Image rotation and mirroring in wavelet domain. Theju Jacob EE department University of Texas at ArlingtonAbstract: JPEG2000, the image codec which gives us the best picture quality at a low bit rate, uses wavelet transform. At present, if we are to achieve mirroring or rotation of images at the receiver side, we have to first convert the image back into time domain and then perform the operation. This thesis put forwards a method in which this operation can be performed in the transform domain itself, without the need for conversion to time domain first.Proposal: A method to achieve rotation and mirroring of images in transform domain, for the wavelet transform, is proposed. Wavelets and Filter Banks: To obtain wavelet domain coefficients of an image, we apply to rows and columns of the image, a set of filters, followed by down sampling. To reconstruct the image from the coefficients, we up sample them, and then subject them to a different set of filters which hold a certain relation with the filters on the transmitter side. It can be represented as follows: Fig 1: Representation of a two channel filter bank for analysis of images. H0 and H1 represent low pass and high pass filters respectively. In the top branch, the rows are filtered by H0, followed by down sampling of rows by 2. Filtering of columns by H0 and down sampling of columns by 2 would lead to cA component. Filtering of columns by H1 followed by down sampling of columns by 2 would lead to cH component. Similarly, filtering of rows by H1, down sampling by 2, followed by filtering by H0 and further down sampling by 2 would produce cV. Filtering of both rows and columns by H1 with down sampling at appropriate places would lead to cD. The synthesis filter bank is as shown below. Each of cA, cH, cV and cD components are given as input to the filter bank. Each of them are up sampled along columns by 2. cA is filtered along the columns by F0, up sampled along rows, and then filtered along rows by F0. cH is filtered along columns by H1, up sampled along rows, and filtered along rows by H0. H0 H0 (↓2) H1 (↓2) (↓2) (↓2) cA cH H0 (↓2) H1 (↓2) H1 (↓2) (↓2) cV cDFig 2: Synthesis filter bank. F0 and F1 are low pass and high pass filters respectively. Similarly, cV would be filtered along columns by F0, up sampled along rows, and filtered along rows by F1. cD would be filtered along columns by F1, up sampled along rows and filtered along rows by F1. Each of the branch outputs, when combined, would give the image back. Property of perfect reconstruction: The relation between filters H0 and F0, or H1 and F1 for that matter, which enables the reconstruction of the image possible, is the property of perfect reconstruction. Derivation of the conditions of perfect reconstruction would ultimately boil down to the fact that [1] the product filter, resulting from a product of each of the analysis-synthesis filter pair, should lead to a half band filter. The half band filter gives us: P(ω) + P(ω+π) = 2, in the frequency domain., where the product filter P(ω) is defined as follows: Pm(ω) = Hm(ω)*Fm(ω), where m = 0,1. The idea: Now, on the analysis side, if X (ω) is our input in frequency domain, after filtering and down sampling, we get: ½*(Hm(ω/2)*X(ω/2) + Hm(ω/2+π)*X(ω/2+π)), where m = 0,1. - (1) On flipping this result in time domain, we get: F0 (↑2) (↑2) F1 (↑2) (↑2) F1 F0 (↑2) (↑2) F1 (↑2) (↑2) F0 cA cH cV cD½*(Hm(-ω/2)*X(-ω/2) + Hm(-ω/2+π)*X(-ω/2+π)), where m = 0,1. - (2) (2) can be looked upon as filtering of reverse sequence by reversed analysis filter. Hence, to maintain the conditions of perfect reconstruction, we can reverse the filters in the synthesis bank as well, and go on with our synthesis – for the general case. But, there is not merit in the process, as it is the same as saying that reversing the input to the synthesis filters, and the synthesis filters themselves, would lead to a reversed result. Consider the case where the filters are symmetric, as in the case of JPEG 2000 9/7 filters and 5/3 filters. For symmetric filters, forward and reverse in the time domain is the same, and so, Hm(ω) = Hm(-ω), for m = 0,1. Hence, (2) is equivalent to the result from filtering of the mirrored input by analysis filter bank. Now, coming to the case of two dimensional images, flipping from left to right of each of cA, cH, cV and cD components would lead to a mirroring of the image – as the condition of perfect reconstruction would still be satisfied in both row and column directions. Similarly, rotating the image by 180 degrees involves flipping from up to down of each of the cA, cH, cV and cD components. Rotation of the image by 90 and 270 are slightly more involved, as we are also switching the order of rows and columns, but they are still possible by switching of cH and CV components. The diagrammatic representation for each is as discussed below. J represents flipped (from left to right) identity matrix of the same dimension as each of cA, cH, cV, and cD. A, H, V and D represents the input terminals for cA, cH, cV and cD components in the original synthesis filter bank. (i) Image Mirroring In the above diagram, SFB represents synthesis filter bank. A H SFB V D Image cA*J cH*J cV*J cD*J(ii) Image rotation by 90 (iii) Image rotation by 180 (iv) Image rotation by 270 A H SFB V D Image (J*cA)T (J*cV)T (J*cH)T (J*cD)T A H SFB V D Image J*cA J*cH J*cV J*cD A H SFB V D Image (cA*J)T (cV*J)T (cH*J)T (cD*J)TConclusion: A method to achieve image mirroring and rotation in the wavelet domain for JPEG2000 filters is proposed. This method could be applied for any symmetric filter bank which has the property of perfect reconstruction. Another interesting part of the problem would be to study the same for asymmetric filter banks and orthogonal filter banks. References: [1] Wavelets and Filter Banks, Gilbert Strang and Truong Nguyen, Wellesley-Cambridge Press, 1997 [2] Digital Signal Processing, Sanjit K Mitra, TaTa McGraw-Hill, 2006 [3] Processing JPEG-compressed images and documents, de Queiroz, R.L, IEEE
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