EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

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1 EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines

2 What is image processing? Image processing is the application of 2D signal processing methods to images Image representation and modeling Image enhancement Image restoration/image filtering Compression (computer vision)

3 Image enhancement Accentuate certain desired features for subsequent analysis or display. Contrast stretching / dynamic range adjustment Color histogram normalization Noise reduction Sharpening Edge detection Corner detection Image interpolation

4 What is interpolation? given a discrete space image g d [n,m] This corresponds to samples of some continuous space image g a (x, y) Compute values of the CS image g a (x, y) at (x,y) locations other than the sample locations. g a (x, y) g d [n,m] g d [n,m]

5 What is interpolation? Interpolation is critical for: Displaying Image zooming Warping Coding Motion estimation

6 Is perfect recovery always possible? No, (never actually), unless some assumptions on g a (x,y) are made There are uncountable infinite collection of functions g a (x, y) that agree with g d [n,m] at the sample points What are these assumptions? g a must be band limited Sampling rate must satisfy Nyquist theorem

7 g a must be band limited There exists such that

8 Sampling rate must satisfy Nyquist theorem v v v

9 Ideal Uniform Rectilinear Sampling 2D uniformly sampled grid

10 Ideal Uniform Rectilinear Sampling g a (x,y) g s (x,y)

11 g a (x,y,) How to recover g a?

12 How to recover g a?

13 Sinc interpolation We can recover g a (x, y) by interpolating the samples g d [n,m] using sinc functions

14 Interpolation Sampled g s MATLAB s interp2

15 What s the problem with sinc interpolation? Real world images need not be exactly bandlimited. Unbounded support Summations require infinitely many samples Computationally very expensive

16 Linear interpolation sinc interpolation formula Linear interpolation: h(x, y) = interpolation kernel Why this is a linear interpolation? Linear function of the samples g d [n,m]

17 Linear interpolation Consider alternative linear interpolation schemes that address issues with: Unbounded support Summations require infinitely many samples Computationally very expensive

18 Basic polynomial interpolation Here kernels that are piecewise polynomials Zero order or nearest neighbor interpolation Use the value at the nearest sample location Kernels are zero order polynomials

19 Nearest neighbor interpolation

20 Linear or bilinear interpolation tri function is a piecewise linear function of its (spatial) arguments Δ/2 Δ/2

21 Linear or bilinear interpolation For any point (x, y), we find the four nearest sample point fit a polynomial of the form Estimate alphas from system of 4 equations in 4 unknowns: Interpolating formula (x [0, 1] [0, 1]):

22 Linear or bilinear interpolation 0 1 If x [ 1, 1] [ 1, 1]: 1 1

23

24 Linear or bilinear interpolation

25 Linear interpolation by Delaunay triangulation Use a linear function within each triangle (three points determine a plane). MATLAB s griddata with the linear opt

26

27

28 What s the problem zero order or bilinear interpolation? neither the zero order or bilinear interpolator are differentiable. They are not continuous and smooth

29 Desirable properties of interpolators self consistency : Continuous and smooth (differentiable): Short spatial extent to minimize computation

30 Desirable properties of interpolators Frequency response approximately rect(). symmetric Shift invariant Minimum sidelobes to avoid ringing artifacts

31 Continuous and smooth Large sidelobes Non smooth Small sidelobes

32 Non smooth; non shift invariant

33 Cubic Polynomial interpolation interpolation kernel: Quadratic B splines

34

35 B spline interpolation Motivation: n order differentiable Short spatial extent to minimize computation

36 Background Splines are piecewise polynomials with pieces that are smoothly connected together The joining points of the polynomials are called knots

37 Background For a spline of degree n, each segment is a polynomial of degree n Do i need n+1 coefficient to describe each piece? Additional smoothness constraint imposes the continuity of the spline and its derivatives up to order (n 1) at the knots only one degree of freedom per segment!

38 B spline expansion Splines are uniquely characterized in terms of a B spline expansion integer shifts of the central B spline of degree n = central B spline = basis (B) spline

39 B splines Symmetrical bell shaped functions constructed from the (n+1) fold convolution

40 B splines

41 B splines n order B spline: Derived using:

42 B spline Using cubic B splines is a popular choice: Important: compactly supported!

43 Cubic B spline and its underlying components support

44 B spline expansion Each spline is unambiguously characterized by its sequence of B spline coefficients c(k)

45 Properties of B spline expansion Discrete signal representation! (even though the underlying model is a continuous representation). Easy to manipulate; E.g., derivatives:

46 B spline interpolation So far: B spline model of a given input signal s(x) s d [n] Interpolation problem: find coefficients c(k) such that spline function goes through the data points exactly That is, reconstruct signal using a spline representation!

47 Reconstruct signal using a spline representation s d [n]

48 B spline interpolation Relationship between coefficients and samples s d [n] = discrete B spline kernel Obtained by sampling the B spline of degree n expanded by a factor of m (m=1)

49 Cardinal B spline interpolation Cardinal splines

50 2D splines in images tensor product basis functions

51 B spline interpolation Conclusion: n order differentiable Short spatial extent to minimize computation

52 General image spatial transformations transform the spatial coordinates of an image f(x, y) so that (after being transformed), it better matches another image Ex: warping brain images into a standard coordinate system to facilitate automatic image analysis

53 image spatial transformations

54 General image spatial transformations Form of transformations: Shift only g(x, y) = f(x x 0, y y 0 ) Affine g(x, y) = f(ax + by, cx + dy) thin plate splines (describes smooth warpings using control points) General spatial transformation (e.g., morphing or warping an image) MATLAB s interp2 or griddata functions

55 Interpolation is critical!

56

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