Recent literature highlights the multiple description coding (MDC) as a promising method to solve the problem of resilient image coding over error-prone networks, where packet losses occur. In this paper, we introduce a novel multiple description wavelet-based image coding scheme using fractal. This scheme exploits the fractal’s ability, which is to describe the different resolution scale similarity (redundancy) among wavelet coefficient blocks. When one description is lost, the lost information can be reconstructed by the proposed iterated function system (IFS) recovering scheme with the similarity and some introduced information. Compared with the referenced methods, the experimental results suggest that the proposed scheme can achieve better performance. Furthermore, it is substantiated to be more robust for images transmission and better subjective quality in reconstructed images even with high packet loss ratios.

Transmission of compressed images over unreliable networks has proven to be a significant challenge. The main problem is the rapid degradation in the reconstructed image quality due to packet loss, which is unavoidable on the networks, such as the Internet. Hence, it is important to propose an effective method to protect compressed images information transmitted over unreliable networks.

Recently, MD coding has emerged as an attractive framework for robust transmission over unreliable networks. It can efficiently combat packet loss without any retransmission thus, satisfying the demand of real-time services and relieving the network congestion [

Lots of MD coding techniques have been developed using different strategies for coding variant data like speech, audio, image, and video. One of the most classical methods is multiple description scalar quantization (MDSQ) [

From the above analysis, the most MD coding approaches are based on traditional coding methods, such as quantization, JPEG, and wavelet. These methods always exploit the redundancy in the neighboring pixels or coefficients. This redundancy is partly added to the descriptions for the recovering in case of some descriptions missed. But the redundancy among coefficient blocks at different resolution scales exists in wavelet domain, and this kind of redundancy is not used in MDC. Fortunately, fractal coding (FC) is an intensively studied image coding method during recent decades [

In this paper, we attempt to design an MD wavelet-based image coding method using fractal. Fractal coding has some advantages, such as high decompression speed, low compression bit rate, and decompression resolution independence. The basic principle of FC is to find an IFS whose fixed point approximates to the original data. Therefore, for the original data, the compressed data is an IFS after FC. As discussed previously in [

In our study, the wavelet transform decomposes the original image into different resolution subbands. At each decomposition level, four subbands are produced. They are named approximation subband (low-pass version) and detail subbands (high-pass version). Using the primary descriptions generation method, two subsets are generated at the encoder. For each subset, we use the extended range blocks fractal method in [

The rest of this paper is organized as follows. In Section

MD coding can tolerate loss from the estimation of missing data with help of a certain amount of introduced redundancy. So, the interests of MD coding are mainly two points; first, in what way the descriptions and redundancy should be created to benefit the lost data estimation; second, how can we efficiently recover the lost data?

Therefore, in this paper, an MD wavelet-based image coding scheme using fractal is proposed. Wavelet is a popular tool for image analysis and compression. The different subbands of wavelet coefficients have similarity, which can be exploited by fractal coding methods. In summary, the proposed scheme partitions the mappings of IFS in wavelet domain with the checkerboard pattern and recovers the lost mappings using IFS recovering schemes.

FC partitions data image

Note that the range-domain matching process initially consists of a shrinking operation in each domain block that averages its pixel intensities forming a block of size

Assuming that

The fixed point theory and the collage theory are two basic theorems for IFS.

Let

For any two points

For any contractive transformations, there must be a fixed point

Let

Constructing the mappings of IFS.

The decoder is simple; it is shown in [

In our study, in case of recovering the lost mappings, a wavelet coefficient image is compressed by the fractal coder with redundancy. It is well known that the different wavelet coefficient subbands have a different kind of similarity, for example, the approximation subband has much similarity in intrasubband and the detail subbands have much similarity in intersubbands. Therefore, in the proposed method, the different FCs are selected according to the different subbands. In the approximation subband, we use the extended range block FC method. And prediction fractal coding method is used in detail subbands.

The basic idea of the extended range block fractal coding is sending the image twice with the same amount of data as the single image and using double-sized range blocks and a new criterion to determine the domain block [

This method extends the original range blocks into new range blocks which are overlapped just like Figure

New range block and partition of extended range block FC.

The process of finding an IFS with new range blocks is just like Section

We use prediction fractal coding [

Coding process of prediction fractal coding method.

In Figure

After wavelet transform, each subband is divided into a group of blocks, the block size may vary for different subbands, then these blocks are partitioned into two nonoverlapping subsets; the mappings corresponding to the blocks in subset are defined as primary descriptions

Primary description generation.

Redundancy which is defined as correlated information

In approximation subband, the extended range block fractal coding method is exploited to compress wavelet coefficients. When only one description is available, the received data is considered as an error IFS of the original image. Because the extended range blocks can cover the lost basic range blocks, we use the basic range blocks in received description and its extended range blocks as new range blocks. In this way, iteratively applying half mappings of IFS with the new range blocks to an initial data image, we can acheive the decoded subband. Note that, in each iterative step, the correlated information is added to the lost basic range blocks.

In Figure

Lost blocks are recovered by extended range blocks.

When both descriptions are available, we use the basic range blocks as the range blocks. With the completed IFS, the decoded subband can be calculated just like the traditional fractal decoding process.

In detail subbands, prediction fractal image coding method is used to compress the wavelet coefficients. When only one description is available, half mappings of IFS are lost in the checkerboard pattern manner. So, the fractal extrapolation method is presented to estimate the lost mappings. This method exploits the similarity in intrasubbands and the introduced correlated information (redundancy).

In detail subbands, there is the similarity in intersubbands at the same direction. So in our study, the lost mapping in subband

Example of fractal extrapolation.

For each estimated block, the correlated information is added on it.

The proposed MDC can be depicted in Figure

Process of proposed MD: (a) MD encoder, (b) MD decoder.

At the encoder, a given input image is decomposed by wavelet transform, and the wavelet coefficients are partitioned into two descriptions by the primary description generation method. In each description, for the approximation subband, extended range block fractal coding method is used to compress the approximation subband coefficients. we use prediction fractal image coding scheme to compress the detail subband. The detail has been given in Section

Let

In the proposed scheme, the central decoded image quality is directly dependent on

Once

Because of the symmetry of both descriptions, we take description 1 as an example to explain how to get the correlated information.

In Figure

For each basic range block in description 2, there exists the quantized difference block. All the quantized difference blocks are coded by entropy coder. The coded data is the correlated information for description 1.

In detail subbands, we still take description 1 as an example. In Figure

When both descriptions are received, the primary coded information from each description is used for the reconstruction. When only one description 1 is available, the correlated information and the primary information from the description are used by the IFS recovering scheme to reconstruct the side decoded image.

Two standard images (

In the first experiment, we show performance results for our proposed MD method. The sample image reconstructions are shown in Figure

Sample image reconstructions, for Lena image, at

PSNR (dB) of reconstruction images using the proposed method and the Salama's linear interpolation method: (a) Zelda, (b) Elain, (c) Barbara, and (d) Lena.

Then, we evaluate the reconstruction performance of the proposed IFS recovery method. Figure

In the linear interpolation scheme, each description only contains the primary information, corresponding to the same coding rate per description as in the proposed scheme. In this way, the redundancy between the two descriptions is minimized, which favors the central decoding performance. When only one description is received, an interpolation method used in [

When only one description is available, the linear interpolation method to recover the lost description yields poor performance. On the other hand, the IFS recovering scheme gives much better results. Note that the redundancy is not used in this experiment for the fair comparison. That is, in this experiment, the IFS recovering scheme and the linear interpolation scheme construct the description by only primary information. It can be seen that the proposed method is more efficient to recover the lost description than directly recovering the description using linear interpolation method.

Figure

Performance comparison of Lena among MDSQ, MD method in [

With the same central reconstruction distortion, the performance gap of the side decoder between the proposed scheme and MDSQ is significant, ranging from 0.4–1 dB, which verifies the efficiency of the proposed scheme. Note that the proposed MD method has outstanding performance when the redundancy is low.

Figure

Performance comparison of Barbara between MDSQ and the proposed method with target bit rates

In fact, the probability that a description is completely lost during the transmission is very low. Hence, it is reasonable to assume that a certain percentage of packets are lost in transmission. Each block mapping and its quanitzed difference is assumed to construct a packet. A uniformly distributed random index of the lost packet is generated for each channel. The lost packet is recovered with Salama's linear interpolation method if its estimation block used for fractal extrapolation is lost too. This experiment is tested with the packet loss ratios of

The target bit rate is fixed at 0.5 bpp. The bitrate of correlated information is fixed at 0.05 bpp. The results are presented in Figure

PSNR with variant packet loss ratios between MDSQ and the proposed method.

We also execute the subjective test at low bitrates. Because FC exploits the different scale redundancy of image, with

Two channel reconstruction for Barbara (

In this paper, we have proposed an MD wavelet image coding scheme using fractal. This scheme exploits the ability of fractal well. Through IFS recovering scheme with the similarity in wavelet coefficient bands, the proposed scheme can reconstruct the lost information well when only one description is received. The results strongly suggest that our method can achieve better performance even when only one description is received. Furthermore, the proposed method is more robust for image transmission.

This research was supported by Science and Technology Project of Beijing Municipal Education Commission (no. KM201311232021).