Image rotation and mirroring in wavelet domain Theju Jacob EE department University of Texas at Arlington Abstract 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 2 H0 2 2 H1 2 2 H0 2 2 H1 2 cA H0 cH cV H1 cD 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 2 cA F0 2 F0 cH 2 F1 2 2 F0 2 cV F1 2 F1 2 cD Fig 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 P 2 in the frequency domain where the product filter P is defined as follows Pm Hm Fm where m 0 1 The idea Now on the analysis side if X is our input in frequency domain after filtering and down sampling we get Hm 2 X 2 Hm 2 X 2 where m 0 1 On flipping this result in time domain we get 1 Hm 2 X 2 Hm 2 X 2 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 Hm 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 cA J cH J A H SFB cV J V cD J D Image In the above diagram SFB represents synthesis filter bank ii Image rotation by 90 cA J T cV J T T iii H SFB cH J V cD J T D Image Image rotation by 180 A J cA H J cH iv A SFB J cV V J cD D Image Image rotation by 270 J cA T J cV T T A H SFB J cH V J cD T D Image Conclusion 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 WellesleyCambridge 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 transactions on image processing Dec 1998
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