Closed-form quality measures for compressed medical images: statistical preliminaries for transform coding

As digital imaging technology advances, the amount of image data we generate increases and the need for compressed images becomes apparent. Because lossy compression will yield higher compression ratios than lossless methods, objective assessment metrics of reconstructed image quality are needed. In medical applications, model observers, especially the channelized Hotelling observer, have been successfully used to predict human observer performance and to evaluate image quality for detection tasks on various backgrounds. To use model observers, however, requires knowledge of noise statistics. This paper finds closed-form expressions for the noise induced by transform coding, one of the most commonly used methods for image compression. Knowledge of the noise enables us to study the effect of image compression on the clinical utility of medical images that have been reconstructed after being compressed using transform coding. In this paper, by analyzing image compression procedures, we propose a block-based transform coding representation in 1-D form, identify the quantization noise as the sole distortion source in transform coding, and derive the compression noise statistics. We show that the probability density function (pdf) of the compression noise is defined as a function of the transform matrix and its corresponding quantization matrix in the transform coding algorithm. We prove that the compression noise is a normal distribution when the dimension of the transform (the block size) is typical. We also provide the pdf of JPEG compression noise as a function of the quantization table and the DCT transform bases. This work provides the theoretical foundation for using the model observers in closed mathematical form, and can be applied to other image compression application areas that require the statistics of compression noise as well.