24 research outputs found
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
On the Hilbert transform of wavelets
A wavelet is a localized function having a prescribed number of vanishing
moments. In this correspondence, we provide precise arguments as to why the
Hilbert transform of a wavelet is again a wavelet. In particular, we provide
sharp estimates of the localization, vanishing moments, and smoothness of the
transformed wavelet. We work in the general setting of non-compactly supported
wavelets. Our main result is that, in the presence of some minimal smoothness
and decay, the Hilbert transform of a wavelet is again as smooth and
oscillating as the original wavelet, whereas its localization is controlled by
the number of vanishing moments of the original wavelet. We motivate our
results using concrete examples.Comment: Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4,
pp. 1890-1894, 201
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
On a registration-based approach to sensor network localization
We consider a registration-based approach for localizing sensor networks from
range measurements. This is based on the assumption that one can find
overlapping cliques spanning the network. That is, for each sensor, one can
identify geometric neighbors for which all inter-sensor ranges are known. Such
cliques can be efficiently localized using multidimensional scaling. However,
since each clique is localized in some local coordinate system, we are required
to register them in a global coordinate system. In other words, our approach is
based on transforming the localization problem into a problem of registration.
In this context, the main contributions are as follows. First, we describe an
efficient method for partitioning the network into overlapping cliques. Second,
we study the problem of registering the localized cliques, and formulate a
necessary rigidity condition for uniquely recovering the global sensor
coordinates. In particular, we present a method for efficiently testing
rigidity, and a proposal for augmenting the partitioned network to enforce
rigidity. A recently proposed semidefinite relaxation of global registration is
used for registering the cliques. We present simulation results on random and
structured sensor networks to demonstrate that the proposed method compares
favourably with state-of-the-art methods in terms of run-time, accuracy, and
scalability
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 adaptive elliptical filtering using box splines
We demonstrate that it is possible to filter an image with an elliptic window
of varying size, elongation and orientation with a fixed computational cost per
pixel. Our method involves the application of a suitable global pre-integrator
followed by a pointwise-adaptive localization mesh. We present the basic theory
for the 1D case using a B-spline formalism and then appropriately extend it to
2D using radially-uniform box splines. The size and ellipticity of these
radially-uniform box splines is adaptively controlled. Moreover, they converge
to Gaussians as the order increases. Finally, we present a fast and practical
directional filtering algorithm that has the capability of adapting to the
local image features.Comment: 9 pages, 1 figur
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