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Abstract: Non-uniform filters are frequently used in many image processing applications to describe regions or to detect specific features. However, non-uniform filtering is a computationally complex task. This paper presents a method to perform fast non-uniform filtering using a reduced number of memory accesses. The idea is based on integral images which are commonly used for box or Haar wavelet filtering. The disadvantage of those filters for several applications is their uniform shape. We describe a method to build Symmetric Weighted Integral Images that are tailored for a variety of kernels and the process to perform fast filtering with them. We show a relevant speedup when compared to Kernel Integral Images and large when compared to conventional non-uniform filtering by reducing the computational complexity.

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- 1. FAST NON-UNIFORM FILTERING WITH SYMMETRIC WEIGHTED INTEGRAL IMAGES David Marimon Telefónica Research and Development Barcelona, Spain marimon@tid.es Telefónica I+D
- 2. Non-uniform filtering is costly Non-uniform filters are frequently used in many image processing applications to describe regions or to detect specific features. However, non-uniform filtering is a computationally complex task. For a generic filtering with a kernel k(x): This requires 2N memory accesses and N multiplications per sample in the input function. Telefónica I+D
- 3. Related Work Fast box filtering on images was introduced by Crow [1] and later used for Haar wavelets by Viola and Jones [2]. The first step consists in pre-computing the integral image: The filtered output of for a box kernel of size NxM can be computed with the following addition: Note that R implies only 4 memory accesses and 3 additions. The limitation of box filtering is the uniform shape of the filter. [1] Telefónica I+D F.Crow. Summed-area tables for texture mapping. In Proc. Computer Graphics (SIGGRAPH), volume 18, pages 207-212,1984. [2] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In Proc. CVPR, vol 1, pages 511-518, 2001.
- 4. Related Work (II) Heckbert [1] proposed to approximate a kernel function by the n times repeated convolution of a box filter. • Filtering can then be computed by only accessing n + 1 samples of the n times integral of f(x). Hussein et al.[2] proposed the Kernel Integral Images (KII). • Find the linear combination of simple functions and the corresponding weighting factors that generate, or approximate, the desired kernel shape. Porikli [3] proposed Reshuffling which exploits the redundancy in the weights of the kernel. • Prior to filtering, two structures must be built from a kernel: the links and the corresponding weights. [1] P. Heckbert. Filtering by repeated integration. In Proc. Computer Graphics (SIGGRAPH), volume 20, pages 315-321, 1986. [2] M. Hussein, F. Porikli, and L. Davis. Kernel integral images: A framework for fast non uniform filtering. In Proc. CVPR, June 2008. [3] F. Porikli. Reshuffling: a fast algorithm for filtering with arbitrary kernels. In SPIE Electronic Imagining Conference on Real- Time Image Processing, volume 6811, 2008 Telefónica I+D
- 5. Symmetric Weighted Integral Images (SWII) Integral images for which the contribution of each sample of the input function is weighted. • Weighting is a slope of increasing or decreasing value. • Designed for filtering with non-uniform kernel shapes defined with slopes of increasing or of decreasing weight. 5 SWII defined for the 2D case: Telefónica I+D
- 6. Filtering with SWII Let us define computational complexity per output sample is indicated as C(a; b; c) where • a is the number of memory accesses, • b the additions, and • c, the multiplications. Kernel with increasing slope in x: • C(6,5,1) Decreasing slope in x: Telefónica I+D
- 7. Filtering with SWII (II) Triangle-shaped kernel: C(10,9,0) Pyramid-shaped kernel built by adding two triangle-shaped kernels: C(20,19,0) Other kernel shapes can be built by translating, overlapping, and adding increasing or decreasing slopes Telefónica I+D
- 8. Experiments Pyramid-shaped kernel Standard filtering: C ( 2·N·M+1, N·M -1, N·M) Reshuffling • Number of redundant coefficients: • C ( 2·N·M+1, N·M, U) Kernel Integral Images (KII) • Three KII needed: • Pre-computation: C(13, 6, 2) • Filtering: C(21, 28, 16) SWII • Pre-computation: C(17, 10, 6) • Filtering: C(21, 19, 0) Telefónica I+D
- 9. Results According to [1], we fix a relative cost of 9 for each memory access (including array indexing and one addition per access) on a 2D array, 4 for an integer multiplication, 1 for an integer addition. Reshuffling is 1.1 to 1.5x faster than conventional filtering, both for different kernels sizes and for multiple scales. Different kernel sizes (N=M) Performance for multi-scale filtering [1] Telefónica“Reshuffling: A fast algorithm for filtering with arbitrary kernels,” in SPIE EI Conf. on Real-Time Image Processing, 2008, F. Porikli, I+D
- 10. Application to keypoint detection Keypoint detection at extrema of Determinant of Hessian in scale space. Mikolajczyk (VGG @ Oxford Univeristy) dataset and Repeatibility measure Telefónica I+D
- 11. Conclusions Contributions: • Symmetric Weighted Integral Images (SWII) can be used to build a variety of kernel shapes. • A novel technique to perform non-uniform filtering. The results show the speed improvement over Kernel Integral Images (especially relevant for multi-scale filtering) and Reshuffling. Successful application to keypoint detection. Telefónica I+D
- 12. Telefónica I+D

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