Digital Image Processing Lectures 11 & 12

Transcription

1 Lectures 11 & 12, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2015

2 Matrix Representation of DFT Consider an -point finite-extent sequence x(n), n [0, 1] with DFT X(k) = 1 1 x(n)w nk k [0, 1] n=0 where W = e 2πj/, then in matrix form X(0). X( 1) = W W W 1 W X = W x W : DFT matrix with elements w kn = 1 The inverse is x = W 1 X. kn mod W x(0). x( 1) 1

3 Properties and Remarks a. Symmetry and Unitary For a 1-D DFT matrix W t = W i.e. symmetry, W 1 = W i.e. unitary Thus W W = I. b. Energy Conservation For a unitary transform the energy is preserved in both domains i.e. To see this 1 X 2 = k=0 = x t x = x 2 X 2 = x 2 X(k) 2 = X t X = x t W t W x

4 c. 2-D DFT in Matrix Form For a 2-D finite-extent image, x(m, n) with ROS,R M, the 2-D DFT is X(k, l) = 1 M 1 M 1 m=0 n=0 x(m, n)wm mk W nl k [0, M 1], l [0, 1 If x and X represent the original image matrix and its DFT image (complex) matrix (both of size M ), respectively, the separable 2-D transform in matrix form can be written as X = W M xw and x = WMXW 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.

5 Basis Images of 2-D DFT Let w k be the kth column of W M and wh l be the lth row of W, then the (k, l)th basis image is Using x = W M XW x = W (k, l) = w k wh l the image can be represented as M 1 k=0 1 l=0 X(k, l)w (k, l) i.e. the image is decomposed onto a space spanned by these M basis images. The coefficients (or projection of image on the basis images) in this representation are given by X(k, l) =< x, W (k, l) > where inner product of two matrices is given by < A, B >= M 1 1 m=0 n=0 a(m, n)b (m, n).

6 Discrete Cosine Transform (DCT) Recall that DFS of any real even symmetric signal contains only real coefficients corresponding to the cosine terms. This can be extended to 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 is 1 1 x(n) k = 0 n=0 X(k) = 1 2 x(n) cos[ (2n+1)kπ ] k [1, 1] n=0

7 The inverse transform is x(n) = 1 2 X(0)+ In matrix form we get 1 k=1 X = Cx, X(k)cos[ (2n + 1)kπ ] n [0, 1] C = [c(k, n)] with forward kernel { 1/ k = 0, n [0, 1] c(k, n) = 2 (2n+1)kπ cos[ ] k [1, 1], n [0, 1] where x is the signal vector X is the DCT vector. The inverse transform in matrix form is Remarks 1 DCT is real and orthogonal i.e. x = C 1 X = C t X C = C, C 1 = C t

8 2 2-D DCT can be performed using 1-D DCT s along columns and row, i.e. separable (see below). 3 DCT is OT the real part of the DFT rather it is related to the DFT of a symmetrically extended signal/image. 4 The energy of signal/image is packed mostly in only a few DCT coefficients (i.e. only a few significant X(k) s), hence making DCT very useful for data compression applications. 2-D DCT of Images The 2-D DCT of a finite-extent image x(m, n) of size is X(k,l)= x(m,n) m=0 n=0 1 1 m=0 n=0 x(m,n) cos[ (2m+1)kπ ] cos[ (2n+1)lπ ] k,l=0,0 m,n [0, 1] k,l 0,0 k,l [0, 1], m,n [0, 1]

9 The inverse 2-D IDCT is x(m,n)= 1 X(0,0)+ 2 1 k=0,l=0 k,l (0,0) X(k,l) cos[ (2m+1)kπ ] cos[ (2n+1)lπ ] The 2-D DCT of an image x in matrix form becomes with forward kernel defined by C(k,l;m,n)= 1 2 X = CxC t k,l=0, m,n [0, 1] (2m+1)kπ cos[ ] cos[ (2n+1)lπ ] k,l 0,0 k,l [0, 1],m,n [0, 1] The inverse 2-D IDCT in matrix form is x = C t XC

10 Figure below shows basis images C(k, l) = c k c t l, k, l [0, 1] of 2-D DCT for = 8 where c k 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 = M 1 k=0 1 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 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?

11 In real applications, however, the original image is typically partitioned into blocks (e.g., 16 16) and DCT coefficients are taken from each block. If a portion of these coefficients is kept for each block and then reconstruction is done block-by-block the following reconstructed image will be obtained, which shows a much better reconstruction.

12 Example 1 Show that the cosine transform matrix C is orthogonal, i.e. prove CC t = I.

13 Consider the k, lth (for k, l 0) element of CC t, = 1 1 n=0 [CC t ] k,l = 2 [e j(2n+1)(k+l)π which simplifies to 1 n=0 cos[ +e j(2n+1)(k+l)π (2n + 1)kπ (2n + 1)lπ ] cos[ ] +e j(2n+1)(k l)π +e j(2n+1)(k l)π ] [CC t (k + l)π l)π ] k,l = cos[ ]δ(k + l ) + cos[(k ]δ(k l) where the first term is always zero and the second term is also zero except when k = l. Thus, { [CC t 1 for k = l ] k,l == 0 otherwise For k, l = 0, [CC t ] 0,0 = 1 n=0 orthogonal transformation matrix. 1 = 1. Hence, CCt = I i.e.

14 KL Transform or Principal Component Analysis (PCA) Unlike the other transforms covered so far this transform applies to random signals/images and has wide applications in data reduction, rotation and data decorrelation applications. Let {x(n)}, n [0, 1] be a finite-extent 1-D random process with zero mean and auto-covariance function r xx (m) = E[x(n)x(n m)] where E[ ] represents the expectation operator (ensemble average). If x is the vector arrangement of {x(n)} i.e. x = [x(0)... x( 1)] t then the covariance matrix is R = E[xx t ] which becomes Toeplitz for a wide-sense stationary process i.e. r xx (0) r xx ( 1) r xx ( + 1) r xx (1) r xx (0) r xx ( + 2) R =.. r xx ( 1) r xx ( 2) r xx (0)

15 ow, the KL transform of x is X = Ψ t x and inverse KL transform x = ΨX where the basis vectors, ξ k s, (i.e. columns of Ψ) of KL transform are the orthonormalized eigenvectors (i.e. ξ t ξ l k = δ(k l)) of matrix R i.e. Rξ k = γ k ξ k, k [0, 1] Ψ = [ξ 0,, ξ 1 ] and Ψ reduces R to its diagonal form Ψ t RΨ = Γ = Diag[γ 0,, γ 1 ] γ 0 > γ 1 > > γ 1 As can be seen, the KL transform is dependent on the 2nd order statistics of the data.