8,315 research outputs found
Recursive filtering of images with symmetric extension
Recursive filters are widely used in image analysis due to their efficiency and simple implementation. However these filters have an initialisation problem which either produces unusable results near the image boundaries or requires costly approximate solutions such as extending the boundary manually. In this paper, we describe a method for the recursive filtering of symmetrically extended images for filters with symmetric denominator. We begin with an analysis of symmetric extensions and their effect on non-recursive filtering operators. Based on the non-recursive case, we derive a formulation of recursive filtering on symmetric domains as a linear but spatially varying implicit operator. We then give an efficient method for decomposing and solving the linear implicit system, along with a proof that this decomposition always exists. This decomposition needs to be performed only once for each dimension of the image. This yields a filtering which is both stable and consistent with the ideal infinite extension. The filter is efficient, requiring less computation than the standard recursive filtering. We give experimental evidence to verify these claims. (c) 2005 Elsevier B.V. All rights reserved
Efficient and Consistent Recursive Filtering of Images with Reflective Extension
Recursive filters are commonly used in scale space construction for their efficiency and simple implementation. However these filters have an initialisation problem which either produces unusable results near the image boundaries or requires costly approximate solutions such as extending the boundary manually. In this paper, we describe a method for the recursive filtering of reflectively extended images for filters with symmetric denominator. We begin with an analysis of reflective extensions and their effect on non-recursive filtering operators. Based on the non-recursive case, we derive a formulation of recursive filtering on reflective domains as a linear but time-varying implicit operator. We then give an efficient method for decomposing and solving the linear implicit system. This decomposition needs to be performed only once for each dimension of the image. This yields a filtering which is both stable and consistent with the ideal infinite extension. The filter is efficient, requiring the same order of computation as the standard recursive filtering. We give experimental evidence to verify these claims
Fast O(1) bilateral filtering using trigonometric range kernels
It is well-known that spatial averaging can be realized (in space or
frequency domain) using algorithms whose complexity does not depend on the size
or shape of the filter. These fast algorithms are generally referred to as
constant-time or O(1) algorithms in the image processing literature. Along with
the spatial filter, the edge-preserving bilateral filter [Tomasi1998] involves
an additional range kernel. This is used to restrict the averaging to those
neighborhood pixels whose intensity are similar or close to that of the pixel
of interest. The range kernel operates by acting on the pixel intensities. This
makes the averaging process non-linear and computationally intensive,
especially when the spatial filter is large. In this paper, we show how the
O(1) averaging algorithms can be leveraged for realizing the bilateral filter
in constant-time, by using trigonometric range kernels. This is done by
generalizing the idea in [Porikli2008] of using polynomial range kernels. The
class of trigonometric kernels turns out to be sufficiently rich, allowing for
the approximation of the standard Gaussian bilateral filter. The attractive
feature of our approach is that, for a fixed number of terms, the quality of
approximation achieved using trigonometric kernels is much superior to that
obtained in [Porikli2008] using polynomials.Comment: Accepted in IEEE Transactions on Image Processing. Also see addendum:
https://sites.google.com/site/kunalspage/home/Addendum.pd
Fast space-variant elliptical filtering using box splines
The efficient realization of linear space-variant (non-convolution) filters
is a challenging computational problem in image processing. In this paper, we
demonstrate that it is possible to filter an image with a Gaussian-like
elliptic window of varying size, elongation and orientation using a fixed
number of computations per pixel. The associated algorithm, which is based on a
family of smooth compactly supported piecewise polynomials, the
radially-uniform box splines, is realized using pre-integration and local
finite-differences. The radially-uniform box splines are constructed through
the repeated convolution of a fixed number of box distributions, which have
been suitably scaled and distributed radially in an uniform fashion. The
attractive features of these box splines are their asymptotic behavior, their
simple covariance structure, and their quasi-separability. They converge to
Gaussians with the increase of their order, and are used to approximate
anisotropic Gaussians of varying covariance simply by controlling the scales of
the constituent box distributions. Based on the second feature, we develop a
technique for continuously controlling the size, elongation and orientation of
these Gaussian-like functions. Finally, the quasi-separable structure, along
with a certain scaling property of box distributions, is used to efficiently
realize the associated space-variant elliptical filtering, which requires O(1)
computations per pixel irrespective of the shape and size of the filter.Comment: 12 figures; IEEE Transactions on Image Processing, vol. 19, 201
Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields
We present \emph{telescoping} recursive representations for both continuous
and discrete indexed noncausal Gauss-Markov random fields. Our recursions start
at the boundary (a hypersurface in , ) and telescope inwards.
For example, for images, the telescoping representation reduce recursions from
to , i.e., to recursions on a single dimension. Under
appropriate conditions, the recursions for the random field are linear
stochastic differential/difference equations driven by white noise, for which
we derive recursive estimation algorithms, that extend standard algorithms,
like the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother, to noncausal
Markov random fields.Comment: To appear in the Transactions on Information Theor
Constant-time filtering using shiftable kernels
It was recently demonstrated in [5] that the non-linear bilateral filter [14]
can be efficiently implemented using a constant-time or O(1) algorithm. At the
heart of this algorithm was the idea of approximating the Gaussian range kernel
of the bilateral filter using trigonometric functions. In this letter, we
explain how the idea in [5] can be extended to few other linear and non-linear
filters [14, 17, 2]. While some of these filters have received a lot of
attention in recent years, they are known to be computationally intensive. To
extend the idea in [5], we identify a central property of trigonometric
functions, called shiftability, that allows us to exploit the redundancy
inherent in the filtering operations. In particular, using shiftable kernels,
we show how certain complex filtering can be reduced to simply that of
computing the moving sum of a stack of images. Each image in the stack is
obtained through an elementary pointwise transform of the input image. This has
a two-fold advantage. First, we can use fast recursive algorithms for computing
the moving sum [15, 6], and, secondly, we can use parallel computation to
further speed up the computation. We also show how shiftable kernels can also
be used to approximate the (non-shiftable) Gaussian kernel that is ubiquitously
used in image filtering.Comment: Accepted in IEEE Signal Processing Letter
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