Edge detection

Edge detection

Edge Detection in Images

• Finding the contour of objects in a scene

Edge Detection in Images

• What is an object?

It is one of the goals of computer vision to identify objects in scenes.

Edge Detection in Images

• Edges have different sources.

What is an Edge

• Lets define an edge to be a discontinuity in image intensity function.

• Edge Models– Step Edge– Ramp Edge– Roof Edge– Spike Edge

Detecting Discontinuities

• Discontinuities in signal can be detected by computing the derivative of the signal.

Differentiation and convolution

• Recall

• Now this is linear and shift invariant, so must be the result of a convolution.

• We could approximate this as

(which is obviously a convolution with Kernel

; it’s not a very good way to do things, as we shall see)

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Definition

Convolution Kernels

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Noise

• Simplest noise model– independent stationary

additive Gaussian noise

– the noise value at each pixel is given by an independent draw from the same normal probability distribution

• Issues– this model allows noise

values that could be greater than maximum camera output or less than zero

– for small standard deviations, this isn’t too much of a problem - it’s a fairly good model

– independence may not be justified (e.g. damage to lens)

– may not be stationary (e.g. thermal gradients in the ccd)

Finite differences and noise

• Finite difference filters respond strongly to noise– obvious reason: image

noise results in pixels that look very different from their neighbours

• Generally, the larger the noise the stronger the response

• What is to be done?– intuitively, most pixels in

images look quite a lot like their neighbours

– this is true even at an edge; along the edge they’re similar, across the edge they’re not

– suggests that smoothing the image should help, by forcing pixels different to their neighbours (=noise pixels?) to look more like neighbours

Finite differences responding to noise

Increasing noise ->(this is zero mean additive gaussian noise)

Smoothing reduces noise

• Generally expect pixels to “be like” their neighbours– surfaces turn slowly– relatively few

reflectance changes

• Generally expect noise processes to be independent from pixel to pixel

• Implies that smoothing suppresses noise, for appropriate noise models

• Scale– the parameter in the

symmetric Gaussian– as this parameter goes

up, more pixels are involved in the average

– and the image gets more blurred

– and noise is more effectively suppressed

The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.

Classical Operators

Prewitt’s Operator

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Classical Operators

Sobel’s Operator

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Gaussian Filter

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• Sobel Edge Detector

Detecting Edges in Image

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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

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.

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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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

Non-Maximum Suppression

• Suppress the pixels in ‘Gradient Magnitude Image’ which are not local maximum

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Hysteresis Thresholding

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

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