| With the fast development of science and technology, color image has moved from natural world into nowadays computer. The key technique that makes this change happen is due to the color image compression method. The reason is that if a certain color image is uncompressed, it may take up quite large memory space up to several gigabits, which would be difficult for people to transmit and storage huge volume of color images in computer network. On the contrary, the color image compressed takes up much lesser space, which makes it possible to transmit and store large amount of images under practical circumstances.The reason that the color image can be compressed lies in the fact that it contains different kinds of redundancies from the viewpoint of information theory. Of course, w e are unable to find the inner principals existed in image data though humans'eyes. However, we can find these redundancies with the help of modern statistics theory. For example, pertinence exists among pixels and also among components of each one pixel, both of which are classical representatives of color image redundancies. Generally speaking, the redundancy of the color image comprises of statistic redundancy, structure redundancy, knowledge redundancy, vision redundancy and color space redundancy, etc.Since Oliver published his PCM coding theory in1948, data compression technique has developed at a rapid speed. There are about one hundred methods until now, such as Huffman Coding, Arithmetic Coding, REL, PCM, DCT, Wavelet Coding, etc. All of these methods have a better effect to wipe off certain kind of image redundancy. But the effect is not so good if we only use one method to compress image. The reason is that there are various kinds of redundancy in one color image. Therefore we usually combine several coding methods together to compress the color image. And one such successful example is JPEG standard.For the sake of fully expression of a color image, our laboratory has published a new expression idea named as 3-D matrix theory, which is capable of expressing the three consecutive frames of a color image into a whole mathematical model to reduce its color redundancy. Unlike the traditional matrix theory, our 3-D matrix theory focuses on the reduction of color space redundancy such that it can further improve the image compression ratio.The main content of this paper is on the color image compression. Unlike the 3-D matrix method and those previous articles, this paper is to compress the color image in the idea of a newly invented concept named as 4-D matrix of\ order n, which extends the prior 3-D matrix model into 4-D matrix field and thus potentially facilitates its application in video compression in the future. Although it has been a time since the 3-D matrix theory's publish and one efficient orthogonal transform method related to that has been made out i.e. WDCT, there is still heavy research on how to express the color image with a better model. As for the 3-D matrix model, the main idea is to apply orthogonal transform on the RGB submatrix after dividing the original image and then compress the transformed coefficients via VQ, i.e. Vector Quantization. Though this does have a comparative high compression ratio, the 3-D matrix multiplication remains confined under the traditional 2-D multiplicaton, which often brings lots of calculation and meanwhile adds complexity of the algorithm because of VQ training process. On the contrary, this paper defines a new matrix multiplication formula which embraces the characteristics of the 2-D matrix multiplication but has a neat and easy rule. On the other hand, inspired by the construction concept of classical Hadamard matrix, we have successfully found a way to find the formula on how to build orthogonal matrix under our 4-D nth order matrix model. Besides, the scalar quantization is used to simplify our algorithm leading to a higher efficiency than 3-D matrix's VQ process.Finally, this paper lists out the performance of the 4-D nth order matrix orthogonal transform by using Visual C ++ 6.0 under Windows environment. The experimental results of our algorithm have shown its competitive energy compression capacity, which definitely proves the effectiveness of our algorithm.However, the 4-D nth order matrix orthogonal transform is still a new image compression algorithm and surely needs further improving. For instance, compared with the DCT's real transformation core, our orthogonal transform matrix is comprised only by integers +1 and -1, which results a relevant low flexibility and a high error rate run through PC. In addition, our algorithm is now confined within the still image compression area while it would be quite promising if we could find its application on the moving picture field and also increase its dimension to a even higher level for future study. |
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