1. Image transformations Digital Image Processing Instructor: Dr. Cheng-Chien LiuCheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 4 September 2003 Chapter 2

2. Introduction  Content: Tools for DIP – linear superposition of elementary images  Elementary image Outer product of two vectors uivjTuivjT  Expand an image g = h c T fh r f = (h c T ) -1 gh r -1 =  g ij u i v j T Example 2.1

3. Unitary matrix  Unitary matrix U U satisfies UU T* = UU H = I  T: transpose  *: conjugate  U T* = U H  Unitary transform of f h c T fh r  If h c and h r are chosen to be unitary  Inverse of a unitary transform f = (h c T ) -1 gh r -1 = h c gh r H = UgV H U  h c ; V  h r

4. Orthogonal matrix  Orthogonal matrix U U is an unitary matrix and its elements are all real U satisfies UU T = I  Construct an unitary matrix U is unitary if its columns form a set of orthonormal vectors

5. Matrix diagonalization  Diagonalize a matrix g g = U  1/2 V T  g is a matrix of rank r  U and V are orthogonal matrices of size N  r  U is made up from the eigenvectors of the matrix gg T  V is made up from the eigenvectors of the matrix g T g   1/2 is a diagonal r  r matrix Example 2.8: compute U and V from g

6. Singular value decomposition  SVD of an image g g =  i 1/2 u i v i T, i =1, 2, …, r  Approximate an image g k =  i 1/2 u i v i T, i =1, 2, …, k; k < r Error: D  g – g k =  i 1/2 u i v i T, i = k+1, 2, …, r ||D|| =  i, i = k+1, 2, …, r  Sum of the omitted eigenvalues Example 2.10  For an arbitrary matrix D, ||D|| = trace[D T D] = sum of all terms squared Minimizing the error  Example 2.11

7. Eigenimages  Eigenimages The base images used to expand the image Intrinsic to each image Determined by the image itself  By the eigenvectors of g T g and gg T Example 2.12, 2.13  Performing SVD and identify eigenimages Example 2.14  Different stages of the SVD

8. Complete and orthogonal set  Orthogonal A set of functions S n (t) is said to be orthogonal over an interval [0,T] with weight function w(t) if  0 T w(t)S n (t)S m (t)dt =  k if n = m  0 if n  m  Orthonormal If k = 1  Complete If we cannot find any other function which is orthogonal to the set and does not belong to the set.

9. Complete sets of orthonormal discrete valued functions  Harr functions Definition  Walsh functions Definition  Harr/Walsh image transformation matrices Scale the independent variable t by the size of the matrix Matrix form of H k (i), W k (i) Normalization (N -1/2 or T -1/2 )

10. Harr transform  Example 2.18 Harr image transformation matrix (4  4)  Example 2.19 Harr transformation of a 4  4 image  Example 2.20 Reconstruction of an image and its square error  Elementary image of Harr transformation Taking the outer product of a discretised Harr function either with itself or with another one Figure 2.3: Harr transform basis images (8  8 case)

11. Walsh transform  Example 2.21 Walsh image transformation matrix (4  4)  Example 2.22 Walsh transformation of a 4  4 image  Hadamard matrices An orthogonal matrix with entries only +1 and –1 Definition Walsh functions can be calculated in terms of Hadamard matrices  Kronecker or lexicographic ordering

12. Hadamard/Walsh transform  Elementary image of Hadamard/Walsh transformation Taking the outer product of a discretised Hadamard/Walsh function either with itself or with another one Figure 2.4: Hadamard/Walsh transform basis images (8  8 case) Example 2.23  Different stages of the Harr transform Example 2.24  Different stages of the Hadamard/Walsh transform

13. Assessment of the Hadamard/Walsh and Harr transform  Higher order basis images Harr: use the same basic pattern  Uniform distribution of the reconstruction error  Allow us to reconstruct with different levels of detail different parts of an image Hadamard/Walsh: approximate the image as a whole, with uniformly distributed details  Don’t take 0  Easier to implement

14. Discrete Fourier transform  1D DFT Definition  2D DFT Definition  Notation of DFT Slot machine  Inverse DFT Definition  Matrix form of DFT Definition

15. Discrete Fourier transform (cont.)  Example 2.25 DFT image transformation matrix (4  4)  Example 2.26 DFT transformation of a 4  4 image  Example 2.27 DFT image transformation matrix (8  8)  Elementary image of DFT transformation Taking the outer product between any two rows of U DFT transform basis images (8  8 case)  Figure 2.7: Real parts  Figure 2.8: Imaginary parts

16. Discrete Fourier transform (cont.)  Example 2.28 DFT transformation of a 4  4 image  Example 2.29 Different stages of DFT transform  Advantages of DFT Obey the convolution theorem Use very detailed basis functions  error   Disadvantage of DFT Retain n basis images requires 2n coefficients for the reconstruction

17. Convolution theorem  Convolution theorem Discrete 2-dimensional functions: g(n, m), w(n, m) u(n, m) =  g(n-n’, m-m’)w(n’, m’)  n’ = 0 ~ N-1  m’ = 0 ~ M-1 Periodic assumptions  g(n, m) = g(n-N, m-M) = g(n-N, m) = g(n, m-M)  w(n, m) = w(n-N, m-M) = w(n-N, m) = w(n, m-M) û(p, q) = (MN) 1/2 ĝ(p, q) ŵ(p, q)  The factor appears because we defined the discrete Fourier transform so that the direct and the inverse ones are entirely symmetric