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Polynomialapproximation of
1D signalPi19404
February 5, 2014
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Contents
Contents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 Local Subspace Approximation Through Convolution . . . . 0.3 Polynomial Approximation of 1D signal . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Polynomial approximation of 1D signal
Polynomial approximation of 1Dsignal
0.1 Introduction
In this article we will look at the concept for polynomial expansion toapproximate a neighborhood of a pixel with a polynomial.
0.2 Local Subspace Approximation ThroughConvolution
Given a discrete signal and filter with N taps we canuse convolution to compute the filter response
This can be also interpreted as an inner product
This provides direct interpretation of convolution as projectionoperation
Convolving
with a series of filters
gives correspond-
ing filter response
.
These filter responses can be interpreted as projection of thesignal onto local filter basis.
For M filters of length N ,the basis matrix B is of size
If we consider a polynomial basis ,the project can be con-sidered as taylors series approximation of signal till order 2.
0.3 Polynomial Approximation of 1D signal
The idea of polynomial expansion is to approximate a neighbor-hood of a pixel with a polynomial.
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Polynomial approximation of 1D signal
Considering only a quadratic polynomial ,pixel values in a neigh-borhood is given by
where A is a symmetric matrix,b is a vector and c is a scalar
The coefficients can be estimated by weighted least square es-timate of pixel values about neighborhood.
Let us consider a sinusoidal signal which needs to be approxi-mated locally using a polynomial function.
Let us consider a polynomial of order.The basis functions of thesubspace where the local signal is being approximated are
The basis function are defined on the discrete grid of -N to
N considering a window/neighborhood of ,for thepresent example let N=3;
using standard least square estimate we can estimate polynomialexpansion of a signal about a neighborhood
The larger the neighborhood ,more samples we have,but it will bemore difficult to fit a general function over this large neigh-borhood.
Let us consider an example
we have sampled the signal at discrete intervals of 1 from
Let us consider a gaussian basis with variance 1 and neighborhoodsize of as the interpolation function.
Let us consider the first sample of signal and perform polynomialapproximation about the neighboorhood
The coefficients we obtain are corresponding to the
basis
Here we have considered x to lie between ,thus in theequation
we need to replace x by
Doing which we obtain
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Polynomial approximation of 1D signal
Thus we have estimated the signal properly
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Polynomial approximation of 1D signal
below are 1D polynomial basis function
(a) Basis and applicability
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Bibliography
Bibliography
http://dblp.uni-trier.de/db/journals/sigpro/sigpro87.html#AnderssonWK07http://dblp.uni-trier.de/db/journals/sigpro/sigpro87.html#AnderssonWK07http://dblp.uni-trier.de/db/conf/icmcs/icme2002-1.html#AnderssonK02http://dblp.uni-trier.de/db/conf/icmcs/icme2002-1.html#AnderssonK02Top Related