3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 International Journal of Wavelets, Multiresolution and Information Processing Vol. 11, No. 5 (2013) 1350032 (25 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S021969131350032X LCT-WAVELET BASED ALGORITHMS FOR DATA COMPRESSION AMIR Z. AVERBUCH∗ , VALERY A. ZHELUDEV† and MOSHE GUTTMANN‡ School of Computer Science Tel Aviv University, Tel Aviv 69978, Israel ∗[email protected] †[email protected] ‡[email protected] DAN D. KOSLOFF Department of Earth and Planetary Sciences Tel Aviv University, Tel Aviv 69978, Israel [email protected] Received 16 August 2012 Revised 31 May 2013 Accepted 1 June 2013 Published 29 August 2013 We present an algorithm that compresses two-dimensional data, which are piece-wise smooth in one direction and have oscillatory events in the other direction. Fine texture, seismic, hyper-spectral and ﬁngerprints have this mixed structure. The transform part of the compression process is an algorithm that combines the application of the wavelet transform in one direction with the local cosine transform (LCT) in the other direction. This is why it is called hybrid compression. The quantization and the entropy coding parts in the compression process were taken from SPIHT codec but it can also be taken from any multiresolution based codec such as EZW. To eﬃciently apply the SPIHT codec to a mixed coeﬃcients array, reordering of the LCT coeﬃcients takes place. When oscillating events are present in diﬀerent directions as in ﬁngerprints or when the image comprises of a ﬁne texture, a 2D LCT with coeﬃcients reordering is applied. These algorithms outperform algorithms that are solely based on the the application of 2D wavelet transforms to each direction with either SPIHT or EZW coding including JPEG2000 compression standard. The proposed algorithms retain ﬁne oscillating events including texture even at a low bitrate. Its compression capabilities are also demonstrated on multimedia images that have a ﬁne texture. The wavelet part in the mixed transform of the hybrid algorithm utilizes the Butterworth wavelet transforms library that outperforms the 9/7 biorthogonal wavelet transform. Keywords: Wavelet transform; local cosine transform (LCT); compression; hybrid compression; SPIHT; hyper-spectral; seismic; ﬁngerprints. AMS Subject Classiﬁcation: 42C40, 68P30, 94A08 1350032-1 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. 1. Introduction 3D seismic data and hyper-spectral images are considered to be very large datasets. Eﬃcient algorithms are needed to compress these datasets in order to store them or transmit them from planes, boats and satellites to base stations for further analysis. The compression ratio should be as high as possible without damaging the analytical capabilities that take place after decompression. The compression of these datasets should preserve ﬁne details. Since wavelet transforms have a successful history in achieving high compression ratios for still images, these techniques were ported to handle seismic compression.14,19,32 However, the outcomes were less satisfactory due to the great variability of seismic data, the inherent noisy background and their oscillatory nature. Moreover, some researchers argued that seismic signals are not wavelet-friendly2,23 due to the oscillatory patterns that are present in seismic data. These oscillatory patterns can be properly handled by cosine transforms. The local cosine transform (LCT),2,23,33 which uses the lapped DCTa -IV transform (LCT)13 with adaptive partition, was applied to compress 2D seismic sections. The LCT catches well oscillatory patterns and sparsiﬁes it as the wavelet transform sparsiﬁes smooth data — see Refs. 6–8. Although these methods provided an excellent energy compaction, the bottleneck was the absence of an eﬃcient quantization — coding scheme comparable with the schemes designed for the wavelet transforms such as EZW,30 SPIHT28 and JPEG2000 standard.18 It was observed in Ref. 34 that the cosine transforms coeﬃcients can be rearranged in a way where their structure becomes similar to the structure of wavelet transform coeﬃcients. This observation paved the way to combine the cosine transforms with the wavelet-oriented coding schemes (WOCS).17,20 The DCT-II–based algorithm coupled with the SPIHT coder for compressing the segmented seismic cube is presented in Ref. 35. An embedded tarp ﬁltering combined with the classiﬁcation of the reordered DCT-II coeﬃcients (TCE) is applied to the image compression in Ref. 29. Evaluation of the image compression quality is given in Ref. 9. Seismic data has a diﬀerent structure in its vertical and horizontal directions. While the horizontal structure is piece-wise smooth, the vertical traces comprise of oscillatory patterns. To some extent, the same can be said on hyper-spectral data where each spatial pixel is represented by a vector (also called multipixel) of the intensities in all the available wavebands (typically ≈ 200). Compression of hyperspectral data should retain the spectral characteristic features of the multipixels. In seismic compression, the oscillating events, which bear the information on the subsurface layers, must be retained. Seismic and hyper-spectral images have diﬀerent structures in diﬀerent directions. This triggered the idea to apply a diﬀerent transform to a diﬀerent direction. a Discrete Cosine Transform. 1350032-2 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression One realization of this approach is presented in Ref. 16 where 3D hyper-spectral data cubes were compressed by the application of a 2D wavelet transform in the spatial direction and the Karhunen–Loéve transform in the spectral direction. The transforms coeﬃcients were coded by JPEG2000. Another scheme for a 3D hyperspectral data compression was presented in Ref. 25 where diﬀerent types of wavelet transforms were applied to the spatial and spectral directions followed by JPEG2000 coding. As was mentioned above, compression of seismic and hyper-spectral data pursues two goals: (1) Transmission of the data from the capturing devices to the analysis station (so-called onboard compression). (2) Storage of these huge data volumes (on-the-ground compression). In this paper, we focus on the compression of 2D rectangles, which comprise one spatial direction and one spectral (or in-depth) direction. Such data rectangles result from data acquisition schemes in hyper-spectral imaging and in seismic exploration. Thus, the proposed algorithms can be directly used for the onboard compression. For on-the-ground compression, 3D methods produce better results but our scheme can be easily extended to 3D setting. We exploit the fact that wavelets provide a sparse representation for such a mixed-structure data in the horizontal direction while the LCT handles well oscillatory patterns. Thus, we propose to apply the wavelet transform to the horizontal direction and the LCT to the vertical direction. The proposed transform produces a mix of LCT and wavelet coeﬃcients. In order to be processed properly by WOCS, the mixed coeﬃcients array should be reshaped to mimic the structure of a 2D array of wavelet coeﬃcients. To achieve it, we supply the LCT coeﬃcients with a spatial meaning by partitioning the data in the vertical direction. The joint coeﬃcients array is organized in a tree-like form by reordering the LCT coeﬃcients. Therefore, this scheme is called an hybrid compression algorithm (HCA). If there are oscillations in both directions, then the LCT is applied to both directions followed by reordering the transform coeﬃcients and then the WOCS is applied. This method is abbreviated by RLCTA. We compare between the performances of HCA and RLCTA where both use the SPIHT codec, and the performance of JPEG2000. In addition, we also compare with the performance of 2D wavelet-based algorithms that use SPIHT codec. In most of the experiments, which operate on hyper-spectral and seismic data, HCA and RLCTA that uses SPIHT outperforms JPEG2000 and wavelet-based algorithms that use SPIHT. In addition, our experiments demonstrated that the biorthogonal wavelet transforms, which are based on inﬁnite impulse response (IIR) Butterworth ﬁlters,3,4 provide better performance in comparison with the transforms that are based on the popular 9/7 transforms,1 which are also utilized in the JPEG2000 standard. This is true for the 2D wavelet transforms as well as for the hybrid transforms. 1350032-3 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Compression of ﬁngerprints images is another successful application of HCA and RLCTA. The FBI uses the compression standard11 that is based on the 9/7 2D wavelet transform. This transform, which uses ﬁnite impulse response (FIR) ﬁlters of lengths 7 and 9, is included also in JPEG2000 compression standard. The HCA and RLCTA produce higher PSNR results and retain better the structure of the ﬁngerprints in comparison to the outputs from the application of 2D wavelet transforms and JPEG2000 to ﬁngerpints. The HCA performance on multimedia type images, which are relatively smooth such as “Lena” and “Fabrics”, was close (sometimes even inferior) to the 2D waveletbased algorithms performance. On the other hand, HCA and RLCTA outperform the 2D wavelet based algorithms on images that have ﬁne texture or oscillating patterns such as “Barbara”, “Elaine”, “Wood fence” etc. HCA and RLCTA retain the texture and the oscillations even in a very low bit rate. To summarize, the contributions of the paper are threefold: (1) We exploit the fact that wavelets provide a sparse representation for such a mixed-structure data in the horizontal direction while the LCT handles well oscillatory patterns. This provides a new way to enhance wavelet based compression. (2) We demonstrate that the biorthogonal wavelet transforms, which are based on inﬁnite impulse response (IIR) Butterworth ﬁlters, provide better performance in comparison with the transforms that are based on the popular 9/7 transforms, which are also utilized in the JPEG2000 standard. (3) The proposed methodology outperforms the existing state-of-the art compression schemes including JPEG2000 applied to ﬁne texture, seismic, hyperspectral and ﬁngerprints images that are characterized by having sharp oscillations. The proposed method does not have special advantages when it is applied to regular rich multimedia images. The paper is organized as follows. Section 2 presents some known facts about LCT, wavelet transforms and SPIHT coding. Section 3 describes the hybrid algorithm. Section 4 presents the experimental compression results of seismic sections, hyper-spectral data, ﬁngerprints and multimedia images. 2. Preliminaries 2.1. Local cosine transform (LCT) The discrete cosine transform of type IV (DCT-IV)26 of the signal f and its inverse −1 N −1 ˆIV π 1 1 2 are deﬁned as fˆIV (k) = N (k) × n=0 fn cos[ N (k + 2 )(n + 2 )], fn = N k=0 f π π (k + 12 )(n + 12 )]. The basis signals {cos[ N (k + 12 )(n + 12 )]}, n = 0, . . . , N − 1, cos[ N of DCT-IV are even on the left side with respect to − 21 and odd on the right side with respect to N − 12 . Therefore, direct application of the DCT-IV to a partitioned data leads to severe boundary discrepancies. However, this transform serves as a base for the so called local cosine bases,13 which are the windowed lapped DCT-IV 1350032-4 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression transforms. These bases were successfully exploited in Refs. 2, 5, 23, 22, 33 for image compression in general and seismic data compression in particular. −1 Assume we have a signal S {sk }N k=0 and some partition P of the interval 0 : N −1. The idea behind the lapped DCT-IV transform, also called the local cosine transform (LCT) or the local Fourier basis of a signal, is to apply overlapped bells to adjacent sub-intervals. Then, the overlapped parts are folded back to sub-intervals across the endpoints of the sub-intervals and each sub-interval is expanded by the application of the DCT-IV transform. In the reconstruction phase, the transform coeﬃcients are unfolded. For details, see Refs. 2, 5, 13, 21. It is important that LCT produces no discrepancies between adjacent intervals. The choice of a bell is discussed in Refs. 5, 22. The bell we chose is given in Sec. 4. There are fast algorithms to implement the LCT. To achieve a sparse representation of the oscillatory kernel a multiscale adaptive local cosine transform rather than a one-level local cosine transform is used as described in Ref. 6. Multiscale adaptive use of the local cosine transform enables us to achieve impressive compact coding description of oscillatory data. It can be used in sparsiﬁcation of operator kernels. 2.2. Wavelet transforms Currently, wavelet transforms constitute a recognized tool for image processing applications. In particular, they have gained a proven success in image compression. We summarize here some other well known facts that are needed later. The multiscale wavelet transform of a signal is implemented via iterated multirate ﬁltering by a pair of L (low-pass) and H (high-pass) ﬁlters. Once implemented, the wavelet transform of a signal produces a partition of the Nyquist frequency band in a logarithmic mode. The diagram of a three-scale wavelet transform and the layout of the transform coeﬃcients are displayed in Fig. 1. On the other hand, the wavelet coeﬃcients have a spatial meaning. A transform coeﬃcient from a certain decomposition scale is a correlation coeﬃcient of the signal with a translation of the waveform (wavelet) related to this scale. The wavelets coeﬃcients from the ﬁrst decomposition scale are translated by two-sample steps, the steps for the second scale are four samples and so on. The wavelets ψ j (k), j = 1, . . . , 4, Φ4 (k) and their frequency responses are displayed in Fig. 2. 2.2.1. Tree structure of the wavelet transform coeﬃcients Assume that the number of scales is J = 3. The high-frequency coeﬃcient h3m from 3 , which is centered around the the coarsest scale is related to the waveform ψ8m+4 2 sample 8m + 4. The high-frequency coeﬃcients h2m and h22m+1 from the second 2 2 and ψ8m+6 , which are centered around the scale are related to the waveforms ψ8m+2 samples 8m + 2 and 8m + 6, respectively. These waveforms occupy, approximately, the same area as the waveform ψ 3 (k)8m+4 . In that sense, we claim that the ﬁner 1350032-5 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 1. Three-scale wavelet transform and the layout of the transform coeﬃcients (color online). 1 ψ ψ2 ψ3 4 ψ 4 φ Fig. 2. Left: The high-frequency wavelets ψj (k), j = 1, . . . , 4, and the low-frequency wavelet Φ4 (k). Right: Their frequency responses (color online). scale coeﬃcients h22m and h22m+1 are the “oﬀsprings” of the coarsest scale coeﬃcient h3m . In turn, their “oﬀsprings” are the ﬁne scale coeﬃcients h14m , h14m+1 , h14m+2 and h12m+3 . Thus, the coarsest scale coeﬃcient h3m is the root of the tree h3m h14m h24m+1 . h14m+2 h22m+1 2 h4m+3 h22m (2.1) The wavelet transform of a two-dimensional array T = {tn,m } of size N × M is implemented as a tensor product of 1D wavelet transform. First, a pair of ﬁlters L and H is applied to the columns of T and the results are downsampled. The coeﬃcients arrays W1L and W1H of size N/2 × M are produced. Then, the ﬁlters 1350032-6 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression L and H are applied to the rows of W1L and W1H . This ﬁltering is followed by downsampling which results in four sub-arrays coeﬃcients W1LL , W1LH , W1HL and W1HH of size N/2 × M/2. The 2D Nyquist frequency domain is split accordingly. Then, the above procedure is applied to the coeﬃcient array W1LL to decompose it into the sub-arrays W2LL , W2LH , W2HL and W2HH of size N/4 × M/4. Then, this procedure is iterated using W2LL instead of W1LL and so on. The layout of the transform coeﬃcients, which corresponds to the Nyquist frequency partition, for the three-scale wavelet transform, is displayed in Fig. 3. J ∈ WJHL and Similarly to the 1D case, each coeﬃcient lhJn,m ∈ WJLH , hln,m J J hhn,m ∈ WHH from the coarse scale has four “oﬀsprings” from the ﬁner scale J − 1. lhJ−1 2n,2m J−1 lh 2n,2m+1 , lhJn,m J−1 lh2n+1,2m lhJ−1 2n+1,2m+1 J−1 hl2n,2m J−1 J hl2n,2m+1 hln,m , J−1 hl2n+1,2m J−1 hl2n+1,2m+1 hhJ−1 2n,2m , J−1 hh 2n,2m+1 hhJn,m . hhJ−1 2n+1,2m hhJ−1 2n+1,2m+1 (2.2) Similar relations exist between the coeﬃcients from scales J − 1 and J − 2, and so on. These relations for J = 3 are illustrated in Fig. 4. J ∈ WJHL or hhJn,m ∈ WJHH from the Thus, each coeﬃcient lhJn,m ∈ WJLH , hln,m coarse scale can be treate as the root of a quad-tree of coeﬃcients. This relationship between wavelet coeﬃcients in diﬀerent scales is exploited in the embedded zerotree wavelet (EZW) codec.30 This codec takes advantage of the self-similarity between wavelet coeﬃcients across the decomposed scales and their decay toward high frequency scales. One of the most eﬃcient algorithms, which is based on the zerotree concept, is the SPIHT algorithm.28 This algorithm combines adaptive quantization of wavelet coeﬃcients with coding. The produced bitstream is further compressed (LL) LH LH (LL) HL (LL) HH HL Fig. 3. HH Three-scale layout coeﬃcients of a 2D wavelet transform. 1350032-7 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 4. Relationship among wavelet coeﬃcients from diﬀerent scales. losslessly by the application of an adaptive arithmetic coding.27 The SPIHT coding procedure is fast and its decoding procedure is even faster. 3. The Hybrid Algorithm We propose to apply the wavelet and the LCT transforms in the horizontal and vertical directions, respectively. The mixed wavelet and LCT transforms coeﬃcients are encoded by the SPIHT algorithm or by JPEG2000 coder. To ﬁt these codecs, the array of the mixed transform coeﬃcients is organized in a wavelet-like way. It requires to reorder the LCT coeﬃcients. The array of DCT coeﬃcients of a signal of length 2k produces a natural logarithmic split of the frequency band once being separated into k + 1 blocks: (c0 | c1 | c2 c3 |c4 c5 c6 c7 |c8 c9 c10 c11 c12 c13 c14 c15 | . . .)T . (3.1) The partition in Eq. (3.1) appears automatically when the coeﬃcients indices are given in a binary representation: (c0 |c1 |c10 c11 |c100 c101 c110 c111 |c1000 c1001 c1010 c1011 c1100 c1101 c1110 c1111 | . . .)T . (3.2) Thus, the array is partitioned according to the number of bits in the coeﬃcients indices. We call this a bit-wise partitioning. Assume we are given an N × M data array T, where N = 2k Q, M = 2J R. We deﬁne the partition P of the interval I [0, 1, . . . , N − 1] by splitting it into Q i k subintervals I = Q i=1 I of length 2 each. We apply the P -based LCT transform to each column of T. Thus, the array T is transformed into the array C of LCT coeﬃcients. 1350032-8 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression For each column, we get the array c of N LCT coeﬃcients, which consists Q k −1 of Q blocks c = i=1 ci , where ci {cin }2n=0 . Each of them can be a bit-wise k i i partitioning c = β=0 bβ as in Eq. (3.2) where bi0 = ci0 , bi1 = ci1 and biβ is the set of coeﬃcients cin , whose indices can be represented by β bits. In order to obtain a wavelet-like structure of the array c of N LCT coeﬃcients, we rearrange it in a bit-wise mode: c→b k bβ , Q bβ biβ . (3.3) i=1 β=0 Q T T 1 2 1 1 2 2 Thus, we get b0 (c10 , c20 , . . . , cQ 0 ) , b1 (c1 , c1 , . . . , c1 ) , b2 (c2 , c3 ; c2 , c3 ; . . . ; Q Q T c2 , c3 ) , and so on. This rearrangement is illustrated in Fig. 5. The structure of the array b is similar to the structure of the coeﬃcients array w of the wavelet transform, where the signal was decomposed into the k − 1 scale. The similarity relations are k−1 k−1 k−2 , b1 ∼ wH , b2 ∼ wH , . . . , bk−1 ∼ w1H . b0 ∼ wL (3.4) The ancestor-descendant relationships in the array b are similar to that in w. We perform this LCT coeﬃcients reordering for all the columns of the array C. Thus, we get C→B= k Bβ , Bβ β=0 Q Biβ . (3.5) i=1 Then, each row in the array B is decomposed into scale J by the application of the wavelet transform. This produces the hybrid LCT-wavelet coeﬃcients array denoted as CW. The structure of CW is similar to the the structure of a 2D wavelet coeﬃcients array, where the transform on the columns decomposed to scale k − 1, while the transform on the rows was decomposed till scale J. A three scales decomposition structure of the CW array is illustrated in Fig. 6. C1 C 2 C B + + + + + + + + + * + * + + * * + + + + * * * * * * * * * * * * Fig. 5. B0 B1 B2 B3 Reordering scheme of the LCT coeﬃcients. 1350032-9 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. WL3 WH3 WH2 WH1 B0 B1 B2 B3 Fig. 6. Three-scales of the LCT-wavelet coeﬃcients layout. The array CW is the input to SPIHT or JPEG2000 coders that implement the quantization followed by the implementation of the entropy coding. If the data arrays have oscillating structures in both vertical and horizontal directions then the LCT application to both directions followed by reordering of the transform coeﬃcients in both directions is recommended. We call this transform the reordered LCT denoted by (RLCTA). 4. Experimental Results The hybrid and the RLCTA algorithms were applied to compress diﬀerent data types: seismic, hyper-spectral, ﬁngerprints and multimedia images. We compared between the performances of these algorithms where the quantization and entropy coding was borrowed from the SPIHT codec. They were compared with the performance of JPEG2000 compression standard. For the latter, we used the MATLAB script imwrite. We also compared between the compression results from HCA and RLCTA algorithms and the results from the application of the 2D wavelet transform using the same SPIHT codec in all the transforms. An additional experimental goal was to compare between the performance of the Butterworth wavelet transforms3,4 and that of the 9/7 biorthogonal wavelet transform.12,1 The choice of a bell in the LCT construction (see Sec. 2.1) has some eﬀect on the performance of the LCT and, consequently, on the hybrid algorithm. A library of bells was introduced in Ref. 22. A comparative study of their eﬀects on the performance of an LCT-based image compression algorithm was given in Ref. 24. However, such a comparison is beyond the scope of this paper and we did not check the eﬀects of diﬀerent bells on the performance of the algorithm. Our goal was to demonstrate the capabilities of the new method. For this purpose, it was suﬃcient to implement LCT with the simple “sine” bell b(x) = sin π2 (x + 1/2).5 1350032-10 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression All the transforms in the paper have the following notation: Jpeg2000 standard is denoted by JP2k. 2D wavelet transforms using 9/7 filters is denoted by W9/7. 2D wavelet transforms of Butterworth wavelet transform with M vanishing moments is denoted by WButt/M. Hybrid transforms are composed from the 1D wavelet transforms in the horizontal direction and from the LCT in the vertical direction, which partitions the vertical section into Q horizontal rectangles. The transform coeﬃcients were reordered to ﬁt SPIHT encoding as was explained in Sec. 3. H9/7/Q denotes the 9/7 wavelet transform in the horizontal direction; HButt/M/Q denotes the Butterworth wavelet transform with M vanishing moments. 2D RLCTA: RLCTA/Q/P denotes the 2D LCT transform, which partitions the image into Q × P rectangles where Q is the partition in one direction and P is the partition in the other direction. The transform coeﬃcients were reordered to ﬁt the SPIHT encoding. All the images, which were used in the experiments, are 8-bit grayscale images. Assume that the image X, whose pixels array is x {xk }N k=1 , was subjected to lossy compression by some method M and, then, reconstructed to the image X̃, whose pixels array is x̃ {x̃k }N k=1 . We compare between the quality of diﬀerent compression methods by the peak signal to noise ratio (PSNR) in decibels N 2552 N (4.1) PSNR 10 log10 dB, (xk − x̃k )2 k=1 in addition to a visual inspection. The SPIHT codec was enhanced by incorporating into it an adaptive arithmetic coder,27 which is a popular entropy coder. The bold numbers in the following tables indicate the best achieved results. 4.1. Seismic sections compression Compression of seismic data, which is used for the reconstruction of subsurface layers structure, should retain all the seismic events in the traces. 4.1.1. Stacked CMP data section In the experiments, we used a stacked common mid point (CMP) data section. We display the original section of size 512 × 512 and a fragment of size 200 × 200 from it in Fig. 7. 1350032-11 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 7. The original stacked CMP section. Left: The whole section. Right: A fragment from the left image. The X-axis corresponds to the horizontal direction along the earth surface while the Y -axis corresponds to the vertical (depth) direction. Table 1. The PSNR values after decompression of the stacked CMP seismic section. bit/pixel JP2k W9/7 WButt/10 H9/7/8 1/8 1/4 1/2 1 2 31.03 33.12 35.91 40.61 48.66 30.86 32.79 35.39 39.83 46.91 31.01 32.99 35.84 40.42 48.82 31.19 33.39 36.68 41.8 51.26 HButt/10/8 31.39 33.79 37 42.26 51.6 RLCTA/8/8 31.46 33.83 37 42.31 51.63 Five types of transforms were used: W9/7, WButt/10, H9/7/8, HButt/ 10/8 and RLCTA/8/8, where the transform coeﬃcients were coded by SPIHT. The results were compared with the results from the application of JP2k. The achieved PSNR values are presented in Table 1. We observe that the PSNR values after the application of the hybrid transforms and the RLCTA are signiﬁcantly higher than those after the application of the 2D wavelet transforms whether JP2k or SPIHT is used. The best PSNR values appear after the application of RLCTA/8/8, which only slightly exceeds the values produced by HButt/10/8. The reason for that is in the presence of oscillations in the vertical and in the horizontal directions. Note that the 2D wavelet and hybrid transforms, which used the Butterworth wavelets, produced higher PSNR in comparison to the 9/7 based transform. The hybrid transforms and the RLCTA retain seismic events much better than the 2D wavelet transforms. It is illustrated in Fig. 8. Figure 8 displays fragments from the reconstruction of the CMP section in Fig. 7 (right) after the application of the JP2k compression algorithm where the compression ratio is 1/4 bit per pixel (left image in Fig. 8) and from the application of the RLCTA/8/8–SPIHT compression (right image in Fig. 8). The RLCTA/8/8– SPIHT compression retains the main structure of the data unlike the outcome from the application of the JP2k algorithm. 1350032-12 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression Fig. 8. A fragment from the reconstructed CMP section in Fig. 7 (right). Left: Output from the application of the JP2k algorithm. Right: The output from the application of the RLCTA/8/8– SPIHT algorithm. The transforms coeﬃcients were encoded by 1/4 bit per pixel. The X-axis corresponds to the horizontal direction along the earth surface while the Y -axis corresponds to the vertical (depth) direction. 170 160 160 150 150 140 140 130 130 120 120 110 110 100 100 90 90 80 150 200 250 300 350 400 450 500 80 150 200 250 300 350 400 450 500 Fig. 9. The diﬀerences between the original line #110 from the vertical trace in the CMP section (Fig. 7 (right)) and the reconstructed line after the applications of diﬀerent compression algorithms. Dotted line: Original. Solid line: Restored. Left: Restored after the application of JP2k algorithm. Right: Restored after the application of the HButt/10/8–SPIHT algorithm. Both were compressed to 1/2 bit per pixel (color online). Figure 9 displays line #110 from the vertical trace from the original section versus the same line from the reconstructed section from 1/2 bit per pixel compression by the applications of JP2k and HButt/10/8–SPIHT based algorithms. We observe that the reconstructed trace in the right image in Fig. 9 is very close to the original trace. This is not the case for the trace in the left image. 1350032-13 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 10. The original marine shot gather (MSG) section. Left: The whole section. Right: A fragment from the section on the left. The X-axis corresponds to the horizontal direction along the earth surface while the Y -axis corresponds to the vertical (depth) direction. 4.1.2. Marine shot gather data section For another seismic experiments, we used a marine shot gather (MSG) data section of size 512×512. As before, we compared between the performance of the 2D wavelet transforms and the hybrid transforms that use diﬀerent wavelets. Each data pixel is quantized to 8 bits. We display the original section of size 512 × 512 and fragment of size 200 × 200 from it in Fig. 10. Five types of transforms were applied: W9/7, WButt/10, H9/7/8, HButt/ 10/8 and RLCTA/8/8, where the transform coeﬃcients were coded by SPIHT. The results were compared with the results from the application of JP2k. The achieved PSNR values are presented in Table 2. The compression results from the applications of W9/7, WButt/10 and JP2k wavelet transforms are close to each other where JP2k has a small advantage over the other two. The HButt/10/8 transform outperforms H9/7/10 and JP2k. Figure 11 displays the fragments of the reconstructed MSG section after the application of JP2k (left image) and Hybrid HButt/10/10 with SPIHT coding (right image). The compression ratio for both is 1 bit per pixel. The HButt/10/8 transform retains the structure of the data much better than JP2k. Figure 12 displays the line #270 from the vertical trace from the original section taken from Fig. 10 (right) versus the same line taken from the reconstructed section after the application of JP2k and HButt/10/8+SPIHT algorithms. Table 2. bit/pixel 1/4 1/2 1 2 The PSNR values after decompression of the MSG seismic section (Fig. 10 (right)). JP2k 32.82 34.65 38 44 W9/7 WButt/10 H9/7/8 HButt/10/8 RLCTA/8/8 32.76 34.42 37.69 43.28 32.69 34.31 37.64 43.25 33.07 35.11 39.10 45.55 33.22 35.23 39.17 45.67 33.11 35.13 39.16 45.62 1350032-14 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression Fig. 11. A fragment from the reconstructed MSG section (Fig. 10 (right)). Left: Reconstruction from the application of JP2k, PSNR = 34.63 dB. Right: The reconstruction from the application of the Hybrid HButt/10/8+SPIHT, PSNR = 35.23 dB. The compression ratio is 1 bit per pixel. The X-axis corresponds to the horizontal direction along the earth surface while the Y -axis corresponds to the vertical (depth) direction. 100 150 200 250 300 350 400 450 500 100 150 200 250 300 350 400 450 500 Fig. 12. Line #270 from the vertical trace in the MSG section. Dotted lines: Original. Solid line: Restored. Left: After the application the JP2k algorithm. Right: The same after the application of the HButt/10/8+SPIHT algorithm. Both are compressed to 1 bit per pixel (color online). We see that the trace, which was reconstructed from HButt/10/8+SPIHT encoding/decoding algorithm, is much closer to the original trace in comparison to the trace reconstructed after the application of the JP2k algorithm that missed many details. Conclusion: For seismic data, the Hybrid compression algorithm signiﬁcantly outperforms compression algorithms that are based on the 2D wavelet transforms including JPEG2000 coding scheme. 4.2. Compression of hyper-spectral images The hybrid compression was applied to hyper-spectral data cubes. These cubes were captured from a plane that took simultaneously pictures in many (≈ 200) 1350032-15 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. wavebands from the ground surface. This type of camera can also be placed in a satellite. We used the SPECIM camera.31 In this camera, the spectrum of intensities for all the wavebands is assigned at once to each spatial pixel that is called multipixel. The proposed compression methodology ﬁts this capturing type and no buﬀering is needed in order to compress the hyper-spectral data. An hyper-spectral image is treated as a 3D cube where X, Y are the spatial axes and Z is the wavelength axis. The camera captures parallel lines of multipixels that are placed in the X/Z planes. The structure of the Z direction is very diﬀerent from that of the X direction. This justiﬁes the application of the hybrid transform. This processing enables a real-time transmission of hyper-spectral data without the need to buﬀer the data before the application of the hybrid compression algorithm. The results are compared with the results from the application of JPEG2000 compression standard. When an hyper-spectral data is compressed, it is important to preserve the spectral characteristic features of the multipixels. We applied the compression algorithm to diﬀerent hyper-spectral data cubes scenarios. Here, we present the results from the application of the compression algorithm to an urban scenery that has many details and thus contains many oscillations. Figure 13 displays an urban ground scene at one of the X/Z multipixel planes. This image was captured by the Specim31 camera placed on a airplane ﬂying 10,000 feet above sea level. The resolution is 1.5 meter/pixel, 3000 × 300 pixels per waveband with 180 wavebands. Five transforms types were applied: W9/7, WButt/4, H9/7/8, HButt/4/8 and RLCTA/8/8, where the transform coeﬃcients were coded by the SPIHT mechanism. The results were compared with the results from the application of Fig. 13. An hyper-spectral image. Left: One horizontal plane from the data cube. Right: A multipixel plane. Collection of all multipixels that belong to one line from the horizontal plane. The X-axis corresponds to the direction along a horizontal line of pixels in the left image while the Y -axis indicates the wavebands. 1350032-16 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression Table 3. PSNR values for the hyper-spectral compression of X/Z plane #200 (plane 200 from the 3000 planes of the whole hyper-spectral cube) from the urban scene Fig. 13 (right). bit/pixel 1/4 1/2 1 2 0 20 JP2k W9/7 36.79 40.9 46 51.65 40 60 35.39 39 43.62 49.97 80 100 WButt/4 H9/7/8 HButt/4/8 RLCTA/8/8 35.32 38.83 43.58 50.01 37.73 42.03 46.64 52.1 38.18 42.25 46.51 52.24 38.03 41.95 46.21 51.69 120 0 20 40 60 80 100 120 Fig. 14. One multipixel from X/Z plane #200. Dotted lines: Original. Solid line: Restored. Left: After the application of JP2k. Right: After the application of HButt/10/8+SPIHT encoding/decoding. The compression ratio for both is 0.25 bits per pixel (color online). JP2k. Table 3 presents the comparison results on the X/Z plane #200. The performances are given in PSNR values. We observe that the hybrid transforms with SPIHT produce PSNR values, which are higher than the results from the applications of the JP2k algorithm, which, in turn, outperforms the wavelet transforms. Figure 14 displays a multipixel from plane #200 from 3000 planes in Fig. 13 (left), which was reconstructed after the application of JP2k (left image) and HButt/4/8 (right image) transforms followed by SPIHT encoding/decoding where the compression ratio is 0.25 bit per pixel. Almost all the “events” in the original data are present in the reconstructed signal after the application of the hybrid transform with even high compression rate. This is not the case following the application of the JP2k algorithm. 4.3. Fingerprints The 2D wavelet, hybrid and LCT transforms followed by SPIHT coding were applied to compress a ﬁngerprint image of size 512 × 512, shown in Fig. 15. It was downloaded from C. Brislawn’s web page http://www.c3.lanl.gov/∼brislawn/index.html. It has an oscillating structure in each direction. Each pixel has 8 bits. 1350032-17 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. 50 50 100 100 150 200 150 250 300 200 350 400 250 450 500 50 Fig. 15. image. 100 150 200 250 300 350 400 450 500 300 260 280 300 320 340 360 380 400 420 440 460 The original ﬁngerprint image. Left: The whole image. Right: A fragment from the left Table 4. PSNR values for ﬁngerprint compression. bit/pixel JP2k W9/7 WButt/10 H9/7/16 HButt/10/16 RLCTA/16/16 1/8 1/4 1/2 1 23.54 26.23 29.84 33.94 23.05 25.51 29.33 33.3 24.03 26.42 29.91 33.83 23.87 26.53 30.06 33.97 24.38 26.77 29.99 34.27 24.55 26.99 30.23 34.83 The performances of the SPIHT based algorithms were compared with the performance of JPEG2000 compression standard. The LCT transforms partition the image into 16 horizontal and vertical rectangles. The achieved PSNR values are given in Table 4. We can see from Table 4 that RLCTA/16/16+SPIHT outperforms all the other algorithms in all bitrates. This is due to better handling of the oscillatory events in comparison to wavelet transforms based algorithms. Figure 16 displays fragments from the reconstructed ﬁngerprint image after the applications of JP2k (left) and RLCTA/16/16+SPIHT (right). The compression ratio is 1/8 bit per pixel. The right fragment retains much better the structure of the data than that in the left fragment. Figure 17 displays the column line #300 from the original section versus the same column line from the reconstructed sections after the applications of JP2k (left) and RLCTA/16/16+SPIHT (right). The compression ratio is 1/8 bit per pixel. We observe that, unlike JP2k, RLCTA/16/16+SPIHT restores the curve to be very close to its original form even when the compression ratio is very high. 4.4. Compression of multimedia images The hybrid compression algorithms also proved to be eﬃcient for multimedia images. Their superior performances are more evident for images that have 1350032-18 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression 50 50 100 100 150 150 200 200 250 250 300 250 300 350 400 300 450 260 280 300 320 340 360 380 400 420 440 460 Fig. 16. A fragment from the reconstructed ﬁngerprint image. Left: After the application of JP2k, PSNR = 23.54. Right: After the application of RLCTA/16/16+SPIHT, PSNR = 24.55. The compression ratio is 1/8 bit per pixel. 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Fig. 17. Column line #300 from the ﬁngerprint image. Dotted line: Original. Solid line: Restored. Left: The two lines after the application of JP2k. Right: The two lines after the application of RLCTA/16/16+SPIHT (right). The compression ratio is 1/8 bit per pixel (color online). oscillating texture. The algorithm restores the texture even at a very low bitrate. On the other hand, it sometimes produces artifacts on the boundaries between smooth and textured areas. Barbara image is of size 512 × 512 and its fragment is of size 280 × 280. They are displayed in Fig. 18. Each pixel in this image has 8 bits. This image comprises areas with oscillatory textures. We compared between the performances of 6 transforms where the hybrid and RLCTA transforms partition the vertical direction into Q = 16 horizontal rectangles each of height 32. The achieved PSNR values are given in Table 5. RLCTA/16/16+SPIHT, which produced the best results except the result for 1 bit per pixel, retains the ﬁne texture of the image much better than JP2k. This 1350032-19 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 18. image. The original “Barbara” image. Left: The whole image. Right: A fragment from the left Table 5. The PSNR values after the compression/decompression of “Barbara”. bit/pixel JP2k W9/7 WButt/10 1/8 1/4 1/2 1 25.78 28.68 32.71 37.95 25.18 27.85 31.57 36.82 25.14 28.29 32.32 37.9 H9/7/16 25.71 28.63 32.72 38 HButt/10/16 RLCTA/16/16 26.08 29.18 33.28 38.38 26.78 29.54 33.28 38.15 fact is illustrated in Fig. 19, which displays fragments of the reconstructed “Barbara” image after the applications of JP2k (left) and RLCTA/16/16+SPIHT (right) algorithms. The compression ratio is 1/8 bit per pixel. Figure 20 displays the column line #246 from the original section versus this column line from the reconstructed sections after the application of JP2k (left) and RLCTA/16/16+SPIHT (right). The compression ratio is 1/4 bit per pixel. We observe that RLCTA/16/16+SPIHT algorithm retains most of the oscillatory events that were missed by JP2k. Elaine image is of size 512 × 512 and its fragment of size 236 × 256 are displayed in Fig. 21. Each pixel has 8 bits. We compared between the performances of 6 transforms where the hybrid and the RLCTA transforms partition the vertical direction into Q = 8 horizontal rectangles each of height 64. The achieved PSNR values are given in Table 6. The highest PSNR values in all bitrates except for 1/8 bit per pixel were produced by RLCTA/8/8+SPIHT, while the 2D wavelet transform with Butterworth wavelet WButt/4+SPIHT was better with 1/8 bit per pixel. At bitrates 1/8 and 1/4 bit per pixel, reconstruction from RLCTA/8/8+SPIHT and hybrid algorithms 1350032-20 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression Fig. 19. A fragment from the reconstructed “Barbara” image after the applications of: Left: JP2k. Right: RLCTA/16/16+SPIHT. The compression ratio is 1/8 bit per pixel. 250 200 150 100 50 0 50 100 150 200 250 300 350 400 450 500 0 0 100 200 300 400 500 Fig. 20. Column line #246 from Barbara image. Dotted line: Original. Solid line: Restored. Left: Restored after the application of JP2k. Right: Restored after the application of RLCTA/16/16+SPIHT (right). The compression ratio is 1/4 bit per pixel (color online). produced a better visual quality of the images in comparison to JP2k, which over smoothed the images. This fact is illustrated in Fig. 22, which displays fragments of “Elaine” image that were reconstructed after the application of JP2k and of RLCTA/8/8+SPIHT. The compression ratio is 1/8 bit per pixel. Over smoothing by JP2k is clearly seen in Fig. 23, which displays the column line #280 from the original section versus this column line from the reconstructed sections after the applications of JP2k (left) and RLCTA/8/8+SPIHT (right). The compression ratio is 1/4 bit per pixel. We observe that the RLCTA/8/8+SPIHT algorithm retains most of the ﬁne texture that was blurred by JP2k. 1350032-21 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. Fig. 21. image. The original “Elaine” image. Left: The whole image. Right: A fragment from the right Table 6. PSNR values after compression/decompression of “Elaine”. bit/pixel JP2k W9/7 WButt/4 H9/7/16 HButt/10/16 RLCTA/8/8 1/8 1/4 1/2 1 31.11 32.29 33.48 36.06 31.1 32.33 33.48 35.75 31.12 32.33 33.36 35.46 30.41 31.96 33.94 36.41 30.65 32.22 34.15 36.79 30.77 32.8 34.9 37.72 Fig. 22. A fragment of the reconstructed “Elaine” image after the applications of: Left: JP2k. Right: RLCTA/8/8+SPIHT. The compression ratio is 1/4 bit per pixel. 1350032-22 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Fig. 23. Column line #280 from the Elaine image. Dotted line: Original. Solid line: Restored. Left: Restored after the application of JP2k. Right: Restored after the application of RLCTA/8/8+SPIHT (right). The compression ratio is 1/4 bit per pixel (color online). 5. Conclusions The experimental results strongly support our assumption that the hybrid wavelet — LCT–SPIHT and 2D LCT algorithms with reordering of the transform coeﬃcients is a new powerful tool to compress seismic data, hyper-spectral images, ﬁngerprints and other data that comprise oscillatory structures and (or) a ﬁne texture. Although all the components of the algorithm are well known, their combined operation via reordering of the transform coeﬃcients outperforms compression schemes that are based on multidimensional wavelet transforms with SPIHT coding and JPEG2000. Additional ﬂexibility of the method stems from the use of the library of Butterworth wavelet transforms of diﬀerent orders. In almost all the experiments, the hybrid transforms with Butterworth wavelets outperform the transforms with the popular 7/9 wavelets based ﬁlters. The 2D LCT algorithm with reordering of the transform coeﬃcients demonstrates an excellent performance when oscillating events are present in diﬀerent directions as in ﬁngerprints or when the image comprises a ﬁne texture as in the “Elaine” image. The algorithm retains ﬁne oscillating events even at a low bitrate. This is important for seismic, hyper-spectral and ﬁngerprint images processing and less critical for multimedia type images. The extension of the algorithm to compress 3D seismic or hyper-spectral cubes is straightforward. In the horizontal planes, the 2D wavelet transform is applied, while the LCT is applied to vertical directions followed by coeﬃcients reordering. The joint array of coeﬃcients is the input to SPIHT encoding. The algorithm is completely automatic and the encoding/decoding operations can be implemented in a fast way. References 1. M. Antonini, M. Barlaud, P. Mathieu and I. Daubechies, Image coding using wavelet transform, IEEE Trans. Image Processing 1(2) (1992) 205–220. 1350032-23 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 A. Z. Averbuch et al. 2. A. Z. Averbuch, F. Meyer, J.-O. Stromberg, R. Coifman and A. Vassiliou, Eﬃcient compression for seismic data, IEEE Trans. Image Processing 10(12) (2001) 1801–1814. 3. A. Z. Averbuch, A. B. Pevnyi and V. A. Zheludev, Butterworth wavelet transforms derived from discrete interpolatory splines: Recursive implementation, Signal Processing 81 (2001) 2363–2382. 4. A. Averbuch and V. Zheludev, Wavelet and frame transforms originated from continuous and discrete splines, in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky (eds.) (Hindawi Publishing Corp., NY, 2007), pp. 1–56. 5. A. Averbuch, G. Aharoni, R. Coifman and M. Israeli, Local cosine transform — A method for the reduction of blocking eﬀects in JPEG, J. Mathematical Imaging and Vision, Special Issue on Wavelets 3 (1993) 7–38. 6. A. Averbuch, L. Braverman, R. Coifman, M. Israeli and A. Sidi, Eﬃcient computation of oscillatory integrals via adaptive multiscale local Fourier bases, Applied and Computational Harmonic Analysis 9(1) (2000) 19–53. 7. A. Averbuch, L. Braverman, M. Israeli and R. Coifman, On eﬃcient computation of multidimensional oscillatory integrals with local Fourier bases, J. Nonlinear Analysis 47 (2001) 3491–3502. 8. A. Averbuch, E. Braverman, R. Coifman and M. Israeli, Fast evaluation of 2-D and 3-D oscillatory integrals with local Fourier bases, Int. J. Pure and Applied Mathematics 3(1) (2002) 1–28. 9. R. Bensalma and M. C. Larabi, A perceptual metric for stereoscopic image quality assessment based on the binocular energy, Multidimensional System Signal Processing, (2012), DOI: 10.1007/s11045-012-0178-3. 10. C. M. Brislawn, Classiﬁcation of nonexpansive symmetric extension transforms for multirate ﬁlter banks, Applied and Computational Harmonic Analysis 3(4) (1996) 337–357. 11. C. M. Brislawn, The FBI ﬁngerprint image compression speciﬁcation, Wavelet Image and Video Compression, P. Topivala (ed.), (Springer, New York, 2002), pp. 271–288. 12. A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. on Pure and Appl. Math. 45 (1992) 485–560. 13. R. R. Coifman and Y. Meyer, Remarques sur l’analyse de Fourier a fenêtre (C. R. Acad. Sci., 1991), 259–261. 14. P. L. Donoho, R. A. Ergas and J. D. Villasenor, High-performance seismic trace compression, in 65th Ann. Int. Mtg., Soc. Expl. Geophys., Expanded Abstracts (1995), pp. 160–163. 15. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992). 16. Q. Du and J. E. Fowler, Hyperspectral image compression using JPEG2000 and principal component analysis, IEEE Geoscience and Remote Sensing Letters 4(4) (2007) 201–205. 17. Y.-A. Jeong and C.-K. Cheong, A DCT-based embedded image coder using wavelet structure of DCT for very low bit rate video codec, IEEE Trans. on Consumer Electronics 44(3) (1998) 500–508. 18. JPEG 2000 Standard, http://www.jpeg.org/jpeg2000/. 19. M. F. Khéne and S. H. Abdul-Jauwad, Eﬃcient seismic compression using the lifting scheme, in 70th Ann. Int. Mtg., Soc. Expl. Geophys., Expanded Abstracts (2000), pp. 2052–2054. 20. H. S. Malvar, Fast progressive image coding without wavelets, in Proc. IEEE Data Compression Conference, J. A. Storer and M. Cohn (eds.) (Snowbird, UT, March 2000), pp. 243–252. 1350032-24 3rd Reading August 26, 2013 14:30 WSPC/S0219-6913 181-IJWMIP 1350032 LCT-Wavelet Based Algorithms for Data Compression 21. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd edn. (Elsevier, 2009). 22. G. Matviyenko, Optimized local trigonometric bases, Applied and Computational Harmonic Analysis 3 (1996) 301–323. 23. F. G. Meyer, Fast compression of seismic data with local trigonometric bases, Proc. SPIE 3813, Wavelet Applications in Signal and Image Processing VII, eds. A. Aldroubi, A. F. Laine and M. A. Unser (1999), pp. 648–658. 24. F. G. Meyer, Image compression with adaptive local cosines: A comparative study, IEEE Trans. on Image Proc. 11(6) (2002) 616–629. 25. B. Penna, T. Tillo, E. Magli and G. Olmo, Progressive 3-D coding of hyperspectral images based on JPEG 2000, IEEE Geoscience and Remote Sensing Letters 3(1) (2006) 125–129. 26. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic Press, 1990). 27. A. Said, Introduction to Arithmetic Coding — Theory and Practice, HP Lab, Published as a chapter in Lossless Compression Handbook by Khalid Sayood (Academic Press, 2004). 28. A. Said and W. W. Pearlman, A new, fast and eﬃcient image codec based on set partitioning in hierarchical trees, IEEE Trans. Circ. and Syst. for Video Tech. 6 (1996) 243–250. 29. V. P. Shah, J. E. Fowler and N. H. Younan, Tarp ﬁltering of block-transform coeﬃcients for embedded image coding, in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Toulouse, France, May 14–16, 2006, Vol. 2 (IEEE, 2006), pp. 21–24. 30. J. M. Shapiro, Embedded image coding using zerotree of wavelet coeﬃcients, IEEE Trans. Sign. Proc. 41 (1993) 3445–3462. 31. Specim camera, http://www.specim.ﬁ/. 32. A. Vassiliou and M. V. Wickerhauser, Comparison of wavelet image coding schemes for seismic data compression, in 67th Ann. Int. Mtg., Soc. Expl. Geophys., Expanded Abstracts (1997), 1334–1337. 33. Y. Wang and R.-S. Wu, Seismic data compression by an adaptive local cosine/sine transform and its eﬀect on migration, Geophysical Prospecting 48 (2000) 1009–1031. 34. Z. Xiong, O. Guleryuz and M. T. Orchard, A DCT-based embedded image coder, IEEE Signal Processing Letters 3 (1996) 289–290. 35. V. A. Zheludev, D. D. Kosloﬀ and E. Y. Ragoza, Compression of segmented 3D seismic data, Int. J. Wavelets, Multiresolution and Information Processing 2(3) (2004) 269–281. 1350032-25

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement