Gopalkrishna Veni. Project 4 (Active Shape Models)

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Marshall French
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1 Gopalkrishna Veni Project 4 (Active Shape Models) Introduction Active shape Model (ASM) is a technique of building a model by learning the variability patterns from training datasets. ASMs try to deform the model in a way that represents within the population of datasets. It uses Principal component analysis (PCA) to determine the principal variations of the population of shapes. Solving PCA provides eigenvalues which determine the magnitude of the variation and eigenvectors which describe the new subspace. The advantage of using PCA is dimensionality reduction. In other words, shapes can be represented by a small set of coefficients. Implementation As instructed, a square shape has been generated by selecting 8 points, 4 on the corners and 4 in between them. They are arranged in the following vector form T For transformation, the above vector has been randomly translated, scaled, translated and scaled, and rotated. Moreover, the shape has also been modified to convert it into trapezoidal form. At the end, all these transformed shapes have been combined and then analyzed. To deform, each transformation has been separately tested by modifying the basic shape 3 times randomly. The rationale behind using more number of shapes than the number of sampling points is to ensure full rank of the covariance matrix. To implement PCA, first consider the mean shape which is given by The shape covariance matrix is given by In order to visualize the correlation of pairs, the correlation matrix has also been calculated which can be said as the normalization of the covariance matrix. To obtain eigenvalues and eigenvectors that define the magnitude and modes of variation, Singular Value Decomposition (SVD) has been used. To display the major modes of variation, shapes are synthesized by deviating from the mean along each eigenvector, defined by a factor s applied to the standard deviations which can range from -2 to +2. Principle_shape [i, s]=mean Image +s* Eig. Value[i]*Eig. Vector[i]

2 Adding noise In order to check the effect of noise on the population of shapes, uniformly distributed noise has been added to the transformed shapes. It can be noticed that the addition of noise to each shape results in increase in dimensionality. In other words, more number of coefficients are required than usual to determine the variations in population of shapes. This has been be illustrated in the results section. Results The results below show the transformed versions of a basic square shape. Next, the correlation matrix, eigenvalue plot, cumulative function of normalized eigenvalues, and principal modes of variation have been shown. Cumulative function plot can be used in selecting an adequate number of principal components that represent smoothed versions of shape. For every transformation, 3 shapes have been considered. The correlation matrix is a square symmetrical MxM matrix with the (ij)th element equal to the correlation coefficient rij between the (i)th and the (j)th variable. It is mathematically defined as The diagonal elements of the correlation matrix are always equal to 1. In a gray level Correlation matrix image, the white pixel represents positive correlation, black represents negative correlation and gray pixel represents no correlation. Translation of basic square shape in X- and Y-axis Translated versions of basic square shape 16x16 Correlation matrix

3 First two modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the translation has two degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only two coefficients are sufficient to represent the principal variations in population of shape. Scaling of basic square shape in X- and Y-axis Translated versions of basic square shape 16x16 Correlation matrix

4 First two modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the scaling also has two degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only two coefficients are sufficient to represent the principal variations in population of shape. Translation and Scaling of basic square shape in X- and Y-axis Translated and scaled versions of basic square shape 16x16 Correlation matrix

four degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only four coefficients

5 First four modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the translation and scaling has four degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only four coefficients are sufficient to represent the principal variations in population of shape. Rotation of basic square shape in X- and Y-axis Rotated versions of basic square shape 16x16 Correlation matrix

6 First two modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the rotation has two degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only two coefficients are sufficient to represent the principal variations in population of shape. Shape variation: Translated versions of basic trapezoidal shape 16x16 Correlation matrix

7 First two modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the translation has two degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only two coefficients are sufficient to represent the principal variations in population of shape. Combination of all the above transformations Combination of all the above transformations 16x16 Correlation matrix

8 First seven modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Since the transfromation has seven degrees of freedom in x and y-axis, it is obvious from the eigenvalue plot that only seven coefficients are sufficient to represent the principal variations in population of shape. Transformation with noise addition As mentioned earlier, by adding noise to each point of the shape, the number of coefficients required to represent the principal variations increase. This can be seen in the following illustrations. Translation of basic square shape in X- and Y-axis and addition of random noise Translated versions of Basic Square shape with noise added 16x16 Correlation matrix

considered to determine the principal variations in population of shapes. However, only first 3 eigenmodes are considered.

9 First three modes of variation. Each parameter is varied independently by ±s Eigen value (s represents From the eigenvalue plot, eigenvalues contributing to the 9% of cumulative function are considered to determine the principal variations in population of shapes. However, only first 3 eigenmodes are considered. This authenticates the fact that by addition of noise, the number of eigenmodes to represent major shape variations increase. This can be seen in every transformation illustrated below. Scaling of basic square shape in X- and Y-axis and addition of random noise scaled versions of Basic Square shape with noise added 16x16 Correlation matrix

10 First two modes of variation. Each parameter is varied independently by ±s Eigen value (s represents From the eigenvalue plot, eigenvalues contributing to the 9% of cumulative function are considered to determine the principal variations in population of shapes. Accordingly, 2 eigenmodes are considered. Combination of all the above transformations with noise added Combined transformed versions of Basic Square shape with noise added 16x16 Correlation matrix

11 First seven modes of variation. Each parameter is varied independently by ±s Eigen value (s represents Thus, the combinational transformation includes translation, scaling, rotation, shape variation on the top of adding noise. From the eigenvalue plot, eigenvalues contributing to the 9% of cumulative function are considered to determine the principal variations in population of shapes. Accordingly, first 7 eigenmodes are considered.

The analysis on each of them is illustrated in the following figures Corpus Calossum For CC, 3 shapes have been considered

12 Additional Exploratory Analysis: Bonus The ASM analysis has been extended to two more shapes namely corpus calossum (CC) and hand shapes. The analysis on each of them is illustrated in the following figures Corpus Calossum For CC, 3 shapes have been considered with 12 points each (x and y) so that 24 coordinates are included for 3 shapes. Since the acquired file has shapes along rows and points along columns. A matrix transpose has been used. Family of shapes for CC 24x24 Correlation matrix

determine the principal variations in population of shapes. Accordingly, first 4 eigenmodes are considered.

13 First four modes of variation. Each parameter is varied independently by ±s Eigen value (s represents From the eigenvalue plot, eigenvalues contributing to the 9% of cumulative function are considered to determine the principal variations in population of shapes. Accordingly, first 4 eigenmodes are considered. Hand shape The hand dataset includes 4 hand shapes with 56 contour points each (x and y). However, the text file provides first 52 x-coordinates representing each shape followed by 56 y-coordinates. So, a small script is required to modify the file such that x and y coordinates of each shape come successively in order to perform further analysis. Family of shapes for hand 112x112 Correlation matrix

14 First mode of variation. Each parameter is varied independently by ±s Eigen value (s represents standard deviation, s=2). Second mode of variation. Each parameter is varied independently by ±s Eigen value (s represents

15 Third mode of variation. Each parameter is varied independently by ±s Eigen value (s represents From the eigenvalue plot, eigenvalues contributing to the 9% of cumulative function are considered to determine the principal variations in population of shapes. Accordingly, first 3 eigenmodes are considered. A list of eigen values obtained for applied deformations on different shapes has been noted down and shown in table below Square translation scaling trans & scale rotation trapezoid combine E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-2 2.E E E E E E E E E-2 2.2E E E E E E E E E E E E E E E E E E E E E E-18 Square with noise added

16 translation scale trans & scale combine CC and hand For convenience, I have enlisted only the first 15 eigen values which represent major principal variations. Others have been neglected since the variation values turn out to be infinitesimal. CC Hands Discussion Thus, ASM has been analyzed on a variety of shapes. It can be thought as a statistical model that describes the mean shape along with a small set of modes of

17 variation that characterizes how the object s shape can change. ASM is an iterative search procedure that is capable of generating a model even in a noisy environment. In square shape example, I have also tried varying the number of points to represent shape. But, the results turned out to be same. However, if there is a smooth structure like hand or CC, I anticipate that by increasing the number of points, one may obtain a smooth generic structure.

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