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Computer Vision: Vision and Modeling. Computer Vision: Vision and Modeling. Lucas-Kanade Extensions Support Maps / Layers: Robust Norm, Layered Motion, Background Subtraction, Color Layers Statistical Models (Forsyth+Ponce Chap. 6, Duda+Hart+Stork: Chap. 1-5)

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## Computer Vision: Vision and Modeling

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**Computer Vision: Vision and Modeling**• Lucas-Kanade Extensions • Support Maps / Layers: • Robust Norm, Layered Motion, Background Subtraction, Color Layers • Statistical Models (Forsyth+Ponce Chap. 6, Duda+Hart+Stork: Chap. 1-5) • - Bayesian Decision Theory • - Density Estimation**A Different View of Lucas-Kanade**2 E = S( ) = I (i) - I(i) v D i t i 2 I (1) - I(1) v D 1 t High Gradient has Higher weight I (2) - I(2) v D 2 t ... D I (n) - I(n) v n t White board**Constrain**- V V Constrained Optimization 2 I (1) - I(1) v D 1 t I (2) - I(2) v D 2 t ... D I (n) - I(n) v n t**Constraints = Subspaces**Constrain - V V E(V) Analytically derived: Affine / Twist/Exponential Map Learned: Linear/non-linear Sub-Spaces**Motion Constraints**• Optical Flow: local constraints • Region Layers: rigid/affine constraints • Articulated: kinematic chain constraints • Nonrigid: implicit /learned constraints**Constrained Function Minimization**Constrain - V V 2 I (1) - I(1) v D 1 t V= M( q) = E(V) I (2) - I(2) v D 2 t ... D I (n) - I(n) v n t**2D Translation: Lucas-Kanade**2D Constrain - V V 2 I (1) - I(1) v D dx, dy 1 t V= = E(V) dx, dy I (2) - I(2) v D 2 t ... ... dx, dy D I (n) - I(n) v n t**a1, a2**a3, a4 2D Affine: Bergen et al, Shi-Tomasi 6D Constrain - V V 2 I (1) - I(1) v D 1 t x dx v = = E(V) i + I (2) - I(2) v D 2 i y dy t i ... D I (n) - I(n) v n t**Affine Extension**• Affine Motion Model: • 2D Translation • 2D Rotation • Scale in X / Y • Shear Matlab demo ->**Affine Extension**Affine Motion Model -> Lucas-Kanade: Matlab demo ->**2D Affine: Bergen et al, Shi-Tomasi**6D Constrain - V V**K-DOF Models**K-DOF Constrain - V V 2 I (1) - I(1) v D 1 t V= M( q) = E(V) I (2) - I(2) v D 2 t ... D I (n) - I(n) v n t**Quadratic Error Norm (SSD) ???**Constrain - V V 2 I (1) - I(1) v D 1 t V= M( q) = E(V) I (2) - I(2) v D 2 t ... D I (n) - I(n) v n t White board (outliers?)**Support Maps / Layers**• L2 Norm vs Robust Norm • Dangers of least square fitting: L2 D**Support Maps / Layers**• L2 Norm vs Robust Norm • Dangers of least square fitting: L2 robust D D**Support Maps / Layers**• Robust Norm -- good for outliers • nonlinear optimization robust D**Support Maps / Layers**• Iterative Technique Add weights to each pixel eq (white board)**Support Maps / Layers**• how to compute weights ? • -> previous iteration: how good does G-warp matches F ? • -> probabilistic distance: Gaussian:**Error Norms / Optimization Techniques**SSD: Lucas-Kanade (1981) Newton-Raphson SSD: Bergen-et al. (1992) Coarse-to-Fine SSD: Shi-Tomasi (1994) Good Features Robust Norm: Jepson-Black (1993) EM Robust Norm: Ayer-Sawhney (1995) EM + MRF MAP: Weiss-Adelson (1996) EM + MRF ML/MAP: Bregler-Malik (1998) Twists / EM ML/MAP: Irani (+Ananadan) (2000) SVD**Computer Vision: Vision and Modeling**• Lucas-Kanade Extensions • Support Maps / Layers: • Robust Norm, Layered Motion, Background Subtraction, Color Layers • Statistical Models (Forsyth+Ponce Chap. 6, Duda+Hart+Stork: Chap. 1-5) • - Bayesian Decision Theory • - Density Estimation**Support Maps / Layers**• Black-Jepson-95**Support Maps / Layers**• More General: Layered Motion (Jepson/Black, Weiss/Adelson, …)**Support Maps / Layers**• Special Cases of Layered Motion: • - Background substraction • - Outlier rejection (== robust norm) • - Simplest Case: Each Layer has uniform color**Support Maps / Layers**• Color Layers: P(skin | F(x,y))**Computer Vision: Vision and Modeling**• Lucas-Kanade Extensions • Support Maps / Layers: • Robust Norm, Layered Motion, Background Subtraction, Color Layers • Statistical Models (Duda+Hart+Stork: Chap. 1-5) • - Bayesian Decision Theory • - Density Estimation**Statistical Models / Probability Theory**• Statistical Models: Represent Uncertainty and Variability • Probability Theory: Proper mechanism for Uncertainty • Basic Facts White Board**General Performance Criteria**Optimal Bayes With Applications to Classification**Bayes Decision Theory**Example: Character Recognition: Goal: Classify new character in a way as to minimize probability of misclassification**Bayes Decision Theory**• 1st Concept: Priors ? P(a)=0.75 P(b)=0.25 a a b a b a a b a b a a a a b a a b a a b a a a a b b a b a b a a b a a**Bayes Decision Theory**• 2nd Concept: Conditional Probability # black pixel # black pixel**Bayes Decision Theory**• Example: X=7**Bayes Decision Theory**• Example: X=8**Bayes Decision Theory**• Example: Well… P(a)=0.75 P(b)=0.25 X=8**Bayes Decision Theory**• Example: P(a)=0.75 P(b)=0.25 X=9**Bayes Decision Theory**• Bayes Theorem:**Bayes Decision Theory**• Bayes Theorem:**Bayes Decision Theory**• Bayes Theorem: Likelihood x prior Posterior = Normalization factor**Bayes Decision Theory**• Example:**Bayes Decision Theory**• Example:**Bayes Decision Theory**• Example: X>8 class b**Bayes Decision Theory**Goal: Classify new character in a way as to minimize probability of misclassification Decision boundaries:**Bayes Decision Theory**Goal: Classify new character in a way as to minimize probability of misclassification Decision boundaries:**Bayes Decision Theory**Decision Regions: R3 R1 R2**Bayes Decision Theory**Goal:minimize probability of misclassification**Bayes Decision Theory**Goal:minimize probability of misclassification**Bayes Decision Theory**Goal:minimize probability of misclassification**Bayes Decision Theory**Goal:minimize probability of misclassification**Bayes Decision Theory**Discriminant functions: • class membership solely based on relative sizes • Reformulate classification process in terms of discriminant functions: x is assigned toCk if**Bayes Decision Theory**Discriminant function examples:

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