Construction Of Tight Frame And Its Application In Coding

To transmit information on packet-based network, data is encapsulated as packets transmitting from a source to a destination through the network. If the network traffic is jammed or the server's buffer is overflowed, the transmitting packets might be delayed or lost. Detection and retransmission will cause huge latency; such delays are unacceptable for many real time applications, such as the transmission of images, audio, and video. How to restore the original data from received packets is a hot topic in research.Frames are redundant sets of vectors. Some frames can remain to be frames by removing arbitrary number of vectors, after that they can still represent all the elements in the space. Therefore, to encode the data by these frames is very robust even if parts of the packets are lost, the receiver can still restore all the original data. There are quantization errors when coding data using frames. Among all the frames, tight frames have minimum quantization errors. There are two methods called approximate reconstruction and perfect reconstruction to restore data. In approximate reconstruction, the uniform tight frame is optimal when there's only one packet lost; if there are two packets lost, the equiangular tight frame is optimal. In perfect reconstruction, the tight frame with maximum robustness to erasure is the best one. The equiangular tight frame can be constructed by graph and signature matrix. Conference Matrix, Hadamard matrix, cube root Seidel matrix and fourth root Seidel matrix are signature matrices.This article constructs equiangular tight frames by sixth root Seidel matrix and depicts the necessary conditions of constructing sixth root Seidel matrix. It also shows the method to construct new signature matrix when two signature matrices satisfying Q~2 = ( M-1) I + 2Q or Q~2 = ( M-1) I+2Q, so as to generate new equiangular tight frame. It will get an orthogonal matrix DFT_M through Discrete Fourier Transform, then chooses 2k or 2k+1 columns in the middle of DFT_M , and then transforms it to real number to get real, tight frames with maximal robustness to erasure. Finally we use tight frame to transmit images, audio and video. Proved by the experiences, the quality of net transmission is tremendously increased by using frame coding.