Discrete sine transform

About: Discrete sine transform is a research topic. Over the lifetime, 3269 publications have been published within this topic receiving 73181 citations.

Digital filtering of images using the discrete sine or cosine transform

Abstract: The discrete sine and cosine transforms (DSTS and DCTS) are powerful tools for image compression. They would be even more useful if they could be used to perform convolution. As we have recently shown, these transforms do possess a convolution-multiplication property and therefore can be used to perform digital filtering. The convolution is a special type, called symmetric corrvo/ution. This paper reviews the concepts of symmetric convolution and then elaborates on how to use the operation, and thereby DSTS and DCTS, to implement digital filters for images. An indication of how to compute the transforms and what is the complexity of their use is also given.

A second-order fast compact scheme with unequal time-steps for subdiffusion problems

Abstract: In consideration of the initial singularity of the solution, a temporally second-order fast compact difference scheme with unequal time-steps is presented and analyzed for simulating the subdiffusion problems in several spatial dimensions. On the basis of sum-of-exponentials technique, a fast Alikhanov formula is derived on general nonuniform meshes to approximate the Caputo’s time derivative. Meanwhile, the spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by a fast discrete sine transform via the FFT algorithm. So the proposed algorithm is computationally efficient with the computational cost about $O(MN\log M\log N)$ and the storage requirement $O(M\log N)$ , where M and N are the total numbers of grids in space and time, respectively. With the aids of discrete fractional Gronwall inequality and global consistency analysis, the unconditional stability and sharp H1-norm error estimate reflecting the regularity of solution are established rigorously by the discrete energy approach. Three numerical experiments are included to confirm the sharpness of our analysis and the effectiveness of our fast algorithm.

Iterative methods for ill-conditioned Toeplitz matrices

Abstract: In this paper we study the use of the Sine Transform for preconditioning linear Toeplitz systems. We consider Toeplitz matrices with a real generating function that is nonnegative with only a small number of zeros. Then we can define a preconditioner of the formS n ΛS n whereS n is the matrix describing the discrete Sine transform and Λ is a diagonal matrix. If we have full knowledge aboutf then we can show that the preconditioned system is of bounded condition number independly ofn. We can obtain the same result for the case that we know only the position and order of the zeros off. If we only know the matrix and its coefficientst j , we present Sine transform preconditioners that show in many examples the same numerical behaviour.

Fractional, canonical, and simplified fractional cosine transforms

Abstract: The Fourier transform can be generalized into the fractional Fourier transform (FRFT), linear canonical transform (LCT), and simplified fractional Fourier transform (SFRFT). They extend the utilities of original Fourier transform, and can solve many problems that can not be solved well by original Fourier transform. We generalize the cosine transform. We derive the fractional cosine transform (FRCT), canonical cosine transform (CCT), and simplified fractional cosine transform (SFRCT). We show that they are very similar to the FRFT, LCT, and SFRFT, but they are much more efficient for dealing with the even, real even functions. For digital implementation, FRCT and CCT can save 1/2 of the real number multiplications, and SFRCT can save 3/4. We also discuss their applications, such as optical system analysis and space-variant pattern recognition.