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CAP5415 Computer VisionSpring 2003

Khurram Hassan-Shafique

� Sobel Edge Detector

Detecting Edges in Image

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Sobel Edge Detector

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Sobel Edge Detector

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Marr and Hildreth Edge Operator

� Smooth by Gaussian

� Use Laplacian to find derivatives

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Marr and Hildreth Edge Operator

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Marr and Hildreth Edge Operator2

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Marr and Hildreth Edge Operator

Zero CrossingsDetection

I ImageσG2* ∆

IG *2σ∆ Edge Image

IG *2σ∆ Zero Crossings

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1=σ

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Quality of an Edge Detector

� Robustness to Noise

� Localization

� Too Many/Too less Responses

Poor robustness to noise Poor localization Too many responses

True Edge

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Canny Edge Detector� Criterion 1: Good Detection: The optimal

detector must minimize the probability of false positives as well as false negatives.

� Criterion 2: Good Localization: The edges detected must be as close as possible to the true edges.

� Single Response Constraint: The detector must return one point only for each edge point.

Canny Edge Detector� Difficult to find closed-form solutions.

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Canny Edge Detector� Convolution with derivative of Gaussian

� Non-maximum Suppression

� Hysteresis Thresholding

Canny Edge Detector� Smooth by Gaussian

� Compute x and y derivatives

� Compute gradient magnitude and orientation

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Canny Edge Operator

( ) IGIGS ** σσ ∆=∆=∆T

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Canny Edge Detector

xS

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Canny Edge Detector

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25=≥∆ ThresholdS

We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?

Non-Maximum Suppression

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Non-Maximum Suppression� Suppress the pixels in ‘Gradient Magnitude

Image’ which are not local maximum

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Non-Maximum Suppression

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Hysteresis Thresholding

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Hysteresis Thresholding� If the gradient at a pixel is above ‘High’ , declare

it an ‘edge pixel’

� If the gradient at a pixel is below ‘Low’ , declare it a ‘non-edge-pixel’

� If the gradient at a pixel is between ‘Low’ and ‘High’ then declare it an ‘edge pixel’ if and only if it is connected to an ‘edge pixel’ directly or via pixels between ‘Low’ and ‘ High’

Hysteresis Thresholding

M 25=≥ ThresholdM

15

35

==

Low

High

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Finding Connected Components� Scan the binary image left to right top to bottom

� If there is an unlabeled pixel p with a value of ‘1’� assign a new label to it

� Recursively check the neighbors of pixel p and assign the same label if they are unlabeled with a value of ‘1’ .

� Stop when all the pixels with value ‘1’ have been labeled.

Connectedness

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Suggested Reading� Chapter 8, David A. Forsyth and Jean Ponce,

"Computer Vision: A Modern Approach“

� Chapter 4, Emanuele Trucco, Alessandro Verri, "Introductory Techniques for 3-D Computer Vision"

� Chapter 2, Mubarak Shah, “Fundamentals of Computer Vision”