Constant-time bilateral filter using spectral decomposition

Abstract

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Fast High-Dimensional Kernel Filtering

“…More recently, it was shown in [15], [17] that fast BLF of grayscale images can be performed using the partial eigendecomposition of the kernel matrix. In fact, the interpretation of BLF (and NLM) as kernel filters goes back to [31]- [33].…”

“…While the Nyström method has widely been used in machine learning [34]- [36], it appears that [31] is the first to apply this for image filtering. Note that, unlike [15], [17], the spatial and range kernel are treated as a single kernel in [31]- [33].…”

“…• As explained in detail in §II, it is difficult to scale [15], [17] for filtering high-dimensional (even color) images, since one needs to populate a huge kernel matrix and compute its eigendecomposition. We propose to use the Nyström method to solve this problem.…”

“…The rest of the paper is organized as follows. In §II, we introduce the notion of kernel filtering, and explain the core problem in relation to the spectral approximations in [15], [17]. We use the Nyström method in §III to overcome this problem.…”

“…For example, κ(t, s) = exp(−θ s−t 2 ), θ > 0, for standard BLF and NLM, where • is the Euclidean norm. It was shown in [15], [17] that the non-linear operations in (1) can be computed using convolutions by approximating κ. For convenience, we will describe this using our notations.…”