# Cap4_Mihaela_all_English.ppt

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Digital image processing Digital image transforms 4. DIGITAL IMAGE TRANSFORMS 4.1. Introduction 4.2. Unitary orthogonal two-dimensional transforms Separable unitary transforms 4.3. Properties of the unitary transforms Energy conservation Energy compaction; the variance of coefficients De-correlation Bas
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Digital image processing   Digital image transforms   4. DIGITAL IMAGE TRANSFORMS 4.1. Introduction 4.2. Unitary orthogonal two-dimensional transforms Separable unitary transforms 4.3. Properties of the unitary transforms  Energy conservation  Energy compaction; the variance of coefficients  De-correlation  Basis functions and basis images 4.4. Sinusoidal transforms The 1-D discrete Fourier transform (1-D DFT)  Properties of the 1-D DFT The 2-D discrete Fourier transform (2-D DFT)  Properties of the 2-D DFT The discrete cosine transform (DCT) The discrete sine transform (DST) The Hartley transform 4.5. Rectangular transforms The Hadamard transform = the Walsh transform The Slant transform The Haar transform 4.6. Eigenvectors-based transforms The Karhunen-Loeve transform (KLT) The fast KLT The SVD 4.7. Image filtering in the transform domain   4.8. Conclusions  4.1 INTRODUCTION Definition:   Image transform   = operation to change the default representation space of a digital image (spatial domain -> another domain) so that: (1)   all the information present in the image is preserved in the transformed domain, but represented differently; (2)   the transform is reversible, i.e., we can revert to the spatial domain Generally : in the transformed domain -> image information is represented in a more compact form => main goal of the transforms: image compression.   Other usage: image analysis - a new type of representation of different types of information present in the image.  Note: Most image transforms =  “generalizations” of frequency transforms  => the representation of the image by a DC component and several AC components. Definition :  “srcinal representation space” of the image U [M×N] = a MN-dimensional space: -   each coordinate of the space = a spatial location (m,n) in the digital image; -   the value of the coordinate of U  on an axis = the grey level in U  in this spatial location (m,n). x 1 =(0,0); x 2 =(0,1); x 3 =(0,2); ... x MN =(M-1,N-1). => A unitary transform of the image U  = a rotation  of the MN-dimensional space, defined by a rotation matrix  A   in MN-dimensions. Digital image processing   Digital image transforms      {u(n), 0 n N 1}     ;  A –  unitary matrix, T*1 AA        1 N 0n1 N k 0 ,n)u(n)a(k,v(k)r o , Auv  (4.1)     1 N 0k *T * 1 N n0 ,n)v(k)(k,au(n)or  vAu  (4.2)  }1 N n0 ,n)(k,{aa  **k    , 4.2 UNITARY ORTHOGONAL TWO-DIMENSIONAL TRANSFORMS   v(k,l) a (m,n) u(m,n), 0 k,l N 1 k,l n 0 N 1m 0 N 1         (4.3) u(m,n) a (m,n) v(k,l), 0 k,l N 1 k,l *l 0 N 1k 0 N 1         (4.4)    orthonormality :     1 N0m1 N0n*'l,'k l,k  )'ll,'k k ()n,m(a)n,m(a   (4.5)      completeness : a m n a m n m m n n k l k l l  N k  N  ,,* (,)(',')(',')          0101   (4.6)   u (m,n) a (m,n) v(k,l), P N,Q N   P,Q k,l *l 0Q 1k 0 P 1        (4.7)   e2n 0 N 1m 0 N 1 P,Q2 [u(m,n) u (m,n)]      (4.8)  N Q P  f 0 2e    i      Digital image processing   Digital image transforms

Jul 23, 2017

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