1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. Input Image
11/14/2018
Ananda M. Mondal

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

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

12. 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

13. 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