Active Shape Models: Their Training and Applications Cootes, Taylor, et al. Robert Tamburo July 6, 2000 Prelim Presentation

Other Deformable Models “Hand Crafted” Models Articulated Models Active Contour Models – “Snakes” Fourier Series Shape Models Statistical Models of Shape Finite Element Models

Motivation – Prior Models Lack of practicality Lack of specificity Lack of generality Nonspecific class deformation Local shape constraints

Goals of Active Shape Model (ASM) Automated Searches images for represented structures Classify shapes Specific to ranges of variation Robust (noisy, cluttered, and occluded image) Deform to characteristics of the class represented “Learn” specific patterns of variability from a training set

Goals of ASM (cont’d.) Utilize iterative refinement algorithm Apply global shape constraints Uncorrelated shape parameters Better test for dependence?

Point Distribution Model (PDM) Captures variability of training set by calculating mean shape and main modes of variation Each mode changes the shape by moving landmarks along straight lines through mean positions New shapes created by modifying mean shape with weighted sums of modes

PDM Construction Manual LabelingAlignment Statistical Analysis Point Distribution Model

Labeling the Training Set Represent example shapes by points Point correspondence between shapes

Aligning the Training Set x i is a vector of n points describing the the i th shape in the set: x i =(x i0, y i0, x i1, y i1,……, x ik, y ik,……,x in-1, y in-1 ) T Minimize: E j = (x i – M(s j,  j )[x k ] – t j ) T W(x i – M(s j,  j )[x k ] – t j ) Weight matrix used:

Alignment Algorithm Align each shape to first shape by rotation, scaling, and translation Repeat – Calculate the mean shape – Normalize the orientation, scale, and origin of the current mean to suitable defaults – Realign every shape with the current mean Until the process converges

Mean Normalization Ensures 4N constraints on 4N variables Equations have unique solutions Guarantees convergence Independent of initial shape aligned to Iterative method vs. direct solution

Aligned Shape Statistics PDM models “cloud” variation in 2n space Assumptions: – Points lie within “Allowable Shape Domain” – Cloud is hyper-ellipsoid (2n-D)

Statistics (cont’d.) Center of hyper-ellipsoid is mean shape Axes are found using PCA – Each axis yields a mode of variation – Defined as, the eigenvectors of covariance matrix, such that,where is the k th eigenvalue of S

Approximation of 2n-D Ellipsoid Most variation described by t-modes Choose t such that a small number of modes accounts for most of the total variance If total variance = and the approximated variance =, then

Generating New Example Shapes Shapes of training set approximated by:, where is the matrix of the first t eigenvectors and is a vector of weights Vary b k within suitable limits for similar shapes

Application of PDMs Applied to: – Resistors – “Heart” – Hand – “Worm” model

Resistor Example 32 points 3 parameters capture variability

Resistor Example (cont.’d) Lacks structure Independence of parameters b 1 and b 2 Will generate “legal” shapes

Resistor Example (cont.’d)

“Heart” Example 66 examples 96 points – Left ventricle – Right ventricle – Left atrium Traced by cardiologists

“Heart” Example (cont.’d)

Varies Width Varies Septum Vary LV Vary Atrium

Hand Example 18 shapes 72 points 12 landmarks at fingertips and joints

Hand Example (cont.’d) 96% of variability due to first 6 modes First 3 modes Vary finger movements

“Worm” Example 84 shapes Fixed width Varying curvature and length

“Worm” Example (cont.’d) Represented by 12 point Breakdown of PDM

“Worm” Example (cont.’d) Curved cloud Mean shape: – Varying width – Improper length

“Worm” Example (cont.’d) Linearly independent Nonlinear dependence

“Worm” Example Effects of varying first 3 parameters: 1 st mode is linear approximation to curvature 2 nd mode is correction to poor linear approximation 3 rd approximates 2 nd order bending

PDM Improvements Automated labeling 3D PDMs Nonlinear PDM – Polynomial Regression PDMs Multi-layered PDMs Hybrid PDMs Chord Length Distribution Model Approximation problem Approximation

PDMs to Search an Image - ASMs Estimate initial position of model Displace points of model to “better fit” data Adjust model parameters Apply global constraints to keep model “legal”

Adjusting Model Points Along normal to model boundary proportional to edge strength Vector of adjustments:

Calculating Changes in Parameters Initial position: Move X as close to new position (X + dX) Calculate dx to move X to dX Update parameters to better fit image Not usually consistent with model constraints Residual adjustments made by deformation where

Model Parameter Space Transforms dx to parameter space giving allowable changes in parameters, db Recall: – Find db such that – - yields Update model parameters within limits

Applications Medical Industrial Surveillance Biometrics

ASM Application to Resistor 64 points (32 type III) Adjustments made finding strongest edge Profile 20 pixels long 5 degrees of freedom 30, 60, 90, 120 iterations

ASM Application to “Heart” Echocardiogram 96 points 12 degrees of freedom Adjustments made finding strongest edge Profile 40 pixels long Infers missing data (top of ventricle)

ASM Application to Hand 72 points Clutter and occlusions 8 degrees of freedom Adjustments made finding strongest edge Profile 35 pixels long 100, 200, 350 iterations

Conclusions Sensitivity to orientation of object in image to model Sensitivity to large changes in scale? Sensitive to outliers (reject or accept) Sensitivity to occlusion Quantitative measures of fit Overtraining Occlusion, cluttering, and noise Dependent on boundary strength Real time Extension to 3 rd dimension Gray level PDM

MR of Brain 2 Improves ASM – Tests several model hypotheses – Outlier detection adjustment/removal 114 landmark points 8 training images Model structures of brain together Model brain structures

MR Brain (cont.’d)

SeparateTogether

MR Brain (cont.’d)

References 1 – Cootes, Taylor, et al., “Active Shape Models: Their Training and Application.” Computer Vision and Image Understanding, V16, N1, January, pp , Duta and Sonka, “Segmentation and Interpretation of MR Brain Images: An Improved Active Shape Model.” IEEE Transactions on Medical Imaging, V17, N6, December 1998

“Hand Crafted” Models Built from subcomponents (circles, lines, arcs) Some degree of freedom May change scale, orientation, size, and position Lacks generality Detailed knowledge of expected shapes Application specific back

Articulated Models Built from rigid components connected by sliding or rotating joints Uses generalized Hough transform Limited to a restricted class of variable shapes back

Active Contours – “Snakes” Energy minimizing spline curvesspline Attracted toward lines and edges Constraints on stiffness and elastic parameters ensure smoothness and limit degree to which they can bend Fit using image evidence and applying force to the model and minimize energy function Uses only local information Vulnerable to initial position and noise back

Spline Curves Splines are piecewise polynomial functions of order d Sum of basis functions with applied weights Spans joined by knots back

Fourier Series Shape Models Models formed from Fourier series Fit by minimizing energy function (parameters) Contains no prior information Not suitable for describing general shapes: – Finite number of terms approximates a square corner – Relationship between variations in shape and parameters is not straightforward back

Statistical Models of Shape Register “landmark” points in N-space to estimate: – Mean shape – Covariance between coordinates Depends on point sequence back

Finite Element Models Model variable objects as physical entities with internal stiffness and elasticity Build shapes from different modes of vibration Easy to construct compact parameterized shapes next