The edge-preserving bilateral filter (BF) is a widely used smoothing tool in many applications. However, its brute-force implementation depends on the size of the box window. The shortcoming causes BF time-consuming for the image processing task with a large window. To make the computational complexity irrelevant to the window size, sparse approximations of the filtering kernels are calculated on a learned cosine dictionary by two steps. First, all possible frequencies are learned (estimated) from the filtering kernel to compose a cosine dictionary. Then, the sparse approximation is conducted on the learned dictionary to seek the optimal cosine approximation for both the range and spatial kernels. By making use of the one-dimensional cosine approximation for the range kernel, the BF is transformed into spatial convolutions. Subsequently, by employing the two-dimensional cosine approximation for the spatial kernel, spatial convolutions are decomposed into box filters of which the computational complexity is $O(1)$ . To the best of our knowledge, our approach is the first method that adaptively constructs a cosine dictionary according to the input kernel. This merit guarantees the best filtering accuracy and efficiency. These advantages are corroborated by several carefully designed experiments.

中文翻译：

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在学习的余弦字典上通过稀疏近似加速双边滤波

边缘保留双边滤波器 (BF) 是许多应用中广泛使用的平滑工具。然而，它的蛮力实现取决于框窗口的大小。缺点是对于大窗口的图像处理任务导致BF耗时。为了使计算复杂度与窗口大小无关，在学习的余弦字典上分两步计算滤波内核的稀疏近似值。首先，从过滤内核中学习（估计）所有可能的频率以组成一个余弦字典。然后，对学习的字典进行稀疏逼近，以寻求范围和空间核的最佳余弦逼近。通过使用范围核的一维余弦近似，将 BF 转换为空间卷积。随后，通过对空间核采用二维余弦近似，空间卷积被分解为盒式滤波器，其计算复杂度为 $O(1)$ 。据我们所知，我们的方法是第一种根据输入核自适应构建余弦字典的方法。这一优点保证了最佳的过滤精度和效率。几个精心设计的实验证实了这些优点。