Canny’s Edge Detector

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Edge Detector
Liangliang Cao
Guest Lecture

• John Canny’s paper: A Computational
Approach to Edge Detection, IEEE Trans
PAMI, 1986
• Peter Kovesi’s Matlab functions


Part I: Mathematical Model

• Canny's aim was to discover the optimal edge
detection algorithm:
– good detection – the algorithm should mark as
many real edges in the image as possible.
– good localization – edges marked should be as
close as possible to the edge in the real image.
– minimal response – a given edge in the image
should only be marked once, and where possible,
image noise should not create false edges.

Detection and Localization criterions
 Impulse response of filter : f(x)
 Edge : G(x)
Assuming edge is centered at x=0, filter’s
finite response bounded by [-W,W]
The response of the filter to the edge at its
center is given by a convolution

Criterion 1: Signal/Noise Ratio
The root mean squared noise will be measured by

Now the first criterion , the output signal-to-noise ratio
(SNR) is given by

Criterion 2: Localization
• For localization we need some measure which
increases as localization increases.

Criterion 3: Multiple Response
• The mean distance between adjacent maxima
in the output is twice the distance between
adjacent zero-crossings in the derivative of the
output operator.
k: the number of
noise maxima
that could lead to
a false response.

General Optimization
It is impossible to directly maximize the SNR,
localization under the multiple response
We use penalty function: non-zero values when
one of the constraints is violated
SNR( f ) * Localizati on( f )   i Pi ( f )
where Pi is a function which has a positive value
only when a constraint is violated.

Optimization for special cases
Three kinds of edges:
• Ridge edges
• Roof edges (rare)
• Step edges (important)
Can be get some analytical solution
for step edges?

Step Edges
• Definition

Optimization for Step Edges
The problem can be solved by variational method

where filter 6 can be approximate by the first derivative
of Gaussian!

• The type of linear operator that provides the
best compromise between noise immunity and
localization is First Derivative of Gaussian
• This operator corresponds to smoothing an
image with Gaussian function and then
computing the gradient

Part II: Implementation

Overview of Procedure
1) Smooth image with a Gaussian
2) Compute the Gradi...
Edge Detector
Liangliang Cao
Guest Lecture
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