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# Chapter 2 - PowerPoint PPT Presentation

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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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

• 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

• 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 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 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

• 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 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.

• 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

• 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

• 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

• Obey the convolution theorem

• Use very detailed basis functions  error 

• 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