sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
30 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES Also let DFT N denote the N-point DFT matrix DFT N = {e −i 2π N kl } k=0,1,2,... ,N−1; , l=0,1,2,... ,N−1 where k is the row index and l is the column index. We have ⃗̂X = DFT N · ⃗X. Apparently DFT N is an N by N square symmetric matrix. Both X ⃗ and ⃗̂X are N- dimensional vectors. It is well known that the matrix DFT N is orthogonal, which is equivalent to saying that the inverse of DFT N is its complex conjugate transpose. Thus, DFT is an orthogonal transform and consequently it is an isometric transform. This result is generally attributed to Parseval. Since the concise statements and proofs can be found in many textbooks, we skip the proof. Why Fourier Transform? The preponderant reason that Fourier analysis is so powerful in analyzing linear time invariant system and cyclic-stationary time series is the fact that Fourier series are the eigenvectors of matrices associated with these transforms. These transforms are ubiquitous in DSP and time series analysis. Before we state the key result, let us first introduce some mathematical notation: Toeplitz matrix, Hankel matrix and circulant matrix. Suppose A N×N is an N by N real-valued matrix: A = {a ij } 1≤i≤N,1≤j≤N with all a ij ∈ R. The matrix A is a Toeplitz matrix when the elements on the diagonal, and the rows that are parallel to the diagonal, are the same, or mathematically a ij = t i−j [74, page 193]. The following is a Toeplitz matrix: ⎛ T = ⎜ ⎝ ⎞ t 0 t −1 ... t −(n−2) t −(n−1) t 1 t 0 ... t −(n−3) t −(n−2) . . . .. . . . t n−2 t n−3 ... t 0 t −1 ⎟ ⎠ t n−1 t n−2 ... t 1 t 0 N×N
3.1. DCT AND HOMOGENEOUS COMPONENTS 31 A matrix A is a Hankel matrix when its elements satisfying a ij = h i+j−1 . A Hankel matrix is symbolically a flipped (left-right) version of a Toeplitz matrix. The following is a Hankel matrix: ⎛ ⎞ h 1 h 2 ... h n−1 h n h 2 h 3 ... h n h n+1 H = . . . .. . . . ⎜ ⎝ h n−1 h n ... h 2n−3 h 2n−2 ⎟ ⎠ h n h n+1 ... h 2n−2 h 2n−1 A matrix A is called a circulant matrix [74, page 201] when a ij = c {i−j mod N} , where x mod N is the non-negative remainder after a modular division. The following is a circulant matrix: ⎛ ⎞ c 0 c N−1 ... c 2 c 1 c 1 c 0 ... c 3 c 2 C = . . ... . . . (3.2) ⎜ ⎝ c N−2 c N−3 ... c 0 c N−1 ⎟ ⎠ c N−1 c N−2 ... c 1 c 0 Obviously, a circulant matrix is a special type of Toeplitz matrix. The key mathematical result that makes Fourier analysis so powerful is the following theorem. Theorem 3.1 For an N × N circulant matrix C as defined in (3.2), the Fourier series {e −i 2π N kl : k =0, 1, 2,... ,N − 1}, forl =0, 1, 2,... ,N − 1, are eigenvectors of the matrix C, and the Fourier transforms of the sequence { √ Nc 0 , √ Nc 1 ,... , √ Nc N−1 } are its eigenvalues. A sketch of the proof is given in Section 3.7. There are two important applications of this theorem. (Basically, they are the two problems that were mentioned at the beginning of this subsubsection.) 1. For a cyclic linear time-invariant (LTI) system, the transform matrix is a circulant matrix. Thus, Fourier series are the eigenvectors of a cyclic linear time invariant system, and the Fourier transform of the square root N amplified impulse response is N×N N×N
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3.1. DCT AND HOMOGENEOUS COMPONENTS 31<br />
A matrix A is a Hankel matrix when its elements satisfying a ij = h i+j−1 . A Hankel<br />
matrix is symbolically a flipped (left-right) version of a Toeplitz matrix. The following is a<br />
Hankel matrix:<br />
⎛<br />
⎞<br />
h 1 h 2 ... h n−1 h n<br />
h 2 h 3 ... h n h n+1<br />
H =<br />
. . . .. . .<br />
.<br />
⎜<br />
⎝ h n−1 h n ... h 2n−3 h 2n−2<br />
⎟<br />
⎠<br />
h n h n+1 ... h 2n−2 h 2n−1<br />
A matrix A is called a circulant matrix [74, page 201] when a ij = c {i−j mod N} , where<br />
x mod N is the non-negative remainder after a modular division. The following is a circulant<br />
matrix:<br />
⎛<br />
⎞<br />
c 0 c N−1 ... c 2 c 1<br />
c 1 c 0 ... c 3 c 2<br />
C =<br />
. . ... . .<br />
. (3.2)<br />
⎜<br />
⎝ c N−2 c N−3 ... c 0 c N−1<br />
⎟<br />
⎠<br />
c N−1 c N−2 ... c 1 c 0<br />
Obviously, a circulant matrix is a special type of Toeplitz matrix.<br />
The key mathematical result that makes Fourier analysis so powerful is the following<br />
theorem.<br />
Theorem 3.1 For an N × N circulant matrix C as defined in (3.2), the Fourier series<br />
{e −i 2π N kl : k =0, 1, 2,... ,N − 1}, forl =0, 1, 2,... ,N − 1, are eigenvectors of the matrix<br />
C, and the Fourier <strong>transforms</strong> of the sequence { √ Nc 0 , √ Nc 1 ,... , √ Nc N−1 } are its<br />
eigenvalues.<br />
A sketch of the proof is given in Section 3.7.<br />
There are two important applications of this theorem. (Basically, they are the two<br />
problems that were mentioned at the beginning of this subsubsection.)<br />
1. For a cyclic linear time-invariant (LTI) system, the transform matrix is a circulant<br />
matrix. Thus, Fourier series are the eigenvectors of a cyclic linear time invariant<br />
system, and the Fourier transform of the square root N amplified impulse response is<br />
N×N<br />
N×N
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