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2-D Discrete Fourier Transform Unified Matrix Representation Other Image Transforms Discrete Cosine Transform (DCT). Digital Image Processing. Lectures 11 & ...
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M.R. Azimi, Professor
Department of Electrical and Computer Engineering Colorado State University
Example 1 (circular convolution): Consider 2-D arrays x(m, n) = 1 0 2 1 ,h(m, n) =^
1 0 1 1. Find^ yˆ(m, n) and y(m, n). The 2 × 2 DFT of these sequences give
X(k, l) =
∑^1 m=
∑^1 n=
x(m, n)e−j^ 2 πmk 2 e−j^ 2 πnl 2
X(0, 0) = x(0, 0) + x(0, 1) + x(1, 0) + x(1, 1) = 4 X(0, 1) = x(0, 0) + x(0, 1)e−jπ^ + x(1, 0) + x(1, 1)e−jπ = x(0, 0) − x(0, 1) + x(1, 0) − x(1, 1) = 1 − 0 + 2 − 1 = 2 X(1, 0) = x(0, 0) + x(0, 1) + x(1, 0)e−jπ^ + x(1, 1)e−jπ = x(0, 0) + x(0, 1) − x(1, 0) − x(1, 1) = − 2
y ˆ(0, 1) = x(0, 0)h(0, 1) + x(0, 1)h(0, 0) + x(1, 0)h(1, 1) +x(1, 1)h(1, 0) = 3 yˆ(1, 0) = x(0, 0)h(1, 0) + x(0, 1)h(1, 1) + x(1, 0)h(0, 0) +x(1, 1)h(0, 1) = 3 yˆ(1, 1) = x(0, 0)h(1, 1) + x(0, 1)h(1, 0) + x(1, 0)h(0, 1) +x(1, 1)h(0, 0) = 2
Now, let us find the result of the desired linear convolution first using the direct method and then using the DFT.
y(0, 0) = x(0, 0)h(0, 0) = 1 y(0, 1) = x(0, 0)h(0, 1) + x(0, 1)h(0, 0) = 0 .. . y(2, 1) = x(1, 1)h(1, 0) + x(1, 0)h(1, 1) = 3 y(2, 2) = x(1, 1)h(1, 1) = 1
y(m, n) =
Note: yˆ(m, n) is the aliased version of this array. Extend arrays to size P = 4 ≥ M 1 + M 2 − 1 , Q = 4 ≥ N 1 + N 2 − 1 i.e.
x′(m, n) =
, h′(m, n) =
Taking 4 × 4 DFT and IDFT of the product gives
y′(m, n) =
which is y(m, n) if the last row and column are removed.
Properties and Remarks a. Symmetry and Unitary For a 1-D DFT matrix
WtN = WN i.e. symmetry, W− N^1 = W∗ N i.e. unitary
Thus W N∗ WN = I. b. Energy Conservation For a unitary transform the energy is preserved in both domains i.e.
‖X‖^2 = ‖x‖^2
To see this
‖X‖^2 =∆
N∑ − 1
k=
|X(k)|^2 = X∗ t X = x∗ t W∗ t N WN^ x
= x∗ t x = ‖x‖^2
c. 2-D DFT in Matrix Form For a 2-D finite-extent image, x(m, n) with ROS,RM N , the 2-D DFT is
X(k, l) =
m=
n=
x(m, n)W (^) Mmk W (^) Nnl 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 separable 2-D transform in matrix form can be written as
X = WM xWN
and x = W M∗ XW N∗
which shows the separability of the 2-D DFT. This is a unified matrix representation since for other image transforms such as sine, cosine, Hadamand, Haar and Slant transforms similar representation hold.
Recall that the DFS of any real even symmetric signal contains only real coefficients corresponding to the cosine terms. This can be extended to the DFT of a symmetrically extended signal/image. There are many ways to symmetrically extend a signal or an image leading to variety of DCT types. 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
X(k) =
√^1 N
N∑ − 1 n=
x(n) k = 0 √ 2 N
N∑ − 1 n=
x(n) cos[ (2n 2 +1)N kπ] k ∈ [1, N − 1]
The inverse transform is
x(n) =
k=
X(k)cos[ (2n + 1)kπ 2 N ] n ∈ [0, N −1]
In matrix form we get
X = Cx, C = [c(k, n)]
with forward kernel
c(k, n) =
√ N^ k^ = 0, n^ ∈^ [0, N^ −^ 1] 2 N cos[^
(2n+1)kπ 2 N ]^ k^ ∈^ [1, N^ −^ 1], n^ ∈^ [0, N^ −^ 1]
where x is the signal vector X is the DCT vector. The inverse transform in matrix form is
x = C−^1 X = CtX
Remarks
(^1) DCT is real and orthogonal i.e. C = C∗, C−^1 = Ct
The inverse 2-D IDCT is
x(m,n)= (^) N^1 X(0,0)+ (^) N^2 ∑^
N∑ − 1 k=0,l= k,l 6 =(0,0)
X(k,l) cos[ (2m 2 +1)N kπ] cos[ (2n 2 +1)N lπ]
The 2-D DCT of an image x in matrix form becomes
X = CxCt
with forward kernel defined by
C(k,l;m,n)=
N^1 k,l=0, m,n∈[0,N^ −1] 2 N cos[^ (2m+1)kπ 2 N ] cos[^ (2n+1)lπ 2 N ]^ k,l^6 =0,^0 k,l∈[0,N^ −1],m,n∈[0,N^ −1]
The inverse 2-D IDCT in matrix form is
x = CtXC
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 is decomposed as a linear combination of these basis images with the DCT coefficients, X(k, l)s, i.e. x =
k=
l=0 X(k, l)C(k, l).
Example To 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?
Example 1 Show that the N × N cosine transform matrix C is orthogonal, i.e. prove CCt^ = IN. Consider the k, lth (for k, l 6 = 0) element of CCt,
[CCt]k,l = 2 N
N∑ − 1
n=
cos[ (2n + 1)kπ 2 N ] cos[ (2n + 1)lπ 2 N ]
= 1 2 N
N∑ − 1
n=
[e −j(2n+1)(^2 N^ k+l)π+e j(2n+1)(^2 Nk^ +l)π+e −j(2n+1)(^2 N^ k−l)π+e j(2n+1)(^2 Nk^ −l)π]
which simplifies to
[CCt]k,l = cos[ (k + l)π 2 N ]δ(k + l − N ) + cos[ (k − l)π 2 N ]δ(k − l)
where the first term is always zero and the second term is also zero except when k = l. Thus,
[CCt]k,l ==
{ 1 f or k = l 0 otherwise
