sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
32 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES the set of eigenvalues of this system. Note: when the signal is infinitely long and the impulse response is relatively short, we can drop the cyclic constraint. Since an LTI system is a widely assumed model in signal processing, the Fourier transform plays an important role in analyzing this type of system. 2. A covariance matrix of a stationary Gaussian (i.e., Normal) time series is Toeplitz and symmetric. If we can assume that the time series does not have long-range dependency 2 , which implies that off-diagonal elements diminish when they are far from the diagonal, then the covariance matrix is almost circulant. Hence, the Fourier series again become the eigenvectors of this matrix. The DFT matrix DFT N nearly diagonalizes the covariance matrix, which implies that after the discrete Fourier transform of the time series, the coefficients are almost independently distributed. This idea can be depicted as DFT N ·{the covariance matrix}·DFT T N ≈{a diagonal matrix}. Generally speaking, it is easier to process independent coefficients than a correlated time series. The eigenvalue amplitudes determine the variance of the associated coefficients. Usually, the eigenvalue amplitudes decay fast: say, exponentially. The coefficients corresponding to the eigenvalues having large amplitudes are important coefficients in analysis. The remaining coefficients could almost be considered constant. Some research in this direction can be found in [58], [59] and [101]. From the above two examples, we see that FT plays an important role in both DSP and time series analysis. Fast Fourier Transform To illustrate the importance of the fast Fourier transform in contemporary computational science, we find that the following sentences (the first paragraph of the Preface in [136]) say exactly what we would like to say. The fast Fourier transform (FFT) is one of the truly great computational developments of this century. It has changed the face of science and engineering so 2 No long-range dependency means that if two locations are far away in the series, then the corresponding two random variables are nearly independent.
3.1. DCT AND HOMOGENEOUS COMPONENTS 33 much so that it is not an exaggeration to say that life as we know it would be very different without the FFT. — Charles Van Loan [136]. There are many ways to derive the FFT. The following theorem, which is sometimes called Butterfly Theorem, seems the most understandable way to describe why FFT works. The author would like to thank Charles Chui [30] for being the first fellow to introduce this theorem to me. The original idea started from Cooley and Tukey [32]. To state the theorem, let us first give some notation for matrices. Let P 1 and P 2 denote two permutation matrices satisfying for any mn-dimensional vector x =(x 0 ,x 1 ,... ,x mn−1 ) ∈ R mn ,wehave, ⎛ P 1 ⎜ ⎝ x 0 x 1 . x mn−1 ⎛ ⎞ ⎟ = ⎠ mn×1 ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ x 0 x n . x (m−1)n x 1 x 1+n . x 1+(m−1)n x (n−1) ⎞ ⎟ ⎠ • • • x (n−1)+n . ⎞ ⎟ ⎠ m×1 m×1 ⎞ ⎟ ⎠ ⎞ ; (3.3) ⎟ ⎠ x (n−1)+(m−1)n m×1
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3.1. DCT AND HOMOGENEOUS COMPONENTS 33<br />
much so that it is not an exaggeration to say that life as we know it would be<br />
very different without the FFT. — Charles Van Loan [136].<br />
There are many ways to derive the FFT. The following theorem, which is sometimes<br />
called Butterfly Theorem, seems the most understandable way to describe why FFT works.<br />
The author would like to thank Charles Chui [30] for being the first fellow to introduce this<br />
theorem to me. The original idea started from Cooley and Tukey [32].<br />
To state the theorem, let us first give some notation for matrices. Let P 1 and P 2 denote<br />
two permutation matrices satisfying for any mn-dimensional vector x =(x 0 ,x 1 ,... ,x mn−1 )<br />
∈ R mn ,wehave,<br />
⎛<br />
P 1 ⎜<br />
⎝<br />
x 0<br />
x 1<br />
.<br />
x mn−1<br />
⎛<br />
⎞<br />
⎟ =<br />
⎠<br />
mn×1<br />
⎜<br />
⎝<br />
⎛<br />
⎜<br />
⎝<br />
⎛<br />
⎜<br />
⎝<br />
⎛<br />
⎜<br />
⎝<br />
x 0<br />
x n<br />
.<br />
x (m−1)n<br />
x 1<br />
x 1+n<br />
.<br />
x 1+(m−1)n<br />
x (n−1)<br />
⎞<br />
⎟<br />
⎠<br />
•<br />
•<br />
•<br />
x (n−1)+n<br />
.<br />
⎞<br />
⎟<br />
⎠<br />
m×1<br />
m×1<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
; (3.3)<br />
⎟<br />
⎠<br />
x (n−1)+(m−1)n<br />
m×1
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