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Chapter 2

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Image transformations

Digital Image Processing

Instructor: Dr. Cheng-Chien Liu

Department of Earth Sciences

National Cheng Kung University

Last updated: 4September 2003

- Content:
- Tools for DIP – linear superposition of elementary images

- Elementary image
- Outer product of two vectors
- uivjT

- Outer product of two vectors
- Expand an image
- g = hcTfhr
- f = (hcT)-1ghr-1 = SSgijuivjT
- Example 2.1

- Unitary matrix U
- U satisfies UUT* = UUH = I
- T: transpose
- *: conjugate
- UT* = UH

- U satisfies UUT* = UUH = I
- Unitary transform of f
- hcTfhr
- If hc and hr are chosen to be unitary

- hcTfhr
- Inverse of a unitary transform
- f = (hcT)-1ghr-1 = hcghrH = UgVH
- U hc; V hr

- Orthogonal matrix U
- U is an unitary matrix and its elements are all real
- U satisfies UUT = I

- Construct an unitary matrix
- U is unitary if its columns form a set of orthonormal vectors

- Diagonalize a matrix g
- g = UL1/2VT
- g is a matrix of rank r
- U and V are orthogonal matrices of size Nr
- U is made up from the eigenvectors of the matrix ggT
- V is made up from the eigenvectors of the matrix gTg

- L1/2 is a diagonal rr matrix

- Example 2.8: compute U and V from g

- g = UL1/2VT

- SVD of an image g
- g = Sli1/2uiviT, i =1, 2, …, r

- Approximate an image
- gk = Sli1/2uiviT, i =1, 2, …, k; k < r
- Error: D g – gk = Sli1/2uiviT, i = k+1, 2, …, r
- ||D|| = Sli , i = k+1, 2, …, r
- Sum of the omitted eigenvalues

- Example 2.10
- For an arbitrary matrix D, ||D|| = trace[DTD] = sum of all terms squared

- Minimizing the error
- Example 2.11

- Eigenimages
- The base images used to expand the image
- Intrinsic to each image
- Determined by the image itself
- By the eigenvectors of gTg and ggT

- Example 2.12, 2.13
- Performing SVD and identify eigenimages

- Example 2.14
- Different stages of the SVD

- Orthogonal
- A set of functions Sn(t) is said to be orthogonal over an interval [0,T] with weight function w(t) if 0Tw(t)Sn(t)Sm(t)dt =
- k if n = m
- 0 if nm

- A set of functions Sn(t) is said to be orthogonal over an interval [0,T] with weight function w(t) if 0Tw(t)Sn(t)Sm(t)dt =
- 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.

- 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 Hk(i), Wk(i)
- Normalization (N-1/2 or T-1/2)

- 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)

- 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

- 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

- 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

- Harr: use the same basic pattern

- 1D DFT
- Definition

- 2D DFT
- Definition

- Notation of DFT
- Slot machine

- Inverse DFT
- Definition

- Matrix form of DFT
- Definition

- 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

- 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

- Convolution theorem
- Discrete 2-dimensional functions: g(n, m), w(n, m)
- u(n, m) = S S 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