Non-Rigid Registration I

Transcription

1 Non-Rigid Registration I CS6240 Multimedia Analysis Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (CS6240) Non-Rigid Registration I 1 / 67

2 Introduction Introduction Non-rigid registration dues with spatial alignment of non-rigid or deformable objects. Can include global rigid alignment, e.g., scaling, rotation, translation. Can be performed with or without known correspondence. Deformable models Can change shape over time. So, transformation is non-linear. Can be represented by equations or connected points. Leow Wee Kheng (CS6240) Non-Rigid Registration I 2 / 67

3 Registration with Known Correspondence Curve Warping Curve Warping Warp a source curve C into a target curve C in 2D space p i C p i C p i = (x i, y i ) denote the points on C. p i = (x i, y i ) denote the corresponding points on C. Warping is nonlinear because the displacement of points cannot be described by a linear transformation. Leow Wee Kheng (CS6240) Non-Rigid Registration I 3 / 67

4 Registration with Known Correspondence Curve Warping A simple representation of a 2D nonlinear transformation is a polynomial function of order m. x i = a pq x p i yq i, p q 0 p + q m, (1) y i = b pq x p i yq i, p q 0 p + q m. (2) Example: 2nd-order Polynomial Function. x i = a 20 x 2 i + a 11 x i y i + a 02 yi 2 + a 10 x + a 01 y + a 00 (3) y i = b 20 x 2 i + b 11 x i y i + b 02 yi 2 + b 10 x + b 01 y + b 00. (4) Eq. 1 and 2 are independent. So, they can be solved individually using model fitting method. Exercise: Curved surface warping by polynomial transformation. Leow Wee Kheng (CS6240) Non-Rigid Registration I 4 / 67

5 Registration with Known Correspondence Image Warping Image Warping Similar method can be used for image warping. p i = (x i, y i ): control points in source image I. p i = (x i, y i ): target locations of control points. Leow Wee Kheng (CS6240) Non-Rigid Registration I 5 / 67

6 Registration with Known Correspondence Image Warping General idea of algorithm: 1 Use known correspondence to compute the nonlinear transformation between p i and p i. 2 For each pixel location p in the target warped image, Compute the real-valued location p is the source image. Use bilinear interpolation to compute the color c of p. Fill pixel p with color c. Notes: In practice, a global transformation applied to the entire source image is not practical may require very high-order polynomial. So, divide the source image into triangular regions using triangulation algorithm (CS5237 Computational Geometry and Applications). Perform a different local transformation within each triangle. Ensure transformations across triangle s edges are smooth. For details, see [Wol90]. Leow Wee Kheng (CS6240) Non-Rigid Registration I 6 / 67

7 Registration with Known Correspondence Image Warping (c) Source image (d) Warped image. Leow Wee Kheng (CS6240) Non-Rigid Registration I 7 / 67

8 Registration with Known Correspondence Further Readings Further Readings 3D volume warping [BMTB02]. Details of image warping [Wol90]. Image morphing (CS5245 lecture notes). Leow Wee Kheng (CS6240) Non-Rigid Registration I 8 / 67

9 Registration with Unknown Correspondence Registration with Unknown Correspondence Registration of a deformable model with unknown correspondence is an ill-posed problem [Had23]. Tikhonov and Arsenin (1977): A function f that maps inputs in X to outputs in Y is well-posed if these conditions are satisfied: 1 Existence. x X, y Y such that y = f(x). 2 Uniqueness. For any pair x, t X, f(x) = f(v) if and only if x = v. 3 Continuity. Let ρ x denote the distance measured in X. For any ǫ > 0, δ = δ(ǫ) such that ρ x (x,v) < δ implies ρ y (f(x), f(v)) < ǫ. Leow Wee Kheng (CS6240) Non-Rigid Registration I 9 / 67

10 Registration with Unknown Correspondence t f f ( t) x δ f ( x) ε If any of the above is not satisfied, then the problem is ill-posed. Leow Wee Kheng (CS6240) Non-Rigid Registration I 10 / 67

11 Registration with Unknown Correspondence Regularization Theory Regularization Theory A general approach to solving ill-posed problem is based on the regularization theory [TA77]. Include information or constraints about possible solutions. Most common constraint: smoothness, i.e., similar inputs map to similar outputs. Tikhonov s regularization theory involves two terms: 1 Standard error term: e.g., mean squared error E S (f) = i (y i f(x i )) 2 (5) 2 Regularization term: E C (f) = Df 2 (6) D is a linear differential operator. Leow Wee Kheng (CS6240) Non-Rigid Registration I 11 / 67

12 Registration with Unknown Correspondence Regularization Theory The total error E to be minimized is E(f) = E S (f) + λ E C (f) = (y i f(x i )) 2 + λ Df 2 (7) i λ is the regularization parameter. E(f) is a functional: function of function. Leow Wee Kheng (CS6240) Non-Rigid Registration I 12 / 67

13 Registration with Unknown Correspondence Regularization Theory The function f(x) that minimizes E(f) takes the form f(x) = i w i G(x,x i ) (8) G(x,x k ) is called the Green s function. w i = (y i f(x i ))/λ is the weight. That is, f is a linear combination of Green s functions. Gaussian function is an example of Green s function. Regularization is frequently used to solve various problems. Leow Wee Kheng (CS6240) Non-Rigid Registration I 13 / 67

14 Medial Axis Fitting Medial Axis Fitting How to fit a curve (1-D model) in 3-D space? Equation of curve in 3-D space does not have the convenient analytic form shown in previous sections. Other ways of representing curves are required. Parametric Equation s = 0 v ( s) s = 1 s s: a parameter, typically 0 s 1 v(s) = (x(s), y(s), z(s)): coordinates of a point on the curve v is a function of s. Leow Wee Kheng (CS6240) Non-Rigid Registration I 14 / 67

15 Medial Axis Fitting First, consider a continuous tubular object. s = 0 v ( s) s p i χ s s = 1 Want to fit a curve to the center line or medial axis. Use parametric equation of curve to represent medial axis. Cross-section χ s at s is perpendicular to the medial axis. v(s) is the centroid of the surface points p i on χ s. v(s) = 1 p i (9) N s p i χ s where N s is the number of points on χ s. Leow Wee Kheng (CS6240) Non-Rigid Registration I 15 / 67

16 Medial Axis Fitting Now, consider a discrete mesh of tubular object. Cross-section χ s may not intersect any mesh point. So, Eq. 9 is not so appropriate. Instead, think of χ s as a thin slab at s that contains a sufficient number of mesh points. That is, a point p i χ s has a normal projection on the medial axis that is close to v(s). Then, can redefine v(s) as v(s) = w i (s)p i p i χ s. (10) w i (s) p i χ s Weight w i (s) is inversely related to the distance between v(s) and the normal projection of p i on the medial axis. Leow Wee Kheng (CS6240) Non-Rigid Registration I 16 / 67

17 Medial Axis Fitting The problem of fitting medial axis becomes one of finding the v(s) that minimize the difference between v(s) and the right-hand-side of Eq. 10. where E(v) = L 0 D(v) = v(s) D(v)ds (11) p i χ s w i (s)p i p i χ s w i (s) 2 (12) Here, 0 s L. Eq. 11 alone is insufficient for optimization because it is ill-posed: there is an infinite number of curves that can fit. To make the problem well-posed, need an additional constraint, e.g., smoothness constraint. Leow Wee Kheng (CS6240) Non-Rigid Registration I 17 / 67

18 Medial Axis Fitting The revised cost function E(v) is [KWL06] E(v) = 1 2 L λ is a regularization weight. v (s) = dv/ds. Minimizing v (s) 2 imposes smoothness. 0 ( D(v) + λ v (s) 2) ds. (13) When E is minimum, v satisfies the following equation, which can be obtained using variational calculus (Appendix A): That is, p i χ s w i (s)[v(s) p i ] 1 2 D(v(s)) λv (s) = 0. (14) ( p i χ s w i (s) ) 1 λv (s) = 0. (15) Leow Wee Kheng (CS6240) Non-Rigid Registration I 18 / 67

19 Medial Axis Fitting Eq. 15 is solved iteratively by looking at change of v over time, i.e., consider v = v(s, t). Then, v t + Then, p i χ s w i (s)[v(s, t) p i ] ( p i χ s w i (s) ) 1 λv (s, t) = 0. (16) v(s, t + 1) = v(s, t) η ( ) 1 w i (s)[v(s, t) p i ] w i (s) λv (s, t) (17) p i χ s p i χ s where η is a small constant time step. Note: v changes in the direction of negative gradient of D. Leow Wee Kheng (CS6240) Non-Rigid Registration I 19 / 67

20 Medial Axis Fitting Example: Fit a curve to the medial axis of a mesh model of aorta. (a) initialization (b) fitting result Demo Exercise: Work out the algorithm/equation for fitting the cross-sectional radii along the medial axis, after fitting the medial axis. This process gives rise to a generalized cylinder model. Leow Wee Kheng (CS6240) Non-Rigid Registration I 20 / 67

21 Active Contour Active Contour Active contour [Coh91, KWT87] also called snake, more like rubber band can be deformed to match any shape The snake model represents a contour v parametrically as v(s) = (x(s), y(s)) 0 s 1 (18) ( x (1), y (1)) ( x ( s ), y ( s )) Leow Wee Kheng (CS6240) Non-Rigid Registration I 21 / 67

22 Active Contour The snake is deformed under the influence of three forces: internal forces: constrain stretching and banding of snake image forces: attract the snake (e.g., edges) The energy E i of the internal forces at point v(s) is given by E i = 1 ( α(s) v (s) 2 + β(s) v (s) 2) (19) 2 v (s) = dv/ds and v (s) = d 2 v/ds 2. α(s) v (s) 2 is the energy associated with stretching or tension or elasticity β(s) v (s) 2 is the energy associated with bending or rigidity Leow Wee Kheng (CS6240) Non-Rigid Registration I 22 / 67

23 Active Contour The energy E associated with the image depends on what you want the snake to be attracted to: dark line: E = I(x, y) bright line: E = I(x, y) edge: E = I(x, y) 2 Marr-Hildreth edge: E = (G σ 2 I) 2 G σ is a Gaussian function Total energy E T of the snake is E T = 1 0 [E i (v(s)) + E(v(s))] ds (20) Want energy E T to be minimum when snake has snapped onto the desired image feature. Leow Wee Kheng (CS6240) Non-Rigid Registration I 23 / 67

24 Active Contour When E T is minimum, v satisfies the following equation, which can be obtained using variational calculus method (Appendix B): (α(s)v (s)) + (β(s)v (s)) + E(v(s)) = 0 (21) To solve Eq. 21, consider change of snake over time, i.e., v(s, t). Then, we have v(s, t) t (α(s)v (s, t)) + (β(s)v (s, t)) + E(v(s, t)) = 0. (22) Leow Wee Kheng (CS6240) Non-Rigid Registration I 24 / 67

25 Active Contour For numerical computation, let s discretize the equation by finite differences. Let h be a small step or difference of s, and v i = v(ih) = (x(ih), y(ih)) α i = α(ih)/h 2 β i = β(ih)/h 4 F = (F x, F y ) = E(v(s)) Apply forward and backward difference alternatively (Why? See Exercises): forward difference : f (x) = f(x + 1) f(x) backward difference : f (x) = f(x) f(x 1) Leow Wee Kheng (CS6240) Non-Rigid Registration I 25 / 67

26 Active Contour Then, Eq. 21 can be discretized into α i (v i v i 1 ) α i+1 (v i+1 v i ) +β i 1 (v i 2v i 1 + v i 2 ) 2β i (v i+1 2v i + v i 1 ) (23) +β i+1 (v i+2 2v i+1 + v i ) + F = 0 Eq. 23 can be written in the matrix form Notes: A is a pentadiagonal banded matrix x = (x 1, x 2,...,x n ) y = (y 1, y 2,...,y n ) F x = (F x1, F x2,...,f xn ) F y = (F y1, F y2,...,f yn ) Ax + F x = 0 (24) Ay + F y = 0 (25) Leow Wee Kheng (CS6240) Non-Rigid Registration I 26 / 67

27 Active Contour For a n-point closed contour (i.e., x n+1 = x 1 ): A = c 1 d 1 e a 1 b 1 b 2 c 2 d 2 e a 2 a 3 b 3 c 3 d 3 e e n a n 1 b n 1 c n 1 d n 1 d n e n 0 0 a n b n c n (26) where a i = β i 1 b i = α i 2β i 1 2β i c i = α i + α i+1 + β i 1 + 4β i + β i+1 d i = α i+1 2β i 2β i+1 e i = β i+1 (27) Leow Wee Kheng (CS6240) Non-Rigid Registration I 27 / 67

28 Active Contour Equations 24 and 25 are solved iteratively by looking at change of v over time. Then, where γ is a small time step. Then, rearranging the terms yield Note: γ(x(t) x(t 1)) + Ax + F x = 0 (28) x(t) = (A + γi) 1 (γx(t 1) F x (t 1)) (29) y(t) = (A + γi) 1 (γy(t 1) F y (t 1)). (30) For a fixed A, the inverse (A + γi) 1 does not change over time. Then, only have to compute it once. Leow Wee Kheng (CS6240) Non-Rigid Registration I 28 / 67

29 Active Contour Example Demo 1 Demo 2 (a) Initial snake points (b) Final snake contour. Additional points are added between initial points before running the snake algorithm to get smoother contour. Have problem snapping onto concave parts of the contour. Leow Wee Kheng (CS6240) Non-Rigid Registration I 29 / 67

30 Active Contour Gradient Vector Flow Gradient Vector Flow The original formulation of snake has two main shortcomings: It does not converge well to concave features Its performance is sensitive to initial guess of snake point positions. Gradient Vector Flow (GVF) overcome the shortcomings [XP97, XP98, XP00]. Based on diffusion of gradient vectors of edge map. Leow Wee Kheng (CS6240) Non-Rigid Registration I 30 / 67

31 Active Contour Gradient Vector Flow Let E(x, y) denote edge map derived from image I(x, y). Then, E is the gradient vector which is normal to the edge. E is maximum at the edge. GVF field g(x, y) = [u(x, y), v(x, y)] is a vector field that minimizes the energy functional E = µ(u 2 x + u 2 y + vx 2 + vy) 2 + E 2 g E 2 dx dy (31) u x = u/ x, similarly for u y, v x, v y. µ: constant that is set according to the amount of noise present. If E is small, the first term dominates, yielding a slowly varying field. If E is large, the second term dominates, keeping g nearly equal to E. Leow Wee Kheng (CS6240) Non-Rigid Registration I 31 / 67

32 Active Contour Gradient Vector Flow Using variational calculus, it can be shown that the GVF field can be found by solving the following Euler equations: µ 2 u (u E x )(E 2 x + E 2 y) = 0 (32) where 2 is the Laplacian operator: µ 2 v (v E y )(E 2 x + E 2 y) = 0 (33) 2 u = 2 u x u y 2 (34) Eq. 32 and 33 can be solved by regarding u and v as functions of time and solving u t = µ 2 u (u E x )(E 2 x + E 2 y) (35) v t = µ 2 v (v E y )(E 2 x + E 2 y). (36) Leow Wee Kheng (CS6240) Non-Rigid Registration I 32 / 67

33 Active Contour Gradient Vector Flow As for the snake model, apply finite difference approximation: u t = 1 (u(k + 1) u(k)) t v t = 1 (v(k + 1) v(k)) t 2 u = u i+1,j + u i,j+1 + u i 1,j + u i,j 1 4u i,j 2 v = v i+1,j + v i,j+1 + v i 1,j + v i,j 1 4v i,j Leow Wee Kheng (CS6240) Non-Rigid Registration I 33 / 67

34 Active Contour Gradient Vector Flow Substituting the approximations into Eq. 35 and 36 yields where u i,j (k + 1) = (1 b i,j t)u i,j (k) + c i,j t+ µ t [u i+1,j (k) + u i,j+1 (k) + u i 1,j (k) + u i,j 1 (k) 4u i,j (k)] v i,j (k + 1) = (1 b i,j t)v i,j (k) + d i,j t+ µ t [v i+1,j (k) + v i,j+1 (k) + v i 1,j (k) + v i,j 1 (k) 4v i,j (k)] Notes: For numerical stability, t should be small. Specifically, t < 1 4µ. For more details, refer to [XP97, XP98]. (37) b = f 2 x + f 2 y (38) c = bf x (39) d = bf y. (40) Leow Wee Kheng (CS6240) Non-Rigid Registration I 34 / 67

35 Active Contour Gradient Vector Flow Combining snake with GVF: Notes: Replace E in Eq. 22 by g. That is, Compute the steady state solutions of u and v, i.e., u(k) and v(k) as k. Replace F x and F y by u and v. Run snake algorithm. GVF is a very general method of propagating edge information to increase the basin of attraction of deformable models. It can be combined with many other deformable models. Leow Wee Kheng (CS6240) Non-Rigid Registration I 35 / 67

36 Active Contour Gradient Vector Flow Example 1: From [XP00]. (a) convergence of contour (b) traditional potential force (c) close-up at concavity: no force to attract the snake towards the bottom of the concavity Leow Wee Kheng (CS6240) Non-Rigid Registration I 36 / 67

37 Active Contour Gradient Vector Flow (a) convergence of contour (b) GVF external force (c) close-up at concavity: forces exist to attract the snake towards the bottom of the concavity Leow Wee Kheng (CS6240) Non-Rigid Registration I 37 / 67

38 Active Contour Gradient Vector Flow Example 2: From [Tia02]. (a) Initial snake. (b) Traditional snake. (c) GVF snake. Demo 1 Demo 2 Leow Wee Kheng (CS6240) Non-Rigid Registration I 38 / 67

39 Active Contour Important Notes Important Notes How to determine the initial snake points? Possible methods: Manual: User provide initial snake points. OK if user inputs are acceptable. Automatic: Use image processing techniques to find approximate locations of image features. User inputs are not required. Semi-automatic: User provide simple inputs, which assist the program to find approximate locations of image features. Need some user inputs. Leow Wee Kheng (CS6240) Non-Rigid Registration I 39 / 67

40 Active Contour Exercises Exercises 1 Derive the discrete estimate of f (x) using forward difference only backward difference only alternative forward and backward differences What are the differences of the estimations derived above? 2 Derive the Euler equations for Gradient Vector Flow using variational calculus. Leow Wee Kheng (CS6240) Non-Rigid Registration I 40 / 67

41 Active Contour Further Readings Further Readings Appendix B: Variational Calculus for Snake Model. Dual contour method [GN94]: One contour expands from inside the target region while the other contracts from outside. This method can reject certain local minima in the snake evolution process. Topology-preserving snakes [HXP01, WJL03]. Snakes with geometric constraints [Ee04, FCH03, SHD01]. Leow Wee Kheng (CS6240) Non-Rigid Registration I 41 / 67

42 Level Set Method Level Set Method Snakes cannot handle applications that require topology changes. (a) (b) (c) Leow Wee Kheng (CS6240) Non-Rigid Registration I 42 / 67

43 Level Set Method (a) Two bubbles expand outward. (b) Later in time, they merge into one, thus changing the topology. (c) With point representation, need to decide which points are located inside the merged bubble; they need to be removed. Doing this in 2D is troublesome. Doing in 3D is very tedious. Level set method [Set96, Set99] solves the problem elegantly by doing it in one higher dimension. Let Γ denote a closed curve in 2D. Define a new function φ(x(t), y(t), t) such that φ(x,y,t = 0) is the distance d of the point (x,y) from Γ. d is positive if (x,y) is outside Γ. d is zero if (x,y) is on Γ. d is negative if (x,y) is inside Γ. Leow Wee Kheng (CS6240) Non-Rigid Registration I 43 / 67

44 Level Set Method The 3D surface (red) φ(x, y, t) is the level set function. Γ at time t = 0 is the intersection of φ(x, y, t = 0) with the xy plane (blue), i.e., the solution of the equation φ(x, y, t = 0) = 0. This is called the zero level set. To compute Γ at a later time, e.g., t = 1, just move the level set function up (or down), and then compute the solution φ(x, y, t = 1) = 0. Leow Wee Kheng (CS6240) Non-Rigid Registration I 44 / 67

45 Level Set Method Now, the problem of propagating Γ over time becomes simpler. F: force that gives the speed of Γ in its normal direction. Then, the change of φ over time t, φ t, is given by the equations φ t + F φ = 0, (41) φ(x, y, t = 0) = given. (42) where φ = (φ 2 x + φ 2 y) 1/2 Leow Wee Kheng (CS6240) Non-Rigid Registration I 45 / 67

46 Level Set Method Notes: Although φ(x, y, t) remains a function, the zero level set φ = 0 corresponds to the propagating contour may change topology and form sharp corners. The idea of level set method is very simple. All the complications come from accurate and stable numerical algorithms for computing φ(x, y, t + t) from φ(x, y, t). See [Set96, Set99] for more details. Leow Wee Kheng (CS6240) Non-Rigid Registration I 46 / 67

47 Level Set Method Force Function Force F is usually formulated as F P + F G. F P is the propagation force: F P = αv(x) (43) α is a positive constant v(x) is a speed function F P controls propagation of surface. F P stops the propagation at the desired boundary. F G is a smoothing force: F G = βv(x)κ(x) (44) β is a positive constant κ(x) is the curvature at x F G smooths out high curvature regions. Leow Wee Kheng (CS6240) Non-Rigid Registration I 47 / 67

48 Level Set Method Possible speed functions Inverse proportion v(x) = (1 + I(x) ) 1 (45) Exponential Sigmoid v(x) = v(x) = exp( γ I(x) ) (46) [ ( 1 + exp I(x) b )] 1 (47) a Leow Wee Kheng (CS6240) Non-Rigid Registration I 48 / 67

49 Level Set Method Notes on numerical implementation: Discretize 3D volume (or 2D plane) into voxels. Compute initial value of level set function at each voxel. Update value at each voxel as the surface propagates. Computationally expensive for high-resolution discretization. Narrow Band Method [Set96, Set99] Update only the voxel values in a narrow band around the propagating surface. So, it s more efficient. Leow Wee Kheng (CS6240) Non-Rigid Registration I 49 / 67

50 Level Set Method Applicaton example: Segmentation of arteries in 2D images. Video demo Leow Wee Kheng (CS6240) Non-Rigid Registration I 50 / 67

51 Level Set Method Application example: Segmentation of great arteries in 3D CT images. 2-D views of segmentation results: Leow Wee Kheng (CS6240) Non-Rigid Registration I 51 / 67

52 Level Set Method 3-D view of segmentation results: Great arteries are segmented using 3D level set. Level set segmentation results have implicit surfaces. Explicit surfaces are obtained using marching cubes algorithm [LC87], followed by surface smoothing. Other video demos: brain, leg Leow Wee Kheng (CS6240) Non-Rigid Registration I 52 / 67

53 Level Set Method Further Readings Further Readings For details of level set method, refer to [Set96, Set99]. Level Set Method Library in ITK: [ITK]. Level Set Method Toolbox: [Mit]. Leow Wee Kheng (CS6240) Non-Rigid Registration I 53 / 67

54 Appendix A: Calculus of Variations Appendix A: Calculus of Variations Calculus of variations or variational calculus is a branch of mathematics that deals with the calculus of functional instead of function. A functional is a function of function, i.e., it maps a function to a real number. Calculus of variations is often used to find the extremum (minimum or maximum) of functional of the form E(v) = b a f(v, v, x)dx (48) where v = v(x) has two continuous derivatives, i.e., v and v exist. E(v) has an extremum only if the Euler-Lagrange equation is satisfied: f v d dx f v = 0. (49) Leow Wee Kheng (CS6240) Non-Rigid Registration I 54 / 67

55 Appendix A: Calculus of Variations Derivation Suppose v is changed to v + δv. Then, v is also changed to v + δv. Since the derivative of v + δv is equal to v + δv, we have d ds (v + δv) = v + δv (50) i.e., d ds δv = δv. (51) Assume that δv and δv are small, and δv = 0 at the boundary points where x = a and x = b. Then, applying Taylor s series expansion yields Therefore, f(v + δv, v + δv, x) = f(v, v, x) + f f δv + v v δv. (52) δe = b a ( ) f f δv + v v δv dx. (53) Leow Wee Kheng (CS6240) Non-Rigid Registration I 55 / 67

56 Appendix A: Calculus of Variations Applying integration-by-parts gives: b a ( ) f v δv dx = = b a = [ f v δv ( ) f d v ds δv dx b a ] b a b a ( d f dx v ( d f dx v ) δv dx ) δv dx (54) So, δe = b a ( f v d dx For δe to be zero for any δv, we need f v d dx f v ) δv dx. (55) f v = 0. (56) Leow Wee Kheng (CS6240) Non-Rigid Registration I 56 / 67

57 Appendix A: Calculus of Variations Application to Medial Axis Fitting Application to Medial Axis Fitting The medial axis is the v(s) that minimizes the cost function E(v): where E(v) = 1 2 and v = v(s), 0 s L. L 0 D(v) = v(s) ( D(v) + λ v 2) ds (57) p i χ s w i (s)p i p i χ s w i (s) 2, (58) Here, f(v,v, s) = 1 2 (D(v) + λ v 2 ). For E(v) to have a minimum, the following must be satisfied, f v d f ds v = 0. (59) Leow Wee Kheng (CS6240) Non-Rigid Registration I 57 / 67

58 Appendix A: Calculus of Variations Application to Medial Axis Fitting Evaluating each partial derivatives of f yields Therefore, f v = 1 D 2 v = 1 2 D f v = λv d f ds v = λv (60) 1 2 D λv = 0. (61) return Leow Wee Kheng (CS6240) Non-Rigid Registration I 58 / 67

59 Appendix B: Variational Calculus for Snake Model Appendix B: Variational Calculus for Snake Model The snake model is given by the equation E T (v) = 1 0 f(v,v,v, s)ds (62) where f(v,v,v, s) = (α v 2 + β v 2 )/2 + E(v), v = v(s), α = α(s), β = β(s), and 0 s 1. Following the derivation in Appendix A, we obtain the condition for E T (v) to have an extremum (Exercise): f v d f ds v + d2 f ds 2 v = 0. (63) Leow Wee Kheng (CS6240) Non-Rigid Registration I 59 / 67

60 Appendix B: Variational Calculus for Snake Model Evaluating each partial derivatives of f yields Therefore, f v = f v = αv E v = E d f ds v = (αv ) f v = βv d 2 ds 2 f v = (βv ). (64) (αv ) + (βv ) + E(v) = 0. (65) return Leow Wee Kheng (CS6240) Non-Rigid Registration I 60 / 67

61 Reference References I L. Balmelli, C. J. Morris, G. Taubin, and F. Bernardini. Volume warping for adaptive isosurface extraction. In Proc. IEEE Visualization, L. D. Cohen. On active contour models and balloons. CVGIP: Image Understanding, 53(2): , citeseer.nj.nec.com/cohen91active.html. X. Ee. Shape-Preserving Elastic Snakes. Honours Year Project Report, Dept. of Computer Science, NUS, Leow Wee Kheng (CS6240) Non-Rigid Registration I 61 / 67

62 Reference References II A. Foulonneau, P. Charbonnier, and F. Heitz. Geometric shape priors for region-based active contours. In Proc. Int. Conf. on Image Processing, S. R. Gunn and M. S. Nixon. A dual active contour for head boundary extraction. In Proc. Colloq. on Image Proc. for Biometric Measurement, pages 6/1 4, J. Hadamard. Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, Leow Wee Kheng (CS6240) Non-Rigid Registration I 62 / 67

63 Reference References III X. Han, C. Xu, and J. L. Prince. A topology preserving deformable model using level set. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, volume 2, pages , ITK: Insight Segmentation and Registration Toolkit, K. J. Kirchberg, A. Wimmer, and C. H. Lorenz. Modeling the human aorta for MR-driven real-time virtual endoscopy. In In Proc. of MICCAI, pages , M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active contour models. Int. Journal of Computer Vision, 1: , Leow Wee Kheng (CS6240) Non-Rigid Registration I 63 / 67

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