polynomial approximation of 1d signal

8/13/2019 Polynomial Approximation of 1D signal

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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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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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Thus we have estimated the signal properly


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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#AnderssonK02

polynomial approximation of 1d signal

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