Fast 2D SeparableSymmetric/Anti-SymmmetricConvolution

Pi19404

February 17, 2014

Contents

Contents

2D Seperable Symmetric/Anti-Symmmetric Convolution 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.1.1 Separable Symmetric/Anti-Symmetric Convolution 30.1.2 Vertical Convolution . . . . . . . . . . . . . . . . . . . . . . . 40.1.3 Horizontal Convolution . . . . . . . . . . . . . . . . . . . . . 60.1.4 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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2D Seperable Symmetric/Anti-Symmmetric Convolution

2D Seperable

Symmetric/Anti-Symmmetric

Convolution

0.1 Introduction

In the article we will look at algorithm for Fast 2D Convolution.

0.1.1 Separable Symmetric/Anti-Symmetric Convolution

� This article presents a convolution algorithm involving a sepa-

rable symmetric/anti symmetric kernel.

� Such kernels are common in image processing like blurring,edge

detection etc

� This optimization helps in speeding up many routines .

� Another requirement being addressed is we may have multiple

kernels that need to be applied to the same image in application

such as computing the basis representation of rectangular path

of the image.

� since we are traversing the image,we can perform computa-

tion of all these kernels simultaneously instead of independent

traversals.

� It is to be noted that this optimization is specific to symmet-

ric/antisymmetric separable filters.

� We will be assuming that all the kernel are of the same size as

well

� Since kernels are separable we will perform vertical convolution

followed by horizontal convolution.

� One of the considerations is to minimize the row accesses .

� Another consideration is to perform a single pass over the

entire source image

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2D Seperable Symmetric/Anti-Symmmetric Convolution

� Let n � 2 + 1 be the kernel size.

� The convolution filter is initialized by accepting all the row and

column filter coefficients as well as the information if the fil-

ter is symmetric or not

� A class called SeperableSConvolution implemented this algorithm

class SeperableSConvolution

{

public:

SeperableSConvolution();

vector<Mat> rowk;//vector of row filter

vector<Mat> colk;//vector of column filters

vector<bool> symmetric; //info if filter is symmetric or not

int N;//number of channels in the output image

//function to set the filter parameters

void setKernel(Mat row,Mat col,bool symm);

//function to clear the filter parameters

void clearKernel();

//apply a set of separable symmetric/anti-symmetric filters on

//the source image and return a multi channel destination image

////function to perform separable filtering

void apply(Mat &src,Mat &dst);

};

0.1.2 Vertical Convolution

� First vertical convolution is considered

� If kernel size is 2N +1 ,we need to access same number of rows

� This if we are applying the filter at row j,we need to access

elements from rows from j � k to j + k where k 2 0::N

dst(x; y) =k=NX

i;k=�N

g(k +N)I(x; y + k) (1)

� The above needs to be processed for each (x; y)

� Since we need to reduce the number of row access,access rows

from j �N; j +N are performed for each j,compute the convo-

lution for each x.

� Since with change in x no new rows are being accessed,hence

we compute convolution for each element of the present row.

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2D Seperable Symmetric/Anti-Symmmetric Convolution

� Each of the rows need to be multiplied with suitable filter co-

efficients.

� since the filter is symmetric or anti symmetric the filter coef-

ficients gk is multiplied with elements of rows j � k and j + k

� The resultant sum of accumulated,this is performed for all the

rows.

for(int y=0;y<s.height;y++)

{

//pointer to the source and destination

float *srow0 = (float*)(src.data + src.step*y),*srow1=0;

float *drow = (float*)(dst.data + dst.step*y);

//performing vertical convolution

//using the row kernel

//computing g[0]*f(x,y)

for( x = 0; x < s.width; x++ )

{

for(int l=0;l<ch;l++)

{

row[x*ch+l] = srow0[x]*rowk[l].at<float>(n);

}

}

//computing g[-n/2]*f(x,y-n/2)

//performing vertical convolution accessing n rows about current row

//accessing a 2 rows at a time,performing computation

for(int k=1;k<=n;k++)

{

//accessing the vertical abouts about the current row

srow0 = (float*)(src.data + src.step*std::max(y-k,0));

srow1 = (float*)(src.data + src.step*std::min(y+k,s.height-1));

for(int x=0;x<s.width;x++)

{

//applying different vertical kernels

//accumulating the sum

for(int l=0;l<ch;l++)

{

float p=srow0[x]+(1*(symmetric[l]?1:-1))*srow1[x];

row[x*ch+l]+=rowk[l].at<float>(k)*(p);

}

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2D Seperable Symmetric/Anti-Symmmetric Convolution

}

}

0.1.3 Horizontal Convolution

� Once vertical convolution is done we proceed to perform hori-

zontal convolution

� Since in horizontal convolution there is only a single row access

,it is relatively simple process.

� The output image is a multi channel image,containing number of

channels as desired number of input kernels being applied to the

source image.

for(x=0;x<s.width;x++)

{

//apply horizontal kernels

//g[0]*f(x,y)

for(int l=0;l<ch;l++)

{

res[l]=row[x*ch+l]*colk[l].at<float>(n);

}

//accumulating the sum for g[-N/2+k]f(x-n/2+k)

for(int k=1;k<=n;k++)

{

for(int l=0;l<ch;l++)

{

float p=(row[(x+k)]+(1*symmetric[l]?1:-1)

*row[(x-k)])*colk[l].at<float>(k);

res[l]=res[l]+p;

}

}

//storing the result of convolution

//output image contains number of channels as different input filters

for(int l=0;l<ch;l++)

{

drow[x*ch+l]=res[l];

}

}

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2D Seperable Symmetric/Anti-Symmmetric Convolution

0.1.4 Code

� The class SeperableSConvolution defines a class for performing

separable symmetric/ant-symmetric convolution/ Code is avail-

able in repository https://github.com/pi19404/OpenVision/

at Improper/convolution.hpp and ImgProc/convolution.cpp files.

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