A Computational Approach to Edge Detection J' Canny

1
A Computational Approach to Edge Detection J.
Canny

• Introduction
• This paper deals with mathematically formulating
  the qualities of an optimal edge detector and
  then deriving an algorithm for finding an optimal
  ( as well as an approximately optimal) edge
  detector for arbitrary edge profiles. One
  dimensional edge profiles are first analyzed and
  the results are then generalized to two ( or
  more) dimensions.

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Overview

• Description of the three criteria for judging an
  edge operators performance and their
  mathematical formulation
• Derivation of an optimal one dimensional edge
  detector for roof, ridge and step edge profiles.
• Approximation of an optimal step edge detector.
• Noise Estimation and Thresholding.
• Derivation and analysis of two dimensional edge
  detectors.
• The need for multiple widths ( of operators ).
• The need for directional operators.

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Edge Detector Performance Indices

• Good edge Detection
• Low false alarm.
• Low missed edges.
• Good edge Localization.
• The edge detector should not give multiple
  responses to a single edge.

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Mathematical Formulation

• Edge Detection
• The probability of false alarm and missed
  detection are minimized when the signal to noise
  ratio of the filter is maximized.
• SNR
• Where,
• G(x) The edge
• f(x) The impulse response of the filter
  bounded by -W,W
• Mean squared noise amplitude per unit

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Mathematical Formulation

• Edge Localization
• We want some measure that increases as
  localization improves. The reciprocal of the
  root-mean-squared distance of the marked edge
  from the center of the true edge is one such
  measure.
• Let be the response of the filter to
  noise only and be its response to the
  edge and let there is a local maximum in the
  total response at point . We have

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Mathematical Formulation

• By
  Taylor expansion about the origin.
• ( By assumption).
• Ignore higher order terms. Hence, we have

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Mathematical Formulation

• Localization
• Multiplying the two expressions, we obtain a
  single performance index which monotonically
  increases as the performance of the filter
  becomes better.
• Elimination of Multiple responses
• The idea is to limit the no. of peaks in the
  response so that there is a low probability of
  declaring more than one edge.

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Mathematical Formulation

• The average distance between zero crossings of
  the response of the function g to Gaussian noise
  is
• In our case we are looking for the mean
  zero-crossing space for the function .
  Hence the mean distance between zero-crossings of
  will be

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Mathematical formulation

• The distance between adjacent maxima in the noise
  response of f, denoted , will be twice
  . This distance is set some fraction k of W.
• Hence, the expected no. of maxima in the region
  is
• The multiple response criterion is invariant with
  respect to spatial scaling of f.

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Optimal detectors by Numerical Optimization

• Solution of J(f) subject to the multiple response
  constraint is normally very difficult to obtain.
  Numerical optimization can be done on the sampled
  operator impulse response.
• Penalty method is used to reduce the constrained
  optimization to one or several unconstrained
  optimizations. The idea is to find f which
  maximizes
• Where is the penalty function with
  weight when the ith constraint is violated.
  It has a zero value is the constraint is not
  violated.

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Step Edge Detector

• For step edge , let input is
• If
• Clearly, there is a trade off between
  localization and detection.

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Step Edge Detector

• J(f)SNRLocalization function for step edge can
  be analytically solved, but it is not discussed
  in this presentation due to spatial and temporal
  constraints!!
• Approximation of step edge detector
• The filter impulse response f(x) of a step edge
  detector can be approximated by the first
  derivative of Gaussian filter
• The performance of the approximate filter is
  worse than the optimal operator but it has the
  advantage that the first derivative of Gaussian
  can be computed with much less effort in two
  dimension.

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Noise Estimation and Thresholding

• To estimate noise from an operator output, we
  need to be able to separate its response to noise
  from the response due to step edges. Wiener
  filtering is a method for estimating one
  component of a two component signal based on the
  knowledge of the autocorrelation function of the
  two components and that of the entire signal.
• In this paper global histogram estimate is used.
  This method exploits the fact that the amplitude
  distribution of the filter response tends to be
  different for edges and noise.
• Even after noise estimation, edge detector will
  be susceptible to streaking. Streaking the
  breaking up of an edge contour caused by operator
  o/p fluctuations above and below the threshold
  along the line of contour. Histeresis (or double
  thresholding) is used to circumvent the problem.

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Two Dimensional Edge Detection

• To detect edges of particular orientation, a 2D
  mask for the orientation is created by convolving
  a linear edge detection function ( aligned normal
  to the edge direction) with a project function
  parallel to the edge direction. If the projection
  function is gaussian with the same variance as
  the Gaussian derivative used in edge detector,
  the computation becomes very efficient.

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Two Dimensional Edge Detection

• An edge is a point of local maximum given by,
• Edge Strength at such a point is
• Non-maxima suppression can be applied to by
  setting all edge candidates which are not local
  maxima to zero.

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The Need for Multiple Width

• Large operators are good for detection and small
  operators are good localization.
• Feature Synthesis All edges from smallest
  operators are marked. From these edges large
  operator outputs that would have been produced
  had there been only edges, are synthesized. The
  actual operator output is compared with the
  synthetic output and only those additional edges
  are chosen which have significantly greater
  response than predicted. This helps to prevent
  multiple detection of the same edge.

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The Need For Directional Operator

• Several directional operators, each corresponding
  to a different orientation, can be used together
  on the image. The orientation of the edge
  corresponds to the operator with the strongest
  response.
• Detection and localization of operator improve as
  the length of the projection function increases.
  An efficient way of forming long directional
  masks is to sample output of non-elongated masks
  with the same direction.