Associations between MSE and SSIM as cost functions in linear decomposition with application to bit allocation for sparse coding

Abstract

The traditional image quality assessments, such as the mean squared error (MSE), the signal-to-noise ratio (SNR), and the Peak signal-to-noise ratio (PSNR), are all based on the absolute error of images. Structural similarity (SSIM) index is another important image quality assessment which has been shown to be more effective in the human vision system (HVS). Although there are many essential differences between MSE and SSIM, some important associations exist between them. In this paper, the associations between MSE and SSIM as cost functions in linear decomposition are investigated. Based on the associations, a bit-allocation algorithm for sparse coding is proposed by considering both the reconstructed image quality and the reconstructed image contrast. In the proposed algorithm, the space occupied by a linear coefficient of a basis in sparse coding is reduced to only 9 to 10 bits, in which 1 bit is used to save the sign of linear coefficient, 3 bits are used to save the number of powers of 10 in scientific notation, and only 5 to 6 bits are used to save the significance digits. The experimental results show that the proposed bit-allocation algorithm for sparse coding can maintain both the image quality and the image contrast well.

Introduction

Sparse coding was proposed to imitate the mammalian visual cortex for image representation in 1996 [1], [2]. Now it has been widely used as an unsupervised learning method for sparse representation in various fields [3], [4], [5]. Although lots of works have been conducted in sparse coding and its applications, few efforts have been taken to discuss bit allocation of the coded data in sparse coding. Bit allocation is very important for some applications of sparse coding, such as compressed sensing [6] and fractal image coding [7]. In these applications, signal is encoded by sparse coding, and then the encoded data need to be saved and used to reconstruct the original signal. As an important part of coding technology, the spaces occupied by the data have great influence on the compression rate and the quality of reconstructed signal.

The absolute error-based image assessments, such as the mean square error (MSE), the signal-to-noise ratio (SNR), and the Peak signal-to-noise ratio (PSNR), are the most commonly used measurements to measure the similarity of images. For two image blocks $\mathbf{x}$ and $\mathbf{y}$, if the pixels in $\mathbf{x}$ are $x_1,x_2,\dots ,x_p$, and the pixels in $\mathbf{y}$ are $y_1,y_2,\dots ,y_p$, then the MSE value between $\mathbf{x}$ and $\mathbf{y}$ can be calculated as following,

$$\mathrm{MSE}(\mathbf{x},\mathbf{y})=\frac{1}{p}{\displaystyle \sum _{i=1}^p}{(y_i-x_i)}^2\text{.}$$

Therefore, MSE is a pixel error-based measurement. SNR and PSNR are both derived from MSE. These absolute error-based assessments are not only used to measure image quality, but also used to measure almost all kinds of signals.

Structural similarity (SSIM) index, proposed by Wang and Bovik [8], aims to improve the effectiveness of image quality assessment (IQA) in human visual systems (HVS). In SSIM, the errors are taken as three parts: the luminance error, the contrast error, and the structure error. For two image blocks $\mathbf{x}$ and $\mathbf{y}$, if ${\mu }_{\mathbf{x}}$ and ${\mu }_{\mathbf{y}}$ are the means of the pixels in the image blocks $\mathbf{x}$ and $\mathbf{y}$, respectively, ${\sigma }_{\mathbf{x}}$ and ${\sigma }_{\mathbf{y}}$ are the standard deviations of the pixels in the image blocks $\mathbf{x}$ and $\mathbf{y}$, respectively, and ${\sigma }_{\mathbf{xy}}$ is the covariance between $\mathbf{x}$ and $\mathbf{y}$, then SSIM gets a form as

$$\mathrm{SSIM}(\mathbf{x},\mathbf{y})=\left[\frac{2{\mu }_{\mathbf{x}}{\mu }_{\mathbf{y}}+{\epsilon }_1}{{\mu }_{\mathbf{x}}^2+{\mu }_{\mathbf{y}}^2+{\epsilon }_1}\right]\left[\frac{2{\sigma }_{\mathbf{xy}}+{\epsilon }_2}{{\sigma }_{\mathbf{x}}^2+{\sigma }_{\mathbf{y}}^2+{\epsilon }_2}\right],$$

where ${\epsilon }_1,{\epsilon }_2<<1$ are two small positive constants. If the variance of a given image block $\mathbf{y}$ is zero, then $\mathbf{y}$ can be losslessly linearly expressed by $\mathbf{1}$ with all ones. Because this paper focuses on linear decomposition and sparse coding, here we only consider the image blocks with non-zero variance. At this case, we can set ${\epsilon }_1={\epsilon }_2=0$ and SSIM gets a simpler form as

$$\mathrm{SSIM}(\mathbf{x},\mathbf{y})=\frac{4{\mu }_{\mathbf{x}}{\mu }_{\mathbf{y}}{\sigma }_{\mathbf{xy}}}{({\mu }_{\mathbf{x}}^2+{\mu }_{\mathbf{y}}^2)({\sigma }_{\mathbf{x}}^2+{\sigma }_{\mathbf{y}}^2)}\text{.}$$

SSIM does not have an absolute advantage over PSNR. For example, Some researches found that PSNR is better than SSIM for Gaussian blur [9], [10]. Dosselmann and Yang showed that SSIM index between two images can be predicted by PSNR between them [11]. Although SSIM has some limitations, it achieves better performance in many synthetic datasets, and has been widely accepted as an effective image quality assessment and applied in many fields [12], [13].

As described above, MSE is a pixel-based image assessment, and SSIM is a structure-based image assessment for HVS. Thus, there are many significant and essential differences between them. Here we do not focus on these important differences. On the contrary, some interesting associations are discussed when we take the image quality assessments as the cost functions in linear decomposition. Firstly, the selected bases from a basis set for a target vector are the same in the linear decomposition schemes with different cost functions MSE and SSIM. Secondly, for a target vector, the ratio of the corresponding linear coefficients of the selected bases in the MSE-based linear decomposition scheme and the SSIM-based scheme is a constant, which is just the value of Pearson’s correlation coefficient between the target vector and its estimated vector.

According to these interesting associations between MSE and SSIM, the absolute value of the linear coefficient of a selected basis in the SSIM-based linear decomposition scheme is always not less than that of the same basis in the MSE-based scheme. The reconstructed image with larger linear coefficients has higher image contrast. By considering both the reconstructed image quality and the reconstructed image contrast, a bit-allocation algorithm for coded data in sparse coding is proposed here. For linear coefficient s of a selected basis, if $|s|=a\times {10}^b$ in scientific notation, we use 1 bit to store the sign of s, 3 bits to store b, and 5 to 6 bits to store a. The experiment results show that the proposed algorithm can maintain image quality and image contrast well.

The rest of this paper is structured as following. In Section 2 we briefly introduce linear decomposition. The associations between MSE and SSIM as cost functions in linear decomposition are studied in Section 3. Then the bit-allocation algorithm for sparse coding is proposed in Section 4. In Section 5, several experiments are conducted to discuss the number of bits used to save the parameters of sparse coding in the proposed bit-allocation algorithm. Finally, the conclusions are drawn in Section 6.