Generalized Newton-Type Method for Energy Formulations in Image Processing

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1 Generalized Newton-Type Method for Energy Formulations in Image Processing Leah Bar and Guillermo Sapiro Department of Electrical and Computer Engineering University of Minnesota

2 Outline Optimization in real functions gradient descent, Newton method Trust -region methods Optimization in variational framewor gradient descent, Newton method generalized Newton method Numerical simulations Conclusions 4-Mar-09

3 Introduction Optimization of a cost functional is a fundamental tas in image processing and computer vision Segmentation, denoising, deblurring, registration etc.. 4-Mar-09 3

4 Problem Statement n minimize f( x) : What is the best path? How to avoid maximum or saddle point? Can we impose some preferences on the path? NEW New optimization approach which incorporates nowledge/information 4-Mar-09 4

5 Descent Methods n minimize f( x) : xdom Given starting point Repeat 1. Compute a search direction d. Line search. Choose step size t >0 3. Update x:=x+td Until stopping criterion is satisfied f d f x 1 ( ) f ( x ) Gradient descent and Newton methods are most widely used in practice 4-Mar-09 5

6 Descent Methods Gradient descent derivation First order Taylor approximation f ( x d) f ( x) f ( x) d directional derivative Minimize w.r.t d As negative as we want 4-Mar-09 6

7 Descent Methods Gradient descent derivation First order Taylor approximation f ( x d) f ( x) f ( x) d d P Quadratic norm z 1/ n : P z P S P L Minimize w.r.t d 1 d P f x ( ) Newton step derivation second order Taylor approximation f ( x d) f ( x) f ( x) d f ( x) d d Quadratic convergence if L m f ( x) 1 mi L f ( x ) f ( x) m f( x) d f( x) 4-Mar-09 7 and ( ) ( ) f x f y L x y

8 Descent Methods The problem of the Newton method is that the solution may be attracted to a local maximum or saddle point if the Hessian is not positive definite Possible solution: Trust-region method. Basic concept Define a trust-region set min f ( x d) : d Define a model m (e.g. Taylor expansion) in the trust region Compute a step d that sufficiently reduces the model s.t. Accept the trial point if n B x : x x x d B f ( x ) f ( x d ) r : (0,0.5) m( x ) m( x d ) Update the trust region radius: if r < 0.5 then decrease if r > 0.75 then increase 4-Mar-09 8

9 Illustration of the Trust-Region Method f ( x, x ) 10x 10x 4sin( x x ) x x Mar-09 9

10 Illustration of the Trust-Region Method Conn, Gould, Toint, Trust-Region Methods, Mar-09 10

11 Convergence Results if f( x) twice-continuously differentiable f ( x) K lbf f ( x) K ufh The sequence f(x ) is strictly decreasing b lim f x 0 f x * 0 Super linear convergence (in CG with trust-region) f x 1 lim 0 f x Sorenson, SIAM J. Numerical Anal, 198 Mor e and Sorenson, SIAM J. Sci. Stat. Comput Steihaug, SIAM J. Num. Anal., 1983 Conn, Gould, Toint, Trust-Region Methods, Mar-09 11

12 Truncated Conjugate Gradients Approach set d 0, r g, v r if return d d ; for j 0,1,,... if Bv v 0 0 j1 j j j j1 1 m f ( x ) g d Bd d find such that d d d and d ; return d; set r r / Bv v ; set d d v ; if r d j j j j j j j find such that d d d and d set r r Bv ; if j1 j j j j1 0 return d d ; j1 set r r / r r set end r r j1 j1 j1 j j v r v j1 j1 j1 j j j j j ; return d; g f ( x ) 4-Mar-09 1 B Steihaug, SIAM J. Num. Anal., 1983 f ( x )

13 So Far Descent methods in real functions (gradient descent, Newton) Trust-region methods for numerical stability 4-Mar-09 13

14 What Next? Descent methods in real functions (gradient descent, Newton) Trust-region methods for numerical stability Can we go further? How can we modify and generalize optimization methods in variational framewor? Can we impose some nowledge by changing the metric of the model? 4-Mar-09 14

15 Optimization in Variational Framewor E( f ) : I x, f ( x), f ( x) dx min Gradient descent In the classical gradient descent 1 d X E( f ) arg min E( f, ) X L X Generalized gradient descent method: A new inner product is defined by u, v u, v 1 ( ) ( ) d E f E f L Symmetric and positive definite operator Prior on the deformation field in shape warping and tracing applications Charpiat, Maurel, Pons,Keriven, Faugeras, IJCV 007. Improved segmentation by Sobolev active contours. Sundaramoothi, Yezzi, Mennucci, VLSM 005, IJCV Mar-09 15

16 Generalized Newton Step Derivation E( f ) : I x, f ( x), f ( x) dx min 1 Q E f E f E f ( ) ( ) (, ) (, ) 1 Q ( ) E( f ) E( f ) Hessian E( f ) L L s.t. L L Is it good enough? 4-Mar-09 16

17 Geometric Active Contour 1 1 F( c, c, ) ( u c ) H( ) ( u c ) 1 H( ) g u H ( ) dx Casseles, Kimmel, Sapiro, IJCV 1997 level set function, u given image, c Chan-Vese, IEEE TIP 001 1, scalars g u 1 u / gradient descent Newton with trust-region 4-Mar-09 17

18 Newton Method with trust-region 4-Mar-09 18

19 Generalized Newton Step Derivation E( f ) : I x, f ( x), f ( x) dx min Q E f E f 1 L L E f ( ) ( ) ( ) Hessian ( ) s.t. Q 1 ( ) E ( f ) ( ) Hessian E f ( ) L L E f s.t. Leads to the following PDE E ( ) ( ) L B (self-adjoint operator) satisfies the convergence conditions! 4-Mar g s.t.

20 Generalized Newton Step Derivation Given starting point f Repeat 1. Compute a search direction : minimizing Q (. ) Solving Euler-Lagrange equation by truncated CG with trust region.. Update f:=f+ 3. Accept/reject f, update, update Until stopping criterion is satisfied 4-Mar-09 0

21 The Second Variation Besides the Euler-Lagrange equations, additional necessary condition for a relative minimum is that the second variation is nonnegative. ( E, ) 0 In the case of D R I, i, j {1,.. N} f xi f xj I I I ff ffx ff y x x x y E( f, ) x y I ffx I f f I f f x I ffy I f y x f I y f y f y Theorem: positive definite R(x) is a necessary condition for a relative minimum (strengthened Legendre condition) The matrix R will indicate the local convexity 4-Mar-09 1

22 Geometric Active Contour 1 1 F( c, c, ) ( u c ) H( ) ( u c ) 1 H( ) g u H ( ) dx g 1 u / u level set function, u given image, c 1, scalars Hessian F( f) " ( ) g ( ) y ' g ( ) g ( ) y g ( ) x yx 3/ 3/ ' g( ) y g( ) yx g ( ) x 3/ 3/ ' ' g ( ) x u c1 u c Indefinite sub-hessian, Legendre condition is not satisfied! 4-Mar-09

23 Geometric Active Contour 1 1 F( c, c, ) ( u c ) H( ) ( u c ) 1 H( ) g u H ( ) dx g 1 u / u level set function, u given image, c 1, scalars repeat c arg min F( ) 1, 1 F 1 c1, s arg min (, ) h * Until convergence criterion By generalized Newton method Smoothing operator (self-adjoint and positive definite) 4-Mar-09 3

24 Results-Geometric Active Contour Gradient descent Newton with trust-region Sobolev active contour Suggested generalized Newton 4-Mar-09 4

25 Results-Geometric Active Contour Gradient descent Newton Sobolev active contour Suggested generalized Newton 4-Mar-09 5

26 Results-Geometric Active Contour Gradient descent Newton with trust-region Sobolev active contour Suggested generalized Newton 4-Mar-09 6

27 Results-Geometric Active Contour Gradient descent Newton with trust-region Sobolev active contour Suggested generalized Newton 4-Mar-09 7

28 Results-Geometric Active Contour Gradient descent Newton with trust-region Sobolev active contour Suggested generalized Newton 4-Mar-09 8

29 Running Time Implementation of the GAC with MATLAB environment, running time in [sec]. image Generalized Newton Newton Gradient descent Sobolev GD shapes dancer newspaper ultrasound Mar-09 9

30 Mumford-Shah Type Color Deblurring c 1 c c ( v 1) F( f, v) h* f g dx v f dx v dx c{ R, G, B} 4 Mumford-Shah, CVPR 1985 J. Shah, CVPR 1996 Bar, Sochen,Kiryati, VLSM 005 h-blur ernel, g-observed image, f-recovered image, v-edge set c c x y c f f f repeat v c arg min F( f 1) By generalized minimal residual method f arg min F( f, v ) c c 1 By generalized Newton method H 1 v ( x) Adaptive edge-based Hamiltonian operator 1 Until convergence criterion (self-adjoint and positive definite) 4-Mar-09 30

31 Color Deblurring h( x)* h( x) 0 0 x Hessian F( f ) 0 v v 0 v f f c c c f f x f x fy 3/ 3/ f f c c c f x f f f y y v 3/ 3/ Indefinite sub-hessian, Legendre condition is not satisfied! 4-Mar-09 31

32 Results - Color Deblurring Blurred Newton E=81 Newton with trust region E=00 smoothing norm E=106 Hamiltonian norm E=14.1,t=47 sec CG method t=176.8 sec 4-Mar-09 3

33 Results - Color Deblurring Blurred Newton E=97 Newton with trust region E=309 smoothing norm E=161 Hamiltonian norm E=4, t= 3sec CG method t= 65sec 4-Mar-09 33

34 Conclusions An efficient generalized Newton-type method with trustregion is suggested Numerically stabilized by the trust-region constraint The method is flexible by designing the inner product in different applications Future research: extending to shape spaces and manifolds 4-Mar-09 34

35 Than you! 4-Mar-09 35