Smoothness priors and quadratic variation (QV) regularization are widely used techniques in many applications ranging from signal and image processing, computer vision, pattern recognition, and many other fields of engineering and science. In this contribution, an extension of such algorithms to band-stop smoothing filters (BSSFs) is investigated. For designing a BSSF, the most important parameters are the order and the cutoff frequencies. In this paper, we show that with the optimization approaches (smoothness priors or QV regularization), the cutoff frequencies are related to the regularized parameters and the order can be directly (and easily) controlled with the number of derivatives. We describe two ways to implement the BSSFs using these approaches. First, we present a parallel structure to BSSF and then illustrate why it is less than ideal. Next, we present a novel approach regarding parallel structure to produce BSSFs with very sharp transition bands for high-performance applications. An improved optimization-based approach to BSSF design is introduced. The performance of the new BSSFs is nearly ideal.

中文翻译：

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带阻平滑滤波器设计

平滑先验和二次变异（QV）正则化是许多应用程序中广泛使用的技术，范围从信号和图像处理，计算机视觉，模式识别以及工程和科学的许多其他领域。在此贡献中，研究了这种算法对带阻平滑滤波器（BSSF）的扩展。对于设计BSSF，最重要的参数是阶数和截止频率。在本文中，我们表明，使用优化方法（先验平滑或QV正则化），截止频率与正则化参数相关，并且可以通过导数的数量直接（轻松地）控制顺序。我们描述了使用这些方法来实现BSSF的两种方法。首先，我们提出了与BSSF并行的结构，然后说明了为什么它不理想。接下来，我们提出一种有关并行结构的新颖方法，以生产具有非常陡峭过渡带的BSSF，以用于高性能应用。介绍了一种改进的基于优化的BSSF设计方法。新的BSSF的性能几乎是理想的。