Modified SPIHT algorithm for wavelet packet image coding

Abstract

This paper introduces a new implementation of wavelet packet decomposition which is combined with SPIHT (Set Partitioning in Hierarchical Trees) compression scheme. We provide the analysis of the problems arising from the application of zerotree quantisation based algorithms (such as SPIHT) to wavelet packet transform coefficients. We established the generalized parent–child relationships for wavelet packets, providing complete tree structures for SPIHT. The proposed algorithm can be used for both wavelet dyadic and Wavelet Packet decomposition (WP-SPIHT). An extensive evaluation of the algorithm was performed and it has been shown that WP-SPIHT significantly outperforms base-line SPIHT coder for texture images. For these images the suboptimal WP cost-function enables good enough energy compaction that is efficiently exploited by the WP-SPIHT.

Introduction

Discrete Wavelet Transform (DWT) provides a multiresolution image representation and has become one of the most important tools in image analysis and coding over the last two decades. Image compression algorithms based on DWT [1], [2], [3], [4], [5], [6], [7], [8] provide high coding efficiency for natural (smooth) images. As dyadic DWT does not adapt to the various space-frequency properties of images, the energy compaction it achieves is generally not optimal. However, the performance can be improved by selecting the transform basis adaptively to the image. Wavelet Packets (WP) represent a generalization of wavelet decomposition scheme. WP image decomposition adaptively selects a transform basis that will be best suited to the particular image. To achieve that, the criterion for best basis selection is needed.

Coifman and Wickerhauser proposed entropy based algorithm for best basis selection [9]. In their work, the best basis is a basis that describes the particular image with the smallest number of basis functions. It is a one-sided metric, which is therefore not optimal in a joint rate-distortion sense. A more practical metric considers the number of bits (rate) needed to approximate an image with a given error (distortion) [10] but this approach and its variation presented in [11] can be computationally too intensive. In [12] a fast numerical implementation of the best wavelet packet algorithm is provided. Coding results show that fast wavelet packet coder can significantly outperform a sophisticated wavelet coder constrained to using only a dyadic decomposition, with a negligible increase in computational load.

The goal of this paper is to demonstrate advantages and disadvantages of using WP decomposition in SPIHT-based codec. SPIHT algorithm was introduced by Said and Pearlman [13], and is improved and extended version of Embedded Zerotree Wavelet (EZW) coding algorithm introduced by Shapiro [14]. Both algorithms work with tree structure, called Spatial Orientation Tree (SOT), that defines the spatial relationships among wavelet coefficients in different decomposition subbands. In this way, an efficient prediction of significance of coefficients based on significance of their “parent” coefficients is enabled. The main contribution of Shapiro's work is zerotree quantization of wavelet coefficients and introduction of special zerotree symbol indicating that all coefficients in a SOT are found to be insignificant with respect to a particular quantization threshold. An embedded zerotree quantizer refines each input coefficient sequentially using a bitplane coding scheme, and it stops when the size of the encoded bitstream reaches the target bit-rate. SPIHT coder provides gain in PSNR over EZW due to introduction of a special symbol that indicates significance of child nodes of a significant parent and separation of child nodes (direct descendants) from second-generation descendants. To date, there have been numerous variants and extensions to SPIHT algorithm, for example: 3-D SPIHT for video coding [15], [16], [17], [18], [19], SPIHT for color image coding [20], [21], and scalable SPIHT for network applications [22], [23], [24], [25].

Since the SPIHT algorithm relies on Spatial Orientation Trees (SOT) defined on dyadic subband structure, there are a few problems that arise from their adaptation to WP decomposition. First is the so-called parental conflict [26], that happens when in the wavelet packet tree one or more of the child nodes are at the coarser scale than the parent node. It must be resolved in order that SOT structure with well-defined parent–child relationships for an arbitrary wavelet decomposition can be created. Xiong et al. [11] avoided the parental conflict by restricting the choice of the basis. In their work the Space-Frequency Quantization (SFQ) algorithm is used. SFQ algorithm employs a rate-distortion (R-D) optimization framework for selecting the best basis and to assign an optimal quantiser to each of the wavelet packet subbands. Rajpoot et al. [26] defined a set of rules to construct the zerotree structure for a given wavelet packet geometry and offered a general structure for an arbitrary WP decomposition. In their work a Compatible Zerotree Quantisation (CZQ) is utilized, and it does not impose restriction on the selection of WP basis. A comparison of PSNR obtained with CZQ-WP and SPIHT shows that SPIHT provides gain in PSNR over CZQ-WP for the standard test images, while CZQ-WP offers better visual quality than SPIHT [26]. This observation motivated us to use SPIHT with WP in order to exploit strengths of both methods. This extension of SPIHT we call Wavelet Packet SPIHT (WP-SPIHT).

This paper is organized as follows: In Section 2 wavelet analysis in the context of WP-SPIHT is described. In Section 3 we explain WP-SPIHT algorithm. Coding results are presented in Section 4, followed by conclusion in Section 5.