2-D Discrete Fourier Transform Unified Matrix Representation Other Image Transforms Discrete Cosine Transform (DCT) Digital Image Processing Lectures 11 & 12 M.R. Azimi, Professor Department of Electrical and Computer Engineering Colorado State University M.R. Azimi Digital Image Processing
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Digital Image Processing Lectures 11 & 12 · 2-D Discrete Fourier Transform Uni ed Matrix RepresentationOther Image Transforms Discrete Cosine Transform (DCT) Digital Image Processing
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c. 2-D DFT in Matrix FormFor a 2-D finite-extent image, x(m,n) with ROS,RMN , the 2-DDFT is
X(k, l) =1√MN
M−1∑m=0
N−1∑n=0
x(m,n)WmkM Wnl
N k ∈ [0,M−1], l ∈ [0, N−1]
If x and X represent the original image matrix and its DFT image(complex) matrix (both of size M ×N), respectively, the separable2-D transform in matrix form can be written as
X = WMxWN
andx = W∗
MXW∗N
which shows the separability of the 2-D DFT. This is a unifiedmatrix representation since for other image transforms such assine, cosine, Hadamand, Haar and Slant transforms similarrepresentation hold.
Recall that the DFS of any real even symmetric signal contains only realcoefficients corresponding to the cosine terms. This can be extended tothe DFT of a symmetrically extended signal/image. There are many waysto symmetrically extend a signal or an image leading to variety of DCTtypes. Here, we present DCT-II which is the most common one (see Fig.).
The 1-D DCT-II of a finite-extent signal x(n) of size N is
2 2-DCT can be performed using 1-D DCT’s along columns androw, i.e. separable.
3 DCT is NOT the real part of the DFT rather it is related tothe DFT of a symmetrically extended signal/image.
4 The energy of signal/image is packed mostly in only a fewDCT coefficients (i.e. only a few significant X(k)’s), hencemaking DCT very useful for data compression applications.
2-D DCT of ImagesThe 2-D DCT of a finite-extent image x(m,n) of size N ×N is
Figure below shows basis images C(k, l) = ckctl , k, l ∈ [0, N − 1] of
2-D DCT for N = 8 where ck is kth column of C. Image x isdecomposed as a linear combination of these basis images with theDCT coefficients, X(k, l)s, i.e. x =
∑M−1k=0
∑N−1l=0 X(k, l)C(k, l).
ExampleTo show the usefulness of the DCT for data reduction and compression
applications, we reconstructed the Peppers image based upon only
64× 64 DCT coefficients. The following figures show the DCT
coefficients of the Peppers image and the reconstructed result, which
exhibits some smearing and ringing artifacts. Why?
In real applications, however, the original image is typicallypartitioned into blocks (e.g., 16× 16) and DCT coefficients aretaken from each block. If a portion of these coefficients is kept foreach block and then reconstruction is done block-by-block thefollowing reconstructed image will be obtained, which shows amuch better reconstruction.