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Chapter 2. Image transformations Digital Image Processing Instructor: Dr. Cheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 4 September 2003. Introduction. Content: Tools for DIP – linear superposition of elementary images Elementary image

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Chapter 2

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Chapter 2

Image transformations

Digital Image Processing

Instructor: Dr. Cheng-Chien Liu

Department of Earth Sciences

National Cheng Kung University

Last updated: 4September 2003


Introduction

  • Content:

    • Tools for DIP – linear superposition of elementary images

  • Elementary image

    • Outer product of two vectors

      • uivjT

  • Expand an image

    • g = hcTfhr

    • f = (hcT)-1ghr-1 = SSgijuivjT

    • Example 2.1


Unitary matrix

  • Unitary matrix U

    • U satisfies UUT* = UUH = I

      • T: transpose

      • *: conjugate

      • UT* = UH

  • Unitary transform of f

    • hcTfhr

      • If hc and hr are chosen to be unitary

  • Inverse of a unitary transform

    • f = (hcT)-1ghr-1 = hcghrH = UgVH

    • U  hc; V  hr


Orthogonal matrix

  • 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


Matrix diagonalization

  • Diagonalize a matrix g

    • g = UL1/2VT

      • 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 ggT

        • V is made up from the eigenvectors of the matrix gTg

      • L1/2 is a diagonal rr matrix

    • Example 2.8: compute U and V from g


Singular value decomposition

  • 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

  • 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


Complete and orthogonal set

  • 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 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.


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 Hk(i), Wk(i)

    • Normalization (N-1/2 or T-1/2)


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)


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


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


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


Discrete Fourier transform

  • 1D DFT

    • Definition

  • 2D DFT

    • Definition

  • Notation of DFT

    • Slot machine

  • Inverse DFT

    • Definition

  • Matrix form of DFT

    • Definition


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


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


Convolution theorem

  • 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


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