . energy minimization Notations General principle Definition (co-area – discrete) Maximum flow / minimal cut Energy representation Let u be a discrete function. Total variation of u is Results More results L − 2 Further results for 3D images X X w p , q | u λ p − u λ u λ TV ( u ) = q | p = 1 { up ≥ λ } where Conclusion λ = 0 { p , q }∈N Conclusion Perspectives Remarks (-) Details suppression (textures). (+) Allows sharp contours. 10
TV models ARS Workshop Lucas Létocart Context Definition Exact total variation minimization Let v ∈ L 1 (Ω) the observed image. The TV model consist of finding Total variation and regularization TV models TV ( u ) + β � u − v � α argmin α ∈ { 1 , 2 } L α Minimization u ∈ BV (Ω) Minimal cut (graph cut) as energy minimization Notations TV + L 2 Model / ROF (Rudin Osher Fatemi 92) General principle Maximum flow / minimal cut (+) Strictly convex ⇒ unicity. Energy representation (-) Lost of contrast (iterative regularization). Results More results Gaussian noise. Further results for 3D images Conclusion TV + L 1 Model (Nikolova 2004) Conclusion Perspectives (-) Convex ⇒ not unicity. (+) No contrast lost. Impulsive noise. 11
Level set approach ARS Workshop Lucas Létocart Context Exact total variation minimization Principle Total variation and regularization 1 Decompose the image in order to solve a succession of quadratic binary TV models u λ (MRF) optimization problems ¯ Minimization u λ where the solution is a level set Solve each problem ¯ 2 Minimal cut (graph cut) as u λ (trivial) energy minimization 3 Reconstruct ¯ u from ¯ Notations General principle Maximum flow / minimal cut Level set decomposition λ Energy representation Upper-set → U λ ( u ) = { p ∈ Ω | u p ≥ λ } Results More results Lower-set → L λ ( u ) = { p ∈ Ω | u p ≤ λ } Further results for 3D images Conclusion Conclusion Reconstruction Perspectives u p = sup { λ ∈ L | p ∈ U λ ( u ) } ∀ p ∈ Ω 12
Level set approach ARS Workshop Lucas Létocart Context Exact total variation minimization Total variation and regularization Reformulation - TV + L 1 TV models Minimization X E λ 1 ( u λ ) = TV ( u λ ) + β [( 1 − y p ) u λ p + y p ( 1 − u λ argmin p )] Minimal cut (graph cut) as u λ ∈{ 0 , 1 } N p ∈ Ω energy minimization Notations with General principle y p = 1 { vp ≥ λ } Maximum flow / minimal cut Energy representation Results Reformulation - TV + L 2 More results “ ” Further results for 3D images X E λ 2 ( u λ ) = TV ( u λ ) + 2 β ( λ − 0 . 5 ) u λ p + v p ( 1 − u λ argmin p ) Conclusion u λ ∈{ 0 , 1 } N p ∈ Ω Conclusion Perspectives 13
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