Acceleration of Gaussian Filter with Short Window Length Using DCT-1

Abstract

This paper presents an accelerated constant-time Gaussian filter $(O(1)$ GF) specialized in short window length where constant-time $(O(1))$ means that computational complexity per pixel does not depend on filter window length. Our method is extensively designed based on the idea of $O(1)$ GF based on Discrete Cosine Transform (DCT). This framework approximates a Gaussian kernel by a linear sum of cosine terms and then convolves each cosine term in $O(1)$ per pixel using sliding transform. Importantly, if window length is short, DCT-1 consists of easily-computable cosine values such as 0, $\pm\frac{1}{2}$ and ±1. This behavior is not satisfied in other DCT types. From this fact, our method accelerates the sliding transform by employing DCT-1 focusing on short window length. Experiments show that our method overcomes naive Gaussian convolution and existing $O(1)$ GF in terms of computational time. Interestingly, the results also reveal that, without truncating negligible terms, our method runs faster than convolution.