## Introduction

The purpose of edge detection in general is to significantly reduce the amount of data in an image, while preserving the structural properties to be used for further image processing. Several algorithms exists, and this worksheet focuses on a particular one developed by John F. Canny (JFC) in 1986. Even though it is quite old, it has become one of the standard edge detection methods and it is still used in research.

The aim of JFC was to develop an algorithm that is optimal with regards to the following criteria:

1. **Detection**: The probability of detecting real edge points should be maximized while the probability of falsely detecting non-edge points should be minimized. This corresponds to maximizing the signal-to-noise ratio.

2. **Localization**: The detected edges should be as close as possible to the real edges.

3. **Number of responses**: One real edge should not result in more than one detected edge (one can argue that this is implicitly included in the first requirement).

With Canny’s mathematical formulation of these criteria, Canny’s Edge Detector is optimal for a certain class of edges (known as step edges). A C# implementation of the algorithm is presented here.

## Background

The readers are advised to do more research on canny edge detection method for detailed theory.

## Using the code

The Canny Edge Detection Algorithm

The algorithm runs in 5 separate steps:

1. Smoothing: Blurring of the image to remove noise.

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<pthe gaussian="" filter="" mask="" is="" generated="" by="" following="" function="" :="" <="" p="">

private void GenerateGaussianKernel(int N, float S ,out int Weight)
{
float Sigma = S ;
float pi;
pi = (float)Math.PI;
int i, j;
int SizeofKernel=N;
float [,] Kernel = new float [N,N];
GaussianKernel = new int [N,N];
float[,] OP = new float[N, N];
float D1,D2;
D1= 1/(2*pi*Sigma*Sigma);
D2= 2*Sigma*Sigma;
float min=1000;
for (i = -SizeofKernel / 2; i <= SizeofKernel / 2; i++)
{
for (j = -SizeofKernel / 2; j <= SizeofKernel / 2; j++)
{
Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] = ((1 / D1) * (float)Math.Exp(-(i * i + j * j) / D2));
if (Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] < min)
min = Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j];
}
}
int mult = (int)(1 / min);
int sum = 0;
if ((min > 0) && (min < 1))
{
for (i = -SizeofKernel / 2; i <= SizeofKernel / 2; i++)
{
for (j = -SizeofKernel / 2; j <= SizeofKernel / 2; j++)
{
Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] = (float)Math.Round(Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] * mult, 0);
GaussianKernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] = (int)Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j];
sum = sum + GaussianKernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j];
}
}
}
else
{
sum = 0;
for (i = -SizeofKernel / 2; i <= SizeofKernel / 2; i++)
{
for (j = -SizeofKernel / 2; j <= SizeofKernel / 2; j++)
{
Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] = (float)Math.Round(Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] , 0);
GaussianKernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j] = (int)Kernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j];
sum = sum + GaussianKernel[SizeofKernel / 2 + i, SizeofKernel / 2 + j];
}
}
}
Weight= sum;
return;
}

Following subroutine removes noise by Gaussian Filtering

private int[,] GaussianFilter(int[,] Data)
{
GenerateGaussianKernel(KernelSize, Sigma,out KernelWeight);
int[,] Output = new int[Width, Height];
int i, j,k,l;
int Limit = KernelSize /2;
float Sum=0;
Output = Data;
for (i = Limit; i <= ((Width - 1) - Limit); i++)
{
for (j = Limit; j <= ((Height - 1) - Limit); j++)
{
Sum = 0;
for (k = -Limit; k <= Limit; k++)
{
for (l = -Limit; l <= Limit; l++)
{
Sum = Sum + ((float)Data[i + k, j + l] * GaussianKernel [Limit + k, Limit + l]);
}
}
Output[i, j] = (int)(Math.Round(Sum/ (float)KernelWeight));
}
}
return Output;
}

2. Finding gradients: The edges should be marked where the gradients of the image haslarge magnitudes.

Sobel
X and Y Masks are used to generate X & Y Gradients of Image; next function
implements differentiation using sobel Filter Mask

private float[,] Differentiate(int[,] Data, int[,] Filter)
{
int i, j,k,l, Fh, Fw;
Fw = Filter.GetLength(0);
Fh = Filter.GetLength(1);
float sum = 0;
float[,] Output = new float[Width, Height];
for (i = Fw / 2; i <= (Width - Fw / 2) - 1; i++)
{
for (j = Fh / 2; j <= (Height - Fh / 2) - 1; j++)
{
sum=0;
for(k=-Fw/2; k<=Fw/2; k++)
{
for(l=-Fh/2; l<=Fh/2; l++)
{
sum=sum + Data[i+k,j+l]*Filter[Fw/2+k,Fh/2+l];
}
}
Output[i,j]=sum;
}
}
return Output;
}

3. Non-maximum suppression: Only local maxima should be marked as edges.

We find gradient direction and using these direction we perform non maxima
suppression (Read “Digital Image Processing- by R Gonzales-Pearson Education)

for (i = 0; i <= (Width - 1); i++)
{
for (j = 0; j <= (Height - 1); j++)
{
NonMax[i, j] = Gradient[i, j];
}
}
int Limit = KernelSize / 2;
int r, c;
float Tangent;
for (i = Limit; i <= (Width - Limit) - 1; i++)
{
for (j = Limit; j <= (Height - Limit) - 1; j++)
{
if (DerivativeX[i, j] == 0)
Tangent = 90F;
else
Tangent = (float)(Math.Atan(DerivativeY[i, j] / DerivativeX[i, j]) * 180 / Math.PI);
if (((-22.5 < Tangent) && (Tangent <= 22.5)) || ((157.5 < Tangent) && (Tangent <= -157.5)))
{
if ((Gradient[i, j] < Gradient[i, j + 1]) || (Gradient[i, j] < Gradient[i, j - 1]))
NonMax[i, j] = 0;
}
if (((-112.5 < Tangent) && (Tangent <= -67.5)) || ((67.5 < Tangent) && (Tangent <= 112.5)))
{
if ((Gradient[i, j] < Gradient[i + 1, j]) || (Gradient[i, j] < Gradient[i - 1, j]))
NonMax[i, j] = 0;
}
if (((-67.5 < Tangent) && (Tangent <= -22.5)) || ((112.5 < Tangent) && (Tangent <= 157.5)))
{
if ((Gradient[i, j] < Gradient[i + 1, j - 1]) || (Gradient[i, j] < Gradient[i - 1, j + 1]))
NonMax[i, j] = 0;
}
if (((-157.5 < Tangent) && (Tangent <= -112.5)) || ((67.5 < Tangent) && (Tangent <= 22.5)))
{
if ((Gradient[i, j] < Gradient[i + 1, j + 1]) || (Gradient[i, j] < Gradient[i - 1, j - 1]))
NonMax[i, j] = 0;
}
}
}

4.Double thresholding: Potential edges are determined by thresholding.

5.Edge tracking by hysteresis: Final edges are determined by suppressing all edges that are not connected to a very certain (strong) edge.

private void HysterisisThresholding(int[,] Edges)
{
int i, j;
int Limit= KernelSize/2;
for (i = Limit; i <= (Width - 1) - Limit; i++)
for (j = Limit; j <= (Height - 1) - Limit; j++)
{
if (Edges[i, j] == 1)
{
EdgeMap[i, j] = 1;
}
}
for (i = Limit; i <= (Width - 1) - Limit; i++)
{
for (j = Limit; j <= (Height - 1) - Limit; j++)
{
if (Edges[i, j] == 1)
{
EdgeMap[i, j] = 1;
Travers(i, j);
VisitedMap[i, j] = 1;
}
}
}
return;
}
private void Travers(int X, int Y)
{
if (VisitedMap[X, Y] == 1)
{
return;
}
if (EdgePoints[X + 1, Y] == 2)
{
EdgeMap[X + 1, Y] = 1;
VisitedMap[X + 1, Y] = 1;
Travers(X + 1, Y);
return;
}
if (EdgePoints[X + 1, Y - 1] == 2)
{
EdgeMap[X + 1, Y - 1] = 1;
VisitedMap[X + 1, Y - 1] = 1;
Travers(X + 1, Y - 1);
return;
}
if (EdgePoints[X, Y - 1] == 2)
{
EdgeMap[X , Y - 1] = 1;
VisitedMap[X , Y - 1] = 1;
Travers(X , Y - 1);
return;
}
if (EdgePoints[X - 1, Y - 1] == 2)
{
EdgeMap[X - 1, Y - 1] = 1;
VisitedMap[X - 1, Y - 1] = 1;
Travers(X - 1, Y - 1);
return;
}
if (EdgePoints[X - 1, Y] == 2)
{
EdgeMap[X - 1, Y ] = 1;
VisitedMap[X - 1, Y ] = 1;
Travers(X - 1, Y );
return;
}
if (EdgePoints[X - 1, Y + 1] == 2)
{
EdgeMap[X - 1, Y + 1] = 1;
VisitedMap[X - 1, Y + 1] = 1;
Travers(X - 1, Y + 1);
return;
}
if (EdgePoints[X, Y + 1] == 2)
{
EdgeMap[X , Y + 1] = 1;
VisitedMap[X, Y + 1] = 1;
Travers(X , Y + 1);
return;
}
if (EdgePoints[X + 1, Y + 1] == 2)
{
EdgeMap[X + 1, Y + 1] = 1;
VisitedMap[X + 1, Y + 1] = 1;
Travers(X + 1, Y + 1);
return;
}
return;
}
}

This is performed by a recursive function which performs double thresholding by two
thresholds High Threshold(TH) and Low Threshold (TL) and 8-connectivity
analysis

## Points of Interest

Please refer to "Digital Image Processing" by Gonzalez & woods, Pearson Education