Interpolation Based Initial Image for Fast Fractal Decoding

Qiang Wang
College of Information Science and Technology
Dalian Maritime University
Linghai Road, No.1, Dalian, 116026, China
Email: wangqiang2011@dlmu.edu.cn

Abstract: Fractal image decoding has been effectively accelerated by using the range-averaged image (RAI) as the initial image, where only one iteration is needed to obtain decoded images of acceptable quality. To further improve the decoded image quality while maintaining real-time decoding, this study proposes an interpolation based initial image (IBII). The primary drawback of RAI is its obvious block artifacts. IBII addresses this by producing a smoother initial image that better approximates the original image, thereby enabling higher decoded image quality with a single iteration under specific decoding strategies. Experimental results demonstrate that the IBII-based method improves decoded image quality by 0.56–1.41 dB in peak signal-to-noise ratio (PSNR) and by 0.0061–0.0173 in mean structural similarity (MSSIM).

Keywords: Fractal image coding; Fast Fractal decoding; Initial image

Introduction

Fractal image coding (FIC) represents a successful application of fractal theory to image compression. The core concept of FIC was pioneered by Barnsley, with Jacquin subsequently proposing the first block-based fractal coding algorithm \cite{1}. Unlike traditional compression techniques such as discrete cosine transform and wavelet methods that achieve compression by removing high-frequency components in the frequency domain, FIC compresses images by exploiting self-similarity redundancy among different local regions \cite{2}. Beyond image compression, FIC has found increasing application in diverse fields including image denoising \cite{3-6}, image encryption \cite{7-9}, and human pose estimation \cite{10,11}.

To enhance the practicality of FIC, researchers have developed fast fractal encoding algorithms that complete the encoding process rapidly while preserving decoded image quality \cite{12,13}. Others have combined FIC with complementary compression techniques to improve overall performance \cite{14,15}. At the decoding stage, although fractal image decoding converges in few iterations, further acceleration remains valuable for real-time applications such as video surveillance. To expedite fractal decoding, researchers have employed better initial images, such as the range-averaged image (RAI), which approximates the input image closely and facilitates convergence to the decoded image under specified iteration strategies \cite{16-18}. Alternative approaches have focused on improved iteration strategies that effectively reduce iteration errors relative to the final decoded image, thereby accelerating the decoding process \cite{19}.

In previous RAI-based decoding, range blocks in the initial image are directly replaced with their averages, bringing the initial image considerably closer to the input image compared to a conventional blank image. This approach enables real-time decoding with a single iteration, producing decoded images of acceptable quality suitable for applications like video surveillance that do not demand high decoding fidelity. Nevertheless, room remains for improving decoded image quality within a single iteration. This study proposes an interpolation based initial image (IBII) that better approximates the input image than RAI, thereby enhancing decoded image quality while preserving real-time decoding and acceptable output quality. The contributions of this work are twofold: (1) IBII is defined to provide a superior initial image that more closely approximates the input image than RAI, enabling higher decoded image quality with one iteration compared to the RAI-based method. (2) The IBII-based method can be integrated with other fast fractal decoding techniques to achieve real-time decoding with improved image quality.

This paper is organized as follows: Section 2 reviews conventional fractal image coding and related work. Section 3 describes the definition of IBII. Section 4 presents the proposed algorithm and experimental results. Section 5 concludes the paper.

2.1 Traditional Fractal Image Coding

In fractal encoding, the input image of size $M \times N$ is partitioned into two block types: range blocks and domain blocks. Range blocks constitute a set of non-overlapping image blocks of size $B \times B$, denoted as $R_i^{B \times B}, i = 1, 2, ..., NumR$, where $NumR$ represents the total number of range blocks. Domain blocks form a set of overlapping image blocks of size $L \times L$, denoted as $D_j^{L \times L}, j = 1, 2, ..., NumD$, where $NumD$ denotes the total number of domain blocks. The domain block pool is constructed by sliding an $L \times L$ window across the image, with $L > B$. Domain block size is typically set to four times that of range blocks, i.e., $L = 2B$. Domain blocks are then scaled to match range block size and extended through eight isometry transforms to facilitate subsequent block matching. For each range block $R_i$, every domain block in the extended pool is compared against $R_i$ to determine intensity contrast and offset parameters by minimizing the following function:

$$\text{argmin} = 1, 2, 3$$

where $I$ denotes a unit vector of dimension $NumR$. The contrast and offset parameters, denoted as $o$, respectively, can be computed via least squares method as:

$$\text{where are the averages of } R_i \text{ and } D_j, \text{ respectively. denotes the inner product.}$$

During decoding, any $M \times N$ image can serve as the initial image, with the decoding process converging to the final decoded image after several iterations.

2.2 Fast Fractal Image Decoding

Fractal image decoding can be accelerated through two primary approaches: (1) Adopting appropriate initial images. For instance, the approximate RAI, which closely approximates the input image, effectively accelerates decoding \cite{16}. The exact RAI is obtained by replacing range blocks in the initial image with their respective averages as:

$$\text{RAI} = 1, 2, 3, ..., NumR$$

Decoding speed under specific iteration strategies can be further improved by using RAI as the initial image \cite{17}. To our knowledge, no superior initial image for fast fractal decoding has been proposed in recent years. Additionally, the collage image established during encoding can effectively shorten the decoding process for fractal image denoising and magnification rather than compression \cite{18}. (2) Better iteration strategies. For example, one-buffer-decoding (OBD) performs mapping operations from a single buffer to itself, unlike conventional decoding that uses two buffers. This allows range blocks under reconstruction to benefit from previously reconstructed blocks, making each iteration's reconstructed image better approximate the final decoded image and effectively accelerating the process \cite{19}.

This study employs peak signal-to-noise ratio (PSNR) to measure decoded image quality:

$$\text{PSNR} = \log_{10} \frac{255^2}{\text{MSE}}$$

where $\text{Decoded}$ represent the input and decoded images, respectively. Previous work combining RAI with OBD effectively enhanced decoding convergence, achieving acceptable decoded image quality with a single iteration. For example, the PSNR of RAI for the Peppers image in Fig. 1 is 23.15 dB, exhibiting obvious block artifacts. After one iteration, the decoded image's visual quality improves significantly, achieving a PSNR of 29.08 dB. This demonstrates that existing RAI-based methods can provide acceptable decoded image quality with a single iteration, simultaneously achieving real-time decoding and acceptable quality. This study aims to further improve decoded image quality while maintaining these benefits.

[FIGURE:1]

3. Interpolation Based Initial Image

[FIGURE:2]

Since natural images typically exhibit smooth intensity transitions, we introduce transition regions between different range blocks. This study illustrates IBII construction using four neighboring $4 \times 4$ range blocks (Fig. 2). These four range blocks are replaced with a series of $2 \times 2$ blocks divided into three categories: (1) Average blocks (ABs, $AB_l, l = 1, 2, 3, 4$) located at range block centers, represented by green dashed blocks. ABs are obtained by replacing central $2 \times 2$ blocks with range block averages as:

$$AB_l = \text{average}(R_l)$$

(2) Transition blocks (TBs, $TB_{pq}, p, q = 1, 2, 3$) located between neighboring ABs, represented by red dashed blocks. TBs are obtained by replacing $2 \times 2$ blocks between neighboring ABs with their averages:

$$TB_{pq} = \frac{AB_p + AB_q}{2}$$

(3) Central blocks (CBs) located at the center of four neighboring ABs, represented by blue dashed blocks. CBs are obtained by replacing $2 \times 2$ blocks with the average of the four surrounding ABs:

$$CB = \frac{AB_1 + AB_2 + AB_3 + AB_4}{4}$$

[FIGURE:3]

For the Peppers image, the corresponding IBII is shown in Fig. 3(a). Compared with RAI in Fig. 1(a), IBII appears smoother and closer to the input image, with PSNR increasing from 23.15 dB to 24.90 dB. This confirms that IBII better approximates the input image than RAI. After one decoding iteration, the decoded image's PSNR increases from RAI's 29.08 dB to IBII's 30.42 dB, demonstrating that IBII provides higher decoded image quality than RAI with a single iteration.

Regarding IBII's computational complexity, division by 2 and 4 in equations (5) and (6) can be implemented via 1-bit and 2-bit right shifts in memory. Constructing one TB requires 4 additions, while one CB requires 3 additions. Since each range block contains one AB, two TBs, and one CB, each range block in IBII requires 11 additions total. These computations are negligible compared to those in each decoding iteration.

4.1 Experimental Setup

The experiments employ six $256 \times 256$ test images: Peppers, House, Lake, Baboon, Boat, and Airplane. Range block size is set to $4 \times 4$ with a sliding step of 8. Jacquin's method and two state-of-the-art fast fractal coding methods—Chaurasia's \cite{12} and Gupta's methods \cite{13}—are used to evaluate the proposed method. Both RAI and IBII are combined with OBD for fractal image decoding. In addition to PSNR, structural similarity (SSIM) is adopted to measure decoded image quality \cite{20}:

$$\text{SSIM}(x, y) = \frac{(2\mu_x\mu_y + C_1)(2\sigma_{xy} + C_2)}{(\mu_x^2 + \mu_y^2 + C_1)(\sigma_x^2 + \sigma_y^2 + C_2)}$$

where $x$ and $y$ represent local image patches at the same location in reference and distorted images $X$ and $Y$, respectively; $\mu_x$ and $\mu_y$ denote mean intensities; $\sigma_x$ and $\sigma_y$ denote standard deviations; and $\sigma_{xy}$ denotes covariance. Constants are set as $C_1 = 0.01L^2$ and $C_2 = 0.03L^2$, where $L = 255$. For the entire image, mean SSIM (MSSIM) evaluates overall quality:

$$\text{MSSIM} = \frac{1}{M} \sum_{k=1}^{M} \text{SSIM}(x_k, y_k)$$

where $M$ is the number of local patches.

[FIGURE:4]

4.2 Experimental Procedures

To compare IBII-based and RAI-based methods, the experimental procedure is designed as follows:

Step 1: Encode the input image.

Step 2: Compute ABs, TBs, and CBs using equations (5), (6), and (7) to obtain IBII. Use IBII as the initial image, perform one OBD iteration, and compute the PSNR and MSSIM of the decoded image.

Step 3: Use RAI as the initial image, perform one OBD iteration, and compute the PSNR and MSSIM of the decoded image.

Step 4: Compare the PSNR and MSSIM quality between RAI-based and IBII-based methods.

4.3 Experimental Results

Compared with RAI in Table 1 [TABLE:1], average PSNR and MSSIM increase from 22.28 dB to 23.26 dB and from 0.6102 to 0.6613, respectively, confirming that IBII better approximates the input image. Furthermore, all PSNR values in Tables 2, 3, and 4 exceed 25 dB, with IBII providing higher PSNR and MSSIM for all test images.

For Jacquin's method in Table 2 [TABLE:2], IBII-based method improves average PSNR from 27.28 dB to 28.04 dB and MSSIM from 0.8598 to 0.8729 compared to the RAI-based method. For Chaurasia's method in Table 3 [TABLE:3], IBII-based method improves average PSNR from 27.04 dB to 27.78 dB and MSSIM from 0.8522 to 0.8664. For Gupta's method in Table 4 [TABLE:4], IBII-based method improves average PSNR from 27.18 dB to 27.92 dB and MSSIM from 0.8536 to 0.8666.

Across all test images in Tables 1–4, decoded image quality improves by 0.56–1.41 dB in PSNR and by 0.0061–0.0173 in MSSIM. These results demonstrate that IBII better approximates the input image than RAI and achieves higher decoded image quality with one iteration across different fractal coding methods. In summary, the proposed IBII-based method effectively improves decoded image quality while maintaining both acceptable quality and real-time decoding.

[TABLE:1]

[TABLE:2]

[TABLE:3]

[TABLE:4]

5. Conclusion

This paper aims to improve decoded image quality while maintaining real-time decoding and acceptable output quality in fractal decoding. Previous RAI-based fractal decoding achieved acceptable quality with a single iteration. To further enhance quality, the proposed IBII better approximates the original image, enabling higher decoded image quality with one iteration under specific decoding strategies. Additionally, IBII can be combined with other fast fractal decoding methods to achieve real-time decoding, making them more practical for applications without strict quality requirements.

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