Efficient and Low-Complexity Image Coding with the Lifting Scheme and Modified SPIHT Hong Pan, W.C. Siu, and N.F. Law Abstract In this paper, we propose an efficient and low complexity image coding algorithm based on the lifting wavelet transform and listless modified SPIHT (LWT-LMSPIHT). LWT-LMSPIHT jointly considers the advantages of progressive transmission and spatial scalability that were not fully provided by the SPIHT algorithm, thus it outperforms the SPIHT at low bit rates coding. The coding efficiency of LWT-LMSPIHT comes from three aspects. The lifting scheme lowers the number of arithmetic operations of the wavelet transform. Moreover, a significance reordering of the modified SPIHT ensures that it codes more significant information earlier in the bit stream belonging to the lower frequency bands than SPIHT to better exploit the energy compaction of the wavelet coefficients. Finally, a listless structure further reduces the amount of memory and improves the speed of compression by more than 47% for a 52 52 image, as compared with the SPIHT algorithm. I. INTRODUCTION ow bit-rate image compression becomes a major role Lin multimedia applications. A variety of powerful and complex wavelet-based image compression schemes [-6] have been developed over the past few years. Among them, Set Partitioning In Hierarchical Trees (SPIHT) [2] is the most well-recognized coding method because of its excellent rate-distortion performance. However, it does not entirely provide the desired features of progressive transmission and spatial scalability in that the algorithm uses an inefficient coefficient partitioning method. Moreover, a larger amount of memory is required to maintain three lists that are used for storing the coordinates of the coefficients and tree sets in the coding and decoding process. A great number of operations to manipulate the memory are also required in the codec scheme, which greatly reduces the speed of coding procedure. In this paper, we propose an efficient and low complexity image coding algorithm based on the lifting wavelet transform and listless modified SPIHT (LWT-LMSPIHT). This algorithm employs the lifting scheme as the transform method and does a breadth first search without using lists. State information is kept in a fixed size array that corresponds to the matrix of coefficient values. By introducing a significance reordering that codes more significant information earlier in the bit stream belonging to the lower frequency bands than that of the SPIHT, our approach can better exploit the energy compaction of the wavelet coefficients. Therefore, the PSNR value of LWT-LMSPIHT Hong Pan, W.C.Siu and N.F.Law are with the Center for Multimedia Signal Processing, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hong Kong Hong Pan is also with School of Automation, Southeast University, China is better than that of SPIHT, at very low bit rate. The remarkable point of our work is that the coding time consumption is reduced by 47%, on average for a 52 52 image, as compared with the SPIHT. In addition, the memory requirement and coding complexity of the LWT-LMSPIHT are also reduced significantly. II. FAST WAVELET TRANSFORM USING THE LIFTING SCHEME As shown in [7], by applying the polyphase representation to analysis filter banks { hz ( ), gz ( )} and synthesis filter banks {h(z), g(z)}, the forward and inverse discrete wavelet transform can be written as () x ( ) ( ) e z ye z T Pz ( ) Pz ( ) z xo( z) zyo( z) () where x and y are the input signal and the reconstructed signal; and are the outputs of lowpass filter hz ( ) and bandpass filter g ( z ); h e( z) g e( z) is the dual polyphase Pz ( ) h o( z) g o( z) he z ge z Pz () ho() z go() z () () matrix and is the polyphase matrix. Daubechies and Sweldens [7] proved that given a complementary filter pair (h, g), there always exist Laurent polynomials s i (z) and t i (z) and a non-zero normalization constant K so that the polyphase matrix P(z) can be factorized as ( ) ( ) m he z ge z si ( z) 0K 0 Pz ( ) ho( z) go( z) i 0 ti( z) 0 K (2) primal lifting dual lifting normalization Then, the conjugate transpose matrix of dual polyphase matrix P ( z) which contains the analysis filter pair becomes, h ( z ) h ( z ) K 0 0 s ( z) T e o i ( ) ( ) ( ) 0 K g i m ti ( z) 0 e z go z normalization dual lifting primal lifting Pz Equation (3) tells us that every forward wavelet transform using FIR filters can be decomposed into a lazy transform and m consecutive primal and dual lifting steps followed by a scaling matrix to preserve the perfect reconstruction property of the filter bank. The main advantage of the lifting wavelet transform is that it is efficient for implementation. The calculation in each step of the transform can be done completely in place. For the Daub 9/7 biorthogonal wavelet, applied in our work, the lifting scheme only requires 4 floating point arithmetic operations, per data point for every decomposition level, while there are 23 arithmetic operations required in the filter bank implementation scheme for the same case [7]. Hence the total computational cost of the lifting scheme is 60.87% of that in the standard filter bank implementation, which greatly improves the speed of the wavelet transform. (3) 960 978--4244-82-3/08/$25.00 c 2008 IEEE
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