Report #4( edge Detection )


Brief Overview

  In working with Edge Detection I have worked with the Sobel Operator. The Sobel Operator works well on local edges. It looks at how much change in intensity exists among local edges present within these images. In particular, the Sobel Operator looks how the image?s intensities change from dark to light and in what direction.

                            These ideas can be realized by using the gradient idea. One can obtain the amplitude with this formula,

sqrt( dy2 + dx2 ) and the orientation from tan-1( dy/dx ). The direction in this case is normal to the edge so some calculations of +/-90 degrees must be done to work with the angle we want.
The dx and dy must also be represented in some fashion. This is where the Sobel Operator comes into play.


        |-1    0    1|

dx= |-2    0    2|

        |-1    0    1|  /  8


        | 1     2     1|

dy= | 0     0     0|

        |-1 -2 -1|  /  8


These matrices convolve locally on each pixel by multiplying the above weights on its immediate neighbors. After summing up the result we then divide by eight thus getting the change in x direction and the change in the y direction.

Now with these tools in place I would like to show my results after varying the amplitudes and directions on the lax image. Also I would like to apply edge detection to these images after becoming influenced with the Gaussian Smoothing filter and the Low Pass filter.


The Initial Images
Original Lax Gradient Intensity Equalized Gradient




These are images of LAX and the gradient of the LAX. I equalized the gradient since it is very dark. This may be due to the lack of edges present in the image. After equalizing the image the edges became more noticeable.


Thresholding Amplitude

As the threshold level increases the edges with the most definition, like those among the buildings, remain. The edges with less intensity diminish proportionaly. I shall use the amplitude threshold of ??? to work on the directional aspect of this assignment since it seems to represent the edges the best.



Thresholding Amplitude & Direction
Amplitude\Direction 100 +/-10 degrees 10 +/-10 degrees 45 +/-10 degrees

With an angle of 100 degrees we get those edges very clearly in that direction. The same holds true for those edges with a 10 degree slant. Taking the directional threshold of 45 degrees really does not show any edges as well as it should not.


Influence of Gaussian Smoothing
Sigma\Misc Before edging After edging Applying amplitude=150 
Sigma = 1.0  
Sigma = 2.7
        The higher the sigma the more blurred the edges become. Also there is less presence of random edges.  The added implementation of thresholding the amplitude became less warrented as the less intense edges disappeared as the sigma rose.

Influence of Low Pass Filtering
Range to Impulse Response\ Before edging After edging Threshold = 10
Close to Impulse Response  
            Likewise with the Low Pass Filter, the threshold of the amplitude became less of a necessity as the frequency came closer to             the  impulse response.  However, the edges became less defined.