Report

The University of CS 4487/9587 Algorithms for Image Analysis Ontario 2D Segmentation (part II) Deformable Models Acknowledgements: many slides from the University of Manchester, demos from Visual Dynamics Group (University of Oxford), 5-1 The University of CS 4487/9587 Algorithms for Image Analysis Deformable Models in 2D Ontario Active Contours or “snakes” • “snakes” vs. “livewire” • (discrete) energy formulations for snakes • relation to Hidden Markov Models (HMM) Optimization (discrete case) • Gradient Descent • Dynamic Programming (DP), Viterbi algorithm • DP versus Dijkstra Extra Reading: Sonka et.al 5.2.5 and 8.2 Active Contours by Blake and Isard 5-2 The University of “Live-wire” vs. “Snakes” Ontario • intelligent scissors [Mortensen, Barrett 1995] • live-wire [Falcao, Udupa, Samarasekera, Sharma 1998] 4 3 1 2 Shortest paths on image-based graph connect seeds placed on object boundary 5-3 The University of “Live-wire” vs. “Snakes” Ontario •Snakes, active contours [Kass, Witkin, Terzopoulos 1987] •In general, deformable models are widely used Given: initial contour (model) near desirable object 5-4 The University of “Live-wire” vs. “Snakes” Ontario •Snakes, active contours [Kass, Witkin, Terzopoulos 1987] •In general, deformable models are widely used Given: initial contour (model) near desirable object Goal: evolve the contour to fit exact object boundary 5-5 The University of Tracking via deformable models Ontario 1. Use final contour/model extracted at frame t as an initial solution for frame t+1 2. Evolve initial contour to fit exact object boundary at frame t+1 3. Repeat steps 1 and 2 for t ‘= t+1 5-6 The University of Tracking via deformable models Ontario Acknowledgements: Visual Dynamics Group, Dept. Engineering Science, University of Oxford. Applications: Traffic monitoring Human-computer interaction Animation Surveillance Computer Assisted Diagnosis in medical imaging 5-7 The University of Tracking via deformable models Ontario Tracking Heart Ventricles 5-8 The University of gradient descent w.r.t. some function describing snake’s quality “Snakes” Ontario A smooth 2D curve which matches to image data Initialized near target, iteratively refined Can restore missing data initial intermediate final Q: How does that work? …. 5-9 The University of Preview f(x) for simplicity, assume that "snake” is a vector (or point) in R1 Ontario assume some energy function f(x) describing snake’s “quality” x R ˆx local minima for f(x) x2 x1 1 x0 gradient descent for 1D functions xi 1 xi t f ' ( xi ) Q: Is snake (contour) a point in some space? ... Yes 5-10 The University of Parametric Curve Representation (continuous case) Ontario A curve can be represented by 2 functions ν ( s ) ( x ( s ), y ( s )) parameter 0s1 closed curve open curve Note: in computer vision and medical imaging the term “snake” is commonly associated with such parametric representation of contours. (Other representations will be discussed later!) Here, contour is a point in R C { ν ( s ) | s [ 0 ,1 ]} (space of functions) 5-11 The University of Parametric Curve Representation (discrete case) Ontario A curve can be represented by a set of 2D points ν i ( xi , y i ) parameter 0in ( xn , yn ) 2n Here, contour is a point in R _ C ( ν i | 0 i n ) ( x 0 , y 0 , x 1 , y 1 ,...., x n 1 , y n 1 ) 5-12 The University of Measuring snake’s quality: Energy function Contours can be seen as points C in R Ontario 2n (or in R ) C ( ν 0 , ν 1 , ν 2 ,...., ν n 1 ) ( x 0 , y 0 , x 1 , y 1 ,...., x n 1 , y n 1 ) We can define some energy function E(C) that assigns some number (quality measure) to all possible snakes R 2n (contours C) WHY?: Somewhat philosophical question, but E(C) R (scalars) specifying a quality function E(C) is an objective way to define what “good” means for contours C. Moreover, one can find “the best” contour (segmentation) by optimizing energy E(C). Q: Did we use any function (energy) to measure quality of segmentation results in 1) image thresholding? NO 2) region growing? NO 3) live-wire? YES (will compare later) 5-13 The University of Energy function Ontario Usually, the total energy of snake is a combination of internal and external energies E E in E ex Internal energy encourages smoothness or any particular shape Internal energy incorporates prior knowledge about object boundary allowing to extract boundary even if some image data is missing External energy encourages curve onto image structures (e.g. image edges) 5-14 The University of Internal Energy (continuous case) Ontario The smoothness energy at contour point v(s) could be evaluated as E in ( ( s )) ( s ) d 2 d 2 (s) ds 2 2 d s Elasticity/stretching Stiffness/bending Then, the interior energy (smoothness) of the whole snake C { ν ( s ) | s [ 0 ,1 ]} is 1 E in E 0 in ( ( s )) ds 5-15 The University of Internal Energy (discrete case) Ontario C ( ν 0 , ν 1 , ν 2 ,...., ν n 1 ) elastic energy (elasticity) d ds v3 v4 v5 2n ν i ( xi , y i ) v2 vi1 i v1 v6 v7 v 10 v9 d v8 bending energy (stiffness) 2 ds 2 ( i 1 i ) ( i i 1 ) i 1 2 i i 1 The University of Internal Energy (discrete case) Ontario C ( ν 0 , ν 1 , ν 2 ,...., ν n 1 ) d ds d 2n ν i ( xi , y i ) vi1 i 2 ds 2 ( i 1 i ) ( i i 1 ) i 1 2 i i 1 n 1 E in | i 1 i | | i 1 2 i i 1 | 2 2 i0 Elasticity Stiffness 5-17 The University of External energy Ontario The external energy describes how well the curve matches the image data locally Numerous forms can be used, attracting the curve toward different image features 5-18 The University of External energy Ontario Suppose we have an image I(x,y) Can compute image gradient I ( I x , I y ) at any point Edge strength at pixel (x,y) is | I ( x , y ) | External energy of a contour point v=(x,y) could be E ex ( v ) | I ( v ) | | I ( x , y ) | 2 2 External energy term for the whole snake is 1 E ex E ex ( ( s )) ds continuous case C { ν ( s ) | s [ 0 ,1 ]} 0 n 1 E ex E i0 ex ( i ) discrete case C { νi |0 i n } 5-19 The University of Basic Elastic Snake Ontario The total energy of a basic elastic snake is 1 E | 0 dv 1 2 | ds ds E | vi 1 vi | 2 elastic smoothness term (interior energy) continuous case C { ν ( s ) | s [ 0 ,1 ]} 0 n 1 i 0 | I ( v ( s )) | ds 2 n 1 | I ( vi ) | 2 discrete case C { νi |0 i n } i 0 image data term (exterior energy) 5-20 The University of Basic Elastic Snake (discrete case) Ontario C ( ν i | 0 i n ) ( x 0 , y 0 , x 1 , y 1 ,...., x n 1 , y n 1 ) n 1 E in Li 2 i This can make a curve shrink i-1 (to a point) i0 Li-1 i+1 Li+1 n 1 Li ( x i 1 x i ) ( y i 1 y i ) 2 i+2 2 C i0 n 1 E ex | I ( xi , yi ) | 2 i0 n 1 | I x ( xi , yi ) | | I y ( xi , yi ) | 2 2 i0 5-21 The University of Basic Elastic Snake (discrete case) Ontario The problem is to find contour that minimizes C ( x 0 , , x n 1 , y 0 , , y n 1 ) R n 1 E ( C ) ( xi1 xi ) ( yi1 yi ) 2 i 0 2n n 1 2 | I x ( xi , yi ) | | I y ( xi , yi ) | 2 2 i 0 Optimization problem for function of 2n variables • can compute local minima via gradient descent (coming soon) • potentially more robust option: dynamic programming (later) 5-22 The University of Basic Elastic Snake Synthetic example Ontario (1) (2) (3) (4) 5-23 The University of Basic Elastic Snake Dealing with missing data Ontario The smoothness constraint can deal with missing data: 5-24 The University of Basic Elastic Snake Relative weighting Ontario Notice that the strength of the internal elastic component can be controlled by a parameter, n 1 E in Li 2 i0 Increasing this increases stiffness of curve large medium small 5-25 The University of Encouraging point spacing Ontario To stop the curve from shrinking to a point n 1 E in 2 ˆ ( Li Li ) i0 • encourages particular point separation 5-26 The University of Simple shape prior Ontario If object is some smooth variation on a known shape, use n 1 E in 2 ˆ ( i i ) i0 – where { ˆi } give points of the basic shape May use a statistical (Gaussian) shape model E in ln N ( | ˆ ) T ˆ ( ) C ( ˆ ) 5-27 The University of Alternative External Energies Ontario Directed gradient measures n 1 E ex u x ,i I x ( ν i ) u y ,i I y ( ν i ) i0 • Where u i ( u x ,i , u y ,i ) is the unit normal to the boundary at contour point i • This gives a good response when the boundary has the same direction as the edge, but weaker responses when it does not 5-28 The University of Additional Constraints Ontario • Snakes originally developed for interactive image segmentation • Initial snake result can be nudged where it goes wrong • Simply add extra external energy terms to – Pull nearby points toward cursor, or – Push nearby points away from cursor 5-29 The University of Interactive (external) forces Ontario Pull points towards cursor: n 1 E pull i0 r 2 | i p | 2 Nearby points get pulled hardest Negative sign gives better energy for positions near p 5-30 The University of Interactive (external) forces Ontario Push points from cursor: n 1 E push | i0 r i 2 p| 2 Nearby points get pushed hardest Positive sign gives better energy for positions far from p 5-31 The University of Dynamic snakes Ontario Adding motion parameters as variables (for each snake node) Introduce energy terms for motion consistency primarily useful for tracking (nodes represent real tissue elements with mass and kinematic energy) 5-32 The University of Open and Closed Curves Ontario 0 0 n open curve n 1 closed curve n2 n 1 E in ( i0 i ) i 1 2 E in ( i ) i 1 2 i0 When using an open curve we can impose constraints on the end points (e.g. end points may have fixed position) – Q: What are similarities and differences with the live-wire if the end points of an open snake are fixed? 5-33 The University of Discrete Snakes Optimization Ontario At each iteration we compute a new snake position within proximity to the previous snake New snake energy should be smaller than the previous one Stop when the energy can not be decreased within local neighborhood of the snake (local energy minima) Optimization Methods 1. Gradient Descent 2. Dynamic Programming 5-34 The University of Gradient Descent Ontario Example: minimization of functions of 2 variables E ( x, y ) x ( x0 , y0 ) y E E x E y negative gradient at point (x,y) gives direction of the steepest descent towards lower values of function E 5-35 The University of Gradient Descent Ontario Example: minimization of functions of 2 variables E ( x, y ) x update equation for a point p=(x,y) p p t E ( x0 , y0 ) y x x t y y E x E y Stop at a local minima where E 0 5-36 The University of Gradient Descent Ontario Example: minimization of functions of 2 variables E ( x, y ) x y High sensitivity wrt. the initialisation !! 5-37 The University of Gradient Descent for Snakes C n 1 E ( x 0 , , x n 1 , y 0 , , y n 1 ) Ontario simple elastic snake energy | I x ( xi , yi ) | | I y ( xi , yi ) | 2 i 0 here, energy is a function of 2n variables n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 update equation for the whole snake C' C E t C E x' 0 x 0 x 0 E y' 0 y 0 y 0 ... ... ... t E x' n 1 x n 1 x n 1 y' n 1 y n 1 y E n 1 5-38 2 The University of Gradient Descent for Snakes C n 1 E ( x 0 , , x n 1 , y 0 , , y n 1 ) Ontario simple elastic snake energy | I x ( xi , yi ) | | I y ( xi , yi ) | 2 i 0 here, energy is a function of 2n variables n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 update equation for each node ν' i ν i Fi t Fi C E x' 0 x 0 x 0 E y' 0 y 0 y 0 ... ... ... t E x' n 1 x n 1 x n 1 y' n 1 y n 1 y E n 1 Fi xEi E yi 5-39 2 The University of Gradient Descent for Snakes C n 1 E ( x 0 , , x n 1 , y 0 , , y n 1 ) Ontario simple elastic snake energy | I x ( xi , yi ) | | I y ( xi , yi ) | 2 2 i 0 here, energy is a function of 2n variables n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 update equation for each node ν' i ν i Fi t Fi C Fi xEi E yi = ? E xi 2 I x I xx 2 I y I yx 2 ( x i 1 x i ) 2 ( x i x i 1 ) E yi 2 I x I xy 2 I y I yy 2 ( y i 1 y i ) 2 ( y i y i 1 ) Q: Do points move independently? NO, motion of point i depends on positions of neighboring points 5-40 The University of Gradient Descent for Snakes C n 1 E ( x 0 , , x n 1 , y 0 , , y n 1 ) Ontario simple elastic snake energy | I x ( xi , yi ) | | I y ( xi , yi ) | 2 2 i 0 here, energy is a function of 2n variables n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 update equation for each node ν' i ν i Fi t Fi C Fi xEi E yi = ? E xi 2 I x I xx 2 I y I yx 2 ( x i 1 x i ) 2 ( x i x i 1 ) E yi 2 I x I xy 2 I y I yy 2 ( y i 1 y i ) 2 ( y i y i 1 ) from exterior (image) energy from interior (smoothness) energy 5-41 The University of Gradient Descent for Snakes C n 1 E ( x 0 , , x n 1 , y 0 , , y n 1 ) Ontario simple elastic snake energy | I x ( xi , yi ) | | I y ( xi , yi ) | 2 2 i 0 here, energy is a function of 2n variables n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 update equation for each node Fi C ν' i ν i Fi t Fi Fi xEi E yi 2 (| I | ) 2 ( xi , yi ) motion of vi towards higher magnitude of image gradients 2 d v 2 d s = ? This term for vi depends on neighbors vi-1 and vi+1 motion of vi reducing contour’s bending 5-42 The University of Discrete Snakes: “Gradient Flow” evolution Ontario dC dt E Contour evolution via “Gradient flow” update equation for each node i C i' ν' i ν i Fi t i 0 , , n 1 Stopping criteria: C’ F i 0 for all i E 0 local minima of energy E 5-43 The University of Difficulties with Gradient Descent Ontario Very difficult to obtain accurate estimates of high-order derivatives on images (discretization errors) • E.g., estimating E ex requires computation of second image derivatives I xx , I xy , I yy Gradient descent is not trivial even for one-dimensional functions. Robust numerical performance for 2ndimensional function could be problematic. • Choice of parameter t is non-trivial – Small t , the algorithm may be too slow – Large t , the algorithm may never converge • Even if “converged” to a good local minima, the snake is likely to oscillate near it 5-44 The University of Alternative solution for 2D snakes: Dynamic Programming Ontario In many cases, snake energy can be written as a sum of pair-wise interaction potentials n 1 E total ( 0 , , n 1 ) E i ( i , i 1 ) i0 More generally, it can be written as a sum of higher-order interaction potentials (e.g. triple interactions). n 1 E total ( 0 , , n 1 ) E i ( i 1 , i , i 1 ) i0 5-45 The University of Snake energy: pair-wise interactions Ontario Example: simple elastic nsnake energy 1 E total ( x 0 , , x n 1 , y 0 , , y n 1 ) | I x ( xi , y i ) | | I y ( xi , y i ) | 2 i 0 n 1 ( x i 1 x i ) ( y i 1 y i ) 2 2 i0 n 1 E total ( 0 , , n 1 ) || I ( i ) || 2 i 0 n 1 || i 1 i || 2 i0 n 1 E total ( 0 , , n 1 ) E i ( i , i 1 ) i0 2 2 where E i ( i , i 1 ) || I ( i ) || || i i 1 || Q: give an example of snake with triple-interaction potentials? 5-46 2 The University of DP Snakes [Amini, Weymouth, Jain, 1990] Ontario v2 v3 v1 v4 v5 v6 control points First-order interactions E ( v1 , v 2 ,..., v n ) E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) Energy E is minimized via Dynamic Programming 5-47 The University of DP Snakes [Amini, Weymouth, Jain, 1990] Ontario v2 v3 v1 v4 v5 v6 control points First-order interactions E ( v1 , v 2 ,..., v n ) E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) Energy E is minimized via Dynamic Programming Iterate until optimal position for each point is the center of the box, i.e. the snake is optimal in the local search space constrained by boxes 5-48 The University of Dynamic Programming (DP) Viterbi Algorithm Ontario Here we will concentrate on first-order interactions sites E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) v1 E 1 ( v1 , v 2 ) v2 E 2 (v 2 , v3 ) states 1 2 … m v3 E 3 (v3 , v 4 ) v4 E 4 (v4 , vn ) vn E1 (1) 0 E 2 (1) E 3 (1) E 4 (1) E n (1) E1 ( 2 ) 0 E 2 (2) E3 (2) E 4 (2) E n (2) E1 (3) 0 E 2 (3) E 3 (3) E 4 (3) E n (3) E1 ( 4 ) 0 E 2 (4) E3 (4) E 4 (4) E n (4) Complexity: O ( nm ), Worst case = Best Case 5-49 2 The University of Dynamic Programming and Hidden Markov Models (HMM) Ontario DP is widely used in speech recognition time word1 word2 word3 word4 ordered (in time) hidden variables (words) to be estimated from observed signal E1 ( v 0 , v1 ) ... E i ( v i 1 , v i ) ... E n ( v n 1 , v n ) ln{Pr( signal ( t i ) | word i ) } ln{Pr( word i | word i 1 )} 5-50 The University of Snakes can also be seen as Hidden Markov Models (HMM) Ontario Positions of snake nodes are hidden variables Timely order is replaced with spatial order Observed audible signal is replaced with image 1 n E1 ( v1 , v 2 ) ... E i ( v i , v i 1 ) ... E n 1 ( v n 1 , v n ) || I ( i ) || E elastic ( i , i 1 ) 5-51 The University of Dynamic Programming for a closed snake? Ontario Clearly, DP can be applied to optimize an open ended snake E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) 1 n Can we use DP for a “looped” energy in case of a closed snake? E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) E n ( v n , v1 ) n 1 n 1 2 4 3 5-52 The University of Dynamic Programming for a closed snake Ontario E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) E n ( v n , v1 ) 1. Can use Viterbi to optimize snake energy in case 1 c is fixed. (in this case the energy above effectively has no loop) 2. Use Viterbi to optimize snake for all possible values of c and choose the best of the obtained m solutions. for exact solution complexity increases to O(nm3) 5-53 The University of Dynamic Programming for a closed snake Ontario E1 ( v1 , v 2 ) E 2 ( v 2 , v 3 ) ... E n 1 ( v n 1 , v n ) E n ( v n , v1 ) DP has problems with “loops” (even one loop increases complexity). However, some approximation tricks can be used in practice… 1. Use DP to optimize snake energy with fixed 1 (according to a given initial snake position). 2. Use DP to optimize snake energy again. This time fix position of an intermediate node n / 2 ˆ n / 2 where ˆ is an optimal position obtained in step 1. This is only an approximation, but complexity is good: O(nm2) 5-54 The University of Dynamic Programming for snakes with higher order interactions Ontario E1 ( v1 , v 2 , v 3 ) E 2 ( v 2 , v 3 , v 4 ) ... E n 2 ( v n 2 , v n 1 , v n ) (e.g. if bending energy is added into the “model” of the snake) Viterbi algorithm can be generalized to 3-clique case but its complexity increases to O(nm3). one approach: combine each pair of neighboring nodes into one super node. Each triple interaction v4 can be represented as a pair-wise interaction v5 between 2 super-nodes. Viterbi algorithm will need m3 operations for each super node (why?) v3 v2 v1 5-55 The University of DP snakes (open case) Summary of Complexity energy type Ontario complexity (order of interactions) n E i (vi ) i 1 n 1 E i ( v i , v i 1 ) i 1 n2 E i ( v i , v i 1 , v i 2 ) i 1 E ( v1 , v 2 ,..., v n ) unary potentials O(nm) (d=1) pair-wise potentials O((n-1)m2)* (d=2) triple potentials O((n-2)m3)* (d=3) complete connectivity O(mn) – exhaustive search (d=n) * - adding a single loop increases complexity by factor md-1 5-56 The University of Problems with snakes Ontario Depends on number and spacing of control points Snake may oversmooth the boundary Not trivial to prevent curve self intersecting Can not follow topological changes of objects 5-57 The University of Problems with snakes Ontario May be sensitive to initialization – may get stuck in a local energy minimum near initial contour Numerical stability can be an issue for gradient descent and variational methods (continuous formulation) • E.g. requires computing second order derivatives The general concept of snakes (deformable models) does generalize to 3D (deformable mesh), but many robust optimization methods suitable for 2D snakes do not apply in 3D • E.g.: dynamic programming only works for 2D snakes 5-58 The University of Problems with snakes Ontario External energy: may need to diffuse image gradients, otherwise the snake does not really “see” object boundaries in the image unless it gets very close to it. image gradients I are large only directly on the boundary 5-59 The University of Diffusing Image Gradients I Ontario image gradients diffused via Gradient Vector Flow (GVF) Chenyang Xu and Jerry Prince, 98 http://iacl.ece.jhu.edu/projects/gvf/ 5-60 The University of Alternative Way to Improve External Energy n 1 Use E ex D( v Ontario n 1 i ) instead of i 0 • Distance Transform E ex | I ( v i ) | where D() is i0 (for detected binary image features, e.g. edges) binary image features (edges) Distance Transform can be visualized as a grayscale image Distance Transform D ( x , y ) • Generalized Distance Transform (directly for image gradients) 5-61 The University of Distance Transform (see p.20-21 of the text book) Ontario Image features (2D) 1 1 1 1 2 3 4 5 Distance Transform 0 1 2 3 4 3 0 1 2 3 3 2 0 1 2 3 2 1 0 0 1 2 1 0 1 1 2 1 0 1 2 2 2 1 0 1 3 3 2 1 0 1 4 4 3 2 1 0 2 1 0 1 2 2 2 1 Distance Transform is a function D () that for each image pixel p assigns a non-negative number D ( p ) corresponding to distance from p to the nearest feature in the image I 5-62 The University of Distance Transform can be very efficiently computed Ontario 5-63 The University of Distance Transform can be very efficiently computed Ontario 5-64 The University of Distance Transform can be very efficiently computed Ontario • Forward-Backward pass algorithm computes shortest paths in O(n) on a grid graph with regular 4-N connectivity and homogeneous edge weights 1 • Alternatively, Dijkstra’s algorithm can also compute a distance map (trivial generalization for multiple sources), but it would take O(n*log(n)). - Dijkstra is slower but it is a more general method applicable to arbitrary weighted graphs 5-65 The University of Distance Transform: an alternative way to think about Assuming Ontario 0 if pixel p is image feature F ( p) O .W . then D ( p ) min {|| p q || F ( q )} min || p q || q q :F ( q ) 0 is standard Distance Transform (of image features) F ( p) D ( p) p Locations of binary image features 5-66 The University of Distance Transform vs. Generalized Distance Transform For general Ontario F ( p) D ( p ) min { || p q || F ( q )} q is called Generalized Distance Transform of F ( p) F ( p) D ( p) D(p) may prefer “strength” of F(p) to proximity q p F(p) may represent non-binary image features (e.g. image intensity gradient) 5-67 The University of Generalized Distance Transforms (see Felzenszwalb and Huttenlocher, IJCV 2005) Ontario The same “Forward-Backward” algorithm can be applied to any initial array • Binary (0 / ) initial values are non-essential. If the initial array contains values of function F(x,y) then the output of the “Forward-Backward” algorithm is a Generalized Distance Transform D ( p ) min ( || p q || F ( q )) q I “Scope of attraction” of image gradients can be extended via external energy E D ( v ) based on a generalized distance transform of n 1 ex i i0 F ( x , y ) g (| I ( x , y ) |) 5-68 The University of Metric properties of discrete Distance Transforms Forward mask Backward mask - 1 0 1 1 0 1 - 1.4 1 1 0 1.4 1.4 Metric Ontario Set of equidistant points Manhattan (L1) metric 0 1 1 1.4 In fact, “exact” Euclidean Distance transform can be computed fairly efficiently (in linear or near-linear time) without bigger masks 1) www.cs.cornell.edu/~dph/matchalgs/ 2) Fast Marching Method –Tsitsiklis, Sethian Better approximation of Euclidean metric Euclidean (L2) metric 5-69 The University of HW assignment 2 Ontario DP Snakes • Use elastic snake model • Compare E E int l E ext (value of l is important) n 1 n 1 E int Li i0 • Compare 2 vs. E int i0 n 1 L vs. E E F ( v ) vs. • Compare generalized distance transform of 2 E int n 1 ext i i 0 i i0 such that |L int i 2 Lˆ | n 1 |L i0 E ext i | n 1 D ( v ) where D is a i i0 F g (| I |) D ( p ) min { || p q || F ( q )} (value of is important) q • Use Viterbi algorithm for optimization • Incorporate edge alignment • Use 3x3 search box for each control point 5-70