Two-Dimensional Orthogonal DCT Expansion in Trapezoid and Triangular Blocks and Modified JPEG Image Compression

Abstract

In the conventional JPEG algorithm, an image is divided into eight by eight blocks and then the 2-D DCT is applied to encode each block. In this paper, we find that, in addition to rectangular blocks, the 2-D DCT is also orthogonal in the trapezoid and triangular blocks. Therefore, instead of eight by eight blocks, we can generalize the JPEG algorithm and divide an image into trapezoid and triangular blocks according to the shapes of objects and achieve higher compression ratio. Compared with the existing shape adaptive compression algorithms, as we do not try to match the shape of each object exactly, the number of bytes used for encoding the edges can be less and the error caused from the high frequency component at the boundary can be avoided. The simulations show that, when the bit rate is fixed, our proposed algorithm can achieve higher PSNR than the JPEG algorithm and other shape adaptive algorithms. Furthermore, in addition to the 2-D DCT, we can also use our proposed method to generate the 2-D complete and orthogonal sine basis, Hartley basis, Walsh basis, and discrete polynomial basis in a trapezoid or a triangular block.

Trapezoidal block split using orthogonal C2 transforms for HEVC video coding

“…It was proven in [5] that the DCT of a rotationally-symmetric image has zeros in positions (i, j) where i + j is odd. Furthermore, the restriction of the DCT basis functions with even i+j to the domain of part A forms an orthogonal basis for that part [5]. In order to get an orthonormal basis we multiply the basis functions by p 2.…”

“…In the appendix we derive some mathematical properties of the C2 transform that were not explained in [5].…”

“…The p 2 factor arises because the b t ijA and b t ijB needs to be scaled compared to b t ij to become unit norm basis functions for P A and P B [5]. However, b t ijA · p A and b t ijB · p B are exactly the C2 coefficient for P A and P B .…”

“…A new orthogonal transform for trapezoidal shapes was developed in [5] where the authors suggested to use a segmentation algorithm to decompose images to trapezoidal, rectangular and triangular blocks, according to the shapes of objects. That approach achieved better compression than JPEG.…”

“…In this paper, following the idea in [5], we suggest an alternative option to encode a rectangular block [6] [7] [8]. We select the best split, among a predefined set of splits, to encode a given rectangular block by two 180-degree rotationallysymmetric parts.…”