Speed Up Bilateral Filtering via Sparse Approximation on A Learned Cosine Dictionary

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.