Recursively implementating the Gaussian and its derivatives

Gaussian filtering is one of the most successfully operation in computer vision in order to reduce noise, calculating the gradient intensity change or performing Laplacian or the second directional derivative of an image. However, it is well known that in a multi-resolution context, where the need for large filters is required, this technique suffers from the fact it is a computationally expensive since the number of operations per point in convolving an image with a Gaussian filter is directly proportional to the width of the operator. We propose in this paper a technique in order to use Gaussian filtering with a reduced and fixed number of operations per output independently of the size of the filter. The key of our approach is to approximate in a mean square sense the prototype Gaussian filters with an exponentially based filter family depending on the same scale factor than the Gaussian filters (i.e. s) and then to implement in an exact and recursive way the approximate filters. An important point of the design presented in this paper is that dealing with Gaussian filters having different scale factor (i.e. s) will not require a new design algorithm as. The coefficients looked for in the recursive realization are determined function of the scale factor of each considered prototype filter, namely the Gaussian filter, its first and second derivative. Some experimental results will be shown to illustrate the efficiency of the approximation process and some applications to edge detection problems and multi-resolution techniques will be considered and discussed

An error estimate of Gaussian Recursive Filter in 3Dvar problem

Computational kernel of the three-dimensional variational data assimilation (3D-Var) problem is a linear system, generally solved by means of an iterative method. The most costly part of each iterative step is a matrix-vector product with a very large covariance matrix having Gaussian correlation structure. This operation may be interpreted as a Gaussian convolution, that is a very expensive numerical kernel. Recursive Filters (RFs) are a well known way to approximate the Gaussian convolution and are intensively applied in the meteorology, in the oceanography and in forecast models. In this paper, we deal with an oceanographic 3D-Var data assimilation scheme, named OceanVar, where the linear system is solved by using the Conjugate Gradient (GC) method by replacing, at each step, the Gaussian convolution with RFs. Here we give theoretical issues on the discrete convolution approximation with a first order (1st-RF) and a third order (3rd-RF) recursive filters. Numerical experiments confirm given error bounds and show the benefits, in terms of accuracy and performance, of the 3-rd RF.Comment: 9 page

Dynamic texture recognition using time-causal and time-recursive spatio-temporal receptive fields

This work presents a first evaluation of using spatio-temporal receptive fields from a recently proposed time-causal spatio-temporal scale-space framework as primitives for video analysis. We propose a new family of video descriptors based on regional statistics of spatio-temporal receptive field responses and evaluate this approach on the problem of dynamic texture recognition. Our approach generalises a previously used method, based on joint histograms of receptive field responses, from the spatial to the spatio-temporal domain and from object recognition to dynamic texture recognition. The time-recursive formulation enables computationally efficient time-causal recognition. The experimental evaluation demonstrates competitive performance compared to state-of-the-art. Especially, it is shown that binary versions of our dynamic texture descriptors achieve improved performance compared to a large range of similar methods using different primitives either handcrafted or learned from data. Further, our qualitative and quantitative investigation into parameter choices and the use of different sets of receptive fields highlights the robustness and flexibility of our approach. Together, these results support the descriptive power of this family of time-causal spatio-temporal receptive fields, validate our approach for dynamic texture recognition and point towards the possibility of designing a range of video analysis methods based on these new time-causal spatio-temporal primitives.Comment: 29 pages, 16 figure