Lecture 10 (Digital Image Processing)
- 1. Numerical Simulation Exercise on B-spline Function Subject: Image Procesing & Computer Vision Dr. Varun Kumar Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 1 / 12
- 2. Outlines 1 Graphical Representation of B-spline Function 2 Relation Between Sampling and Interpolation 3 B-spline function 4 References Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 2 / 12
- 3. Resampling and interpolation 1D sampling operation In this given figure, no value of amplitude exist for intermediate location in time axis. We need interpolation for solving such a problem. Desirable properties of interpolation: 1 Finite region of support 2 Smooth interpolation no-discontinuity 3 Shift invariant Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 3 / 12
- 4. B-spline function: It is a piece wise polynomial function. It is useful for local approximation of a curve. Mathematical representation: x(t) = n i=0 pi Bi,k(t) (1) pi → control point how the B-spline function should be guided for smooth curve. Bi,k(t) → Normalized B-spline of order k Bi,1(t) = 1 ∀ ti < t < 1 = 0 otherwise (2) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 4 / 12
- 5. Continued– Bi,k(t) = (t − ti )Bi,k−1(t) ti+k−1 − ti + (ti+1 − 1)Bi+1,k1(t) ti+k − ti+1 (3) Conversion Bi,k(t) = B0,k(t − i) or B0,1(t) = 1 ∀ 0 ≤ t < 1 = 0 otherwise (4) or B0,2(t) = t ∀ 0 ≤ t < 1 = 2 − t ∀ 1 ≤ t < 2 = 0 otherwise (5) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 5 / 12
- 6. Continued– B0,3(t) = t2 /2 ∀ 0 ≤ t < 1 = −t2 + 3t − 1.5 ∀ 1 ≤ t < 2 = (3 − t)2 /2 ∀ 2 ≤ t < 3 = 0 otherwise (6) or B0,4(t) = t3 /6 ∀ 0 ≤ t < 1 = −3t3 + 12t2 − 12t + 4 6 ∀ 1 ≤ t < 2 = 3t3 − 24t2 + 60t − 44 6 ∀ 2 ≤ t < 3 = (4 − t)3 6 ∀ 3 ≤ t < 4 = 0 otherwise (7) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 6 / 12
- 7. Different interpolation Constant interpolation Linear interpolation Quadratic interpolation Cubic and higher order interpolation f (t) = n i=0 pi Bi,1(t) ⇒ Bi,1 = 1 Q A sampled data is given as x(n)= {1, 2, 0.75, 0.55, 0.8, 0.75, 0.9, 1.5, 1.1, 0.45, 0.35} ∀ 0 ≤ n ≤ 10 = 0 otherwise Convert it into a continuous time signal x(t) ∀ 0 ≤ t ≤ 10 through the different B-spline interpolation function. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 7 / 12
- 8. Graphical representation of B-spline function 0 5 10 0 0.5 1 1.5 B0,1 (t) 0 5 10 0 0.5 1 1.5 B0,2 (t) 0 5 10 0 0.5 1 1.5 B0,3 (t) 0 5 10 0 0.5 1 1.5 B0,4 (t) Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 8 / 12
- 9. Constant Interpolation 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Sampled data 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Constant Interpolation Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 9 / 12
- 10. Linear Interpolation 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Sample Data 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Linear Interpolation Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 10 / 12
- 11. Cubic Interpolation 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Sample Data 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 Cubic Interpolation Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 11 / 12
- 12. References M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision. Cengage Learning, 2014. D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern approach, vol. 17, pp. 21–48, 2003. L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey, 2001. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB. Pearson Education India, 2004. Subject: Image Procesing & Computer Vision Dr. Varun KumarLecture 10 12 / 12