Chapter 2
This presentation is the property of its rightful owner.
Sponsored Links
1 / 17

Chapter 2 PowerPoint PPT Presentation


  • 58 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Chapter 2

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Chapter 2

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

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

  • 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

  • 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

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

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

  • 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

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

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

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

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

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

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

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

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 cont1

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

  • 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


  • Login