This letter proposes a fast implementation of the regularity-constrained discrete sine transform (R-DST). The original DST leaks the lowest frequency (DC: direct current) components of signals into high frequency (AC: alternating current) subbands. This property is not desired in many applications, particularly image processing, since most of the frequency components in natural images concentrate in DC subband. The characteristic of filter banks whereby they do not leak DC components into the AC subbands is called regularity. While an R-DST has been proposed, it has no fast implementation because of the singular value decomposition (SVD) in its internal algorithm. In contrast, the proposed regularity-constrained fast sine transform (R-FST) is obtained by just appending a regularity constraint matrix as a postprocessing of the original DST. When the DST size is $M\times M$ ($M=2^\ell$, $\ell \in \mathbb {N}_{\geq 1}$), the regularity constraint matrix is constructed from only $M/2-1$ rotation matrices with the angles derived from the output of the DST for the constant-valued signal (i.e., the DC signal). Since it does not require SVD, the computation is simpler and faster than the R-DST while keeping all of its beneficial properties. An image processing example shows that the R-FST has fine frequency selectivity with no DC leakage and higher coding gain than the original DST. Also, in the case of $M=8$, the R-FST saved approximately 0.126 seconds in a 2-D transformation of $512\times 512$ signals compared with the R-DST because of fewer extra operations.

中文翻译：

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规则约束的快速正弦变换

这封信提出了一种快速实施规则约束离散正弦变换 (R-DST)。原始 DST将信号的最低频率（DC：直流）分量泄漏到高频（AC：交流）子带中。在许多应用中，尤其是图像处理中并不需要这种特性，因为自然图像中的大多数频率分量都集中在 DC 子带中。滤波器组不会将直流分量泄漏到交流子带中的特性称为规律性. 虽然已经提出了 R-DST，但由于其内部算法中的奇异值分解 (SVD)，它没有快速实现。相比之下，所提出的规则约束快速正弦变换 (R-FST) 仅通过附加规则约束矩阵作为原始 DST 的后处理来获得。当 DST 大小为$M\倍 M$($M=2^\ell$,$\ell \in \mathbb {N}_{\geq 1}$)，正则性约束矩阵仅由$M/2-1$旋转矩阵，其角度来自恒定值信号（即直流信号）的 DST 输出。由于它不需要 SVD，因此计算比 R-DST 更简单、更快，同时保留了其所有有益特性。一个图像处理示例表明，R-FST 具有良好的频率选择性，没有直流泄漏，并且比原始 DST 具有更高的编码增益。此外，在$M=8$，R-FST 在 2-D 变换中节省了大约 0.126 秒$512\乘以 512$信号与 R-DST 相比，因为额外的操作更少。