Edge detection from truncated Fourier data using spectral mollifiers
Edge detection from truncated Fourier data using spectral mollifiers
复制标题
使用光谱缓和器从截断的傅立叶数据中进行边缘检测
DOI：
10.1007/s10444-011-9258-4
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发表时间：
2011
影响因子：
1.7
通讯作者：
Yang Wang
中科院分区：
文献类型：
--
作者：
D. Cochran;Anne Gelb;Yang Wang
从有限数量的傅立叶系数的边缘检测是具有挑战性的，因为它需要从全局数据中提取局部信息。当输入数据有噪声时，该问题加剧，因为准确的高频信息对于检测边缘是至关重要的。噪声还增加了分段光滑函数的傅立叶重构中的振荡，特别是在不连续性附近。Gelb和Tadmor（Appl Comput Harmon Anal 7：101-135，1999，SIAM J Numer Anal 38（4）：1389-1408，2000）中的边缘检测方法引入了“浓度核”的思想作为收敛到分段平滑函数的奇异支撑的方式。然而，那里使用的内核，以及随后的修改，以减少噪声的影响，一般是振荡的，因此振荡总是普遍存在于跳跃不连续性的邻域。本文重新讨论了浓度核，但坚持一致收敛于函数的“尖峰”，即边缘检测方法在远离跳跃的地方收敛到零，而不会在跳跃附近引入新的振荡。我们表明，这是可以实现的，通过一个容许类的光谱软化。我们的方法还抑制了由添加的噪声引起的振荡。
Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generallyoscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists onuniform convergenceto the “sharp peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.