Normalizedconvolution for

Image Interpolation

Pi19404

February 4, 2014

Contents

Contents

Normalized convolution for Image Interpolation 3

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Normalized convolution for Image Interpolation

Normalized convolution for

Image Interpolation

0.1 Introduction

In this article we will look at the concept for normalized convolution

for image interpolation and how a sparsely and ununiformaly sampled

grid can be used for image reconstruction.

� Convolution is a neighborhood operation.

� For 1D case,the convolution of a signal f(t) with a filter h(t) can

be expressed as

y(n) =X

k

h(k)f(n� k) (1)

� Thus result of convolution at point n is weighted sum of samples

in the neighborhood of sample point n.

� Interpolation is process of estimation of value of signal at un-

known point based on set of known points.

� Many times we require analysis of irregularily sampled data, which

is more compilated than regularily sampled data.It is often re-

quired to reconstruct the irregularily sampled signal or resample

it onto a regular grid.

� One method to do this is to use interpolation techniques obtain

a regularily sampled signal.The missing values in the regularily sam-

pled grid are computed using intepolation which is implemented

using convolution operations.

� Let us consider a signal 1D signal x(t) = [x1; 0; 0; x4; x5; 0; 0] and h(t) =[1=3; 1=3; 1=3].

� The result of convolution is given by y(t) = [x1=3; x1=3; x4=3; x4 +x5=3; x4 + x5=3; x5=3; x1=3]

� convolution can be expressed as weighted average about a local

neighborhood.

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Normalized convolution for Image Interpolation

� In general Let x1; : : : ; xn be set of values and w1; : : : ; wn be the

associated weights

� The convolution is given by

xavg =

Piw(i)f(x� i)P

iw(i)(2)

� The convolution can be made more effective by a normalized

operation which takes into account the missing samples.One of

applications of normalized convolution which is a method of in-

terpolating irregularily sampled data points.The conceptual basis

for the method is the signal/certainty philosophy separating the

values of a signal from the certainty of the measurements.

� Normalized Convolution can, for each neighborhood of the sig-

nal, geometrically be interpreted as a projection into a subspace

which is spanned by some analysis functions

� The idea of normalized convolution is to associate each signal

with a certainty component which expresses the level of confi-

dence in the reliability of each measure of the signal.

� Certainty associated with missing samples is 0 ,while that of

known samples is 1.

� Thus we can express a map c(t) associated with signal f(t) which

has the same dimensions as the signal f(t).The certanity map

associated with a signal is simple the locations at which samples

are to be found.

� In case of the above examples the certainty map is given by

c(t) = [1; 0; 0; 1; 1; 0; 0]

� Having both signal and associated certainty map leads to main

concept behind normalized convolution

� Let us consider the convolution of certainty map by filter h(t)yc(t) = [1=3; 1=3; 1=3; 2=3; 2=3; 1=3; 1=3]

� In normalized convolution we have two distinct weights ,a cer-

tainty c(t) which is associate with the signal and applicability a(t)which is associatedd with neighborhood.

� the applicability function determines neighborhood of convolu-

tion as well as weights associated with neighborhood pixel.

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Normalized convolution for Image Interpolation

� For 1D signals it can be expressed as

xavg =sumkf(x� k) � c(x� k) � a(k)P

k c(x� k)a(k)(3)

xavg =(fc) � x

c � a(4)

� in case of above examples we get

y(t) = [x1; x1; x4; (x4 + x5)=2; (x4 + x5)=2; x5; x1]

� This is an approximation of original signal,where components

x1; x5 have been retained while the other components are inter-

polated values in the neighborhood.

� The division of certainty matrix ensures than the signal values

remain within a valid range and is primary reason for better

performance.

� when signal value increases due to applicability function ,the cer-

tainty values also increases by the same ammount and this will

compensate for the increase in the signal values and provide

a normalized measure.This behavior is also observed when the

signal value is reduced.

� Let us consider a gaussian applicability function.Only parameter

that neeeds to be controlled is the standard deviation � of the

gaussian function and the aperture/neighborhood size of the

gaussian.

� Due to normalization property we get a much better result

than a standard convolution would yeild

� The matlab code for the 2D normalized convolution is given

below.

1 im=double(imread('lena.png'));

2 figure(1);colormap(gray);imagesc(im);

3 cert = double(rand(size(im)) > 0.8); imcert = im.*cert;

4 figure(2);colormap(gray);imagesc(imcert);

5

6 %applicability function 3x3 gaussian with variance of 1

7 x = ones(7,1)*(-3:3)

8 y = x';

9 a = exp(-(x.^2+y.^2)/4);

10 figure(3);mesh(a);

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Normalized convolution for Image Interpolation

(a) orignal (b) sampled (c) filter

(d) gaussian interpola-

tion

(e) normalized convo-

lution

11

12 imlp = conv2(imcert, a, 'same');

13 figure(4);colormap(gray);imagesc(imlp);

14

15 %convolution with certainty map

16 G=conv2(cert,a,'same');

17

18 %normalized convolution

19 c = imlp./G;

20 figure(5);colormap(gray);imagesc(c);

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