Final Project CSCE 790E (Medical Image Processing) Contour-Based Shape Matching - Quadratic Polynomial Interpolation of Adjacent Landmarks Final Project CSCE 790E (Medical Image Processing) Ananda M. Mondal

Objective Present a new technique for shape refinement,QPIAL, of contour representation Implementation of QPIAL before extracting shape descriptor such as Fourier descriptor Compare the shape matching results using Fourier descriptor with and without QPIAL 11/14/2018 Ananda M. Mondal

Contour Representation Set of landmark points Set of parametric curves Canny Edge Detector Contour is represented by a set of landmark points and parametric curves interpolating them Landmark points are often extracted by some low level operator such as Canny Edge Detector. Canny Edge Detector takes as input a gray scale image, and produces as output an image showing the positions of tracked intensity discontinuities. 11/14/2018 Ananda M. Mondal

Formulation Contour segment around a feature point pi = (xi, yi) Curve penetrates two adjacent feature points at t = -1 and t = 1 After obtaining the landmark points using canny edge detector, We employ quadratic polynomials to interpolate two adjacent landmark points. So, the contour segment around a feature point pi = (xi, yi) is represented parametrically as: Where, t is a parameter and ai, bi, ci, and di are coefficients. Note that the curve passes pi at t = 0. We made an assumption that the curve passes through two adjacent feature points at t = -1 and t = 1. So, we can find the coefficients as: 11/14/2018 Ananda M. Mondal

Formulation (cont.) Constraint: Two interpolating polynomials adjacent to each other have the same first order derivative at the landmark points. A constraint is suggested that two interpolating polynomials adjacent to each other have the same first order derivative at the landmark points. This means that the derivative of polynomial pi at t = -1 is equal to the derivative of polynomial pi-1 at t =0. Here polynomial pi represents the polynomial passing through the point pi. Using this constraints, we get the following relations----- 11/14/2018 Ananda M. Mondal

Formulation(cont.) Constraint Measure for pi 11/14/2018 Using this constraints, we get the following relations----- Now using these relations, we can formulate the constraint measure as: Note that: the measure is non-positive . Why? The measure is zero when pi satisfies the constraint. 11/14/2018 Ananda M. Mondal

Formulation(cont.) Update rule for pi 11/14/2018 Ananda M. Mondal Now the update rule for feature point pi can be obtained by differentiating the constraint by each attribute. So, these are the update rules 11/14/2018 Ananda M. Mondal

Cubic Spline vs. Quadratic Original Cubic Spline Quadratic Three sets of feature points are generated around a circle, a straight line, and a sine curve, the first row of this figure. Second row shows the results obtained using a cubic spline interpolator. The bottom row represents the results obtained by quadratic polynomial interpolation. The amount of noise is reduced significantly in the quadratic interpolation method. From this interpretation we can think that the new technique can be applied for shape refinement before extracting the descriptors. And this is the motivation for the present work. 11/14/2018 Ananda M. Mondal

Input Image 11/14/2018 Ananda M. Mondal

Results With Canny 11/14/2018 Ananda M. Mondal

Results with QPIAL 11/14/2018 Ananda M. Mondal

Conclusion QPIAL reduces the noise more effectively than than cubic spline does Introduction of QPIAL before extracting shape descriptor might improve the contour based shape matching 11/14/2018 Ananda M. Mondal

Future Work Apply the QPIAL method before extracting the Fourier descriptor Use the Fourier descriptor for contour-based shape matching with and without QPIAL Use SQUID database for experiment 11/14/2018 Ananda M. Mondal