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EDGE DETECTION – FUNDAMENTALS


The derivatives of a digital function are defined in terms of differences.
The above statement made me to analyze about derivatives and how it is used for edge detection.  The first time when I came across the edge detection operation [Example: edge(Image,’sobel’)], I wondered how it worked. 
Consider a single dimensional array,
A =
5
4
3
2
2
2
2
8
8
8
6
6
5
4
0

MATLAB CODE:

x=1:15;
y=[5 4 3 2 2 2 2 8 8 8 6 6 5 4 0];

figure,
plot(x,y,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','y');
title('Input Array');




First-order Derivative for one dimensional function f(x):




MATLAB  CODE:

x1=1:14;
y1=diff(y,1);
figure,
plot(x1,y1,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','r');


-1
-1
-1
0
0
0
6
0
0
-2
0
-1
-1
-4
































NOTE: The contiguous values are zero. Since the values are nonzero for non-contiguous values, the result will be thick edges.
The first-order derivative produces thicker edges.

Second-order Derivative for one dimensional function f(x):

















MATLAB CODE:

x2=1:13;
y2=diff(y,2);
figure,
plot(x2,y2,'-o','LineWidth',3,'MarkerEdgeColor','k','Color','g');


0
0
1
0
0
6
-6
0
-2
2
-1
0
-3































The Second-order derivative gives finer result compared to first-order derivative. It gives fine detailed thin lines and isolated points.  Let’s see how the second-order derivative used for Image sharpening (Laplacian) in my upcoming post.
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2 comments:

Unknown said... Reply to comment

Very useful
Thank you

miss finaz radcliffe said... Reply to comment

thank you for your explanation :)

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