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Survey of Gaussian-Based Edge-Detection Methods Mitra Basu

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Motivation

- What is the edge detection? And Why we need it?
- Edge detection is the process which detects the presence and locations of intensity transitions.
- drastically reduces the amount of data
- important information about the shapes of objects
- easy to integrate into a large number of object recognition algorithms

Problem of edge detection

- The addition of noise to an image can cause the position of the detected edge to be shifted from its true location.
- Any linear filtering or smoothing performed on these edges to suppress noise will also blur the significant transitions.
- Solution?

Earlier methods:

- Some of the earlier methods, such as the Sobel and Prewitt detectors, used local gradient operators which only detected edges having certain orientations and performed poorly when the edges were blurred and noisy.
- Sobel operator:

Sobel Operator

Figures adapted from: http://en.wikipedia.org/wiki/Sobel_operator

Problems of methods based on local gradient

- Effects of noise

Figures adapted from: http://en.wikipedia.org/wiki/Sobel_operator

Smoothing filter

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Gaussian derivatives

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Laplacian of Gaussian

Laplacian of Gaussian

operator

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Scale-space representation

For a given image f(x,y), its linear (Gaussian) scale-space representation is a family of derived signals L(x,y;t) defined by the convolution of f(x,y) with the Gaussian kernel

Such that

Figures adapted from:

http://en.wikipedia.org/wiki/Scale-space

Multiscale edge detection

- Procedure
- Applying smoothing operators of different sizes
- Extracting the edges at each scale
- Combining the recovered edge information to create a single edge map.
- Problems to be solved
- how many filters should be used
- how to determine the scales of the filters
- how to combine the responses from each filter so as to create a single edge map.

SIGNIFICANCE OF THE GAUSSIAN FILTER

- Babaudet al. proved that when one-dimensional (1-D) signals are smoothed with a Gaussian filter, the scale space representation of their second derivatives shows that new zero-crossings are never created.
- Yuilleet al. extended this work to 2-D signals (proved that with Laplacian)
- The best tradeoff between the conflicting goals of the localization in spatial and frequency domains
- The only rotationally symmetric filter that is separable in Cartesian coordinates.

2D edge detection filters

Laplacian of Gaussian

Gaussian

derivative of Gaussian

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Marr-Hildreth method

- Consider the Gaussian operator in two dimensions given by
- Applied Gaussian filters of different scales to an image.
- They find the zero-crossings of their second derivatives using the LOG function
- The Marr-Hildreth operator formally introduced Gaussian filter into the edge-detection process. This is a turning point in the low-level image processing research area.

Marr-Hildreth method’s problems

- Zero-crossings are only reliable in locating edges if they are well separated and the SNR in the image is high.
- The location shifts from the true edge location for the finite-width case.
- Detection of false edges. Zero-crossings correspond to local maxima and minima.
- Missing edges

Marr-Hildreth method’s problems

- it is very difficult to combine LOG zero-crossings from different scales, because:
- a physically significant edge does not match a zero-crossing for more than a few and very limited number of scales
- zero-crossings in larger scales move very far away from the true edge position due to poor localization of the LOG operator
- there are too many zero-crossings in the small scales of a LOG filtered image, most of which is due to noise.

Canny edge detector - Formulation

Figures from:

Canny edge detector - Formulation

- Canny developed an operator, based on optimizing three criteria
- good detection
- good localization
- only one response to a single edge.

Canny’s method – Optimal filter

- By variational methods, Canny showed that the optimal filter given these assumptions is a sum of four exponential terms. He also showed that this filter can be well approximated by first-order derivatives of Gaussians. For example for a 1-D step edge:

Figures from:

Canny’s method – Optimal filter

- an example of a 5x5 Gaussian filter

Canny’s method – image gradient

- The edge detection operator (Roberts, Prewitt, Sobel for example) returns a value for the first derivative in the horizontal direction (Gy) and the vertical direction (Gx).
- Magnitude and direction:

Canny’s method – image gradient

original image (Lena)

norm of the gradient

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Canny’s method – Non-maxima suppression

- Derivative directions are rounded to four angles
- At each point, compute its edge gradient, compare with the gradients of its

neighbors along the gradient

direction. If smaller,

turn 0; if largest, keep it.

http://www.pages.drexel.edu/~weg22/can_tut.html

Figures from:

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Canny’s method – Non-maxima suppression

thresholding

thinning

(non-maximum suppression)

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Canny’s method – hysteresis thresholding

- Therefore we begin by applying a high threshold. This marks out the edges we can be fairly sure are genuine. Starting from these, using the directional information derived earlier, edges can be traced through the image. While tracing an edge, we apply the lower threshold, allowing us to trace faint sections of edges as long as we find a starting point.

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Effect of (Gaussian kernel size)

original

Canny with

Canny with

- The choice of depends on desired behavior
- large detects large scale edges
- small detects fine features

adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Problems with Canny edge detector

- The algorithm marks a point as an edge if its amplitude is larger than that of its neighbors without checking that the differences between this point and its neighbors are higher than what is expected for random noise.
- The technique causes the algorithm to be slightly more sensitive to weak edges, but it also makes it more susceptible to spurious and unstable boundaries wherever there is an insignificant change in intensity (e.g., on smoothly shaded objects and on blurred boundaries).

Schunck method

- The initial steps of Schunck’s algorithm are based on Canny’s method.
- The gradient magnitudes over the chosen range of scales are multiplied to produce a composite magnitude image.
- Ridges that appear at the smallest scale and correspond to major edges will be reinforced by the ridges at larger scales. Those that do not, will be attenuated by the absence of ridges at larger scales.

Schunck method - problems

- he did not discuss how to determine the number of filters to use.
- He chooses the width of the smallest Gaussian filter to be around 7. Choosing such a large size for the smallest filter, Schunck’s technique loses a lot of important details which may exist at smaller scales.

Witkin’s representation

- Idea:
- examine the smoothed signal at various scales
- The zero-crossings of the second derivative are marked.
- This scale-space representation of a signal contains the location of a zero-crossing at all scales starting from the smallest scale to the scale at which it disappears.

Witkin’s representation

larger

Gaussian filtered signal

Properties of scale space (w/ Gaussian smoothing)

- edge position may shift with increasing scale ()
- two edges may merge with increasing scale
- an edge may not split into two with increasing scale

adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

Bergholm’s method

- Bergholm proposed an algorithm which combines edge information moving from a coarse-to-fine scale. His method is called edge focusing.
- The idea behind edge focusing is to reverse the effect of the blurring caused by the Gaussian operator. The most obvious way of undoing the blurring process is to start with edges detected at the coarse scale and gradually track or focus these edges back to their original locations in the fine scale.

Bergholm’s method - problems

- how to determine the starting and ending scales of the Gaussian filter? This is a parameter which is critical in determining how well the algorithm performs
- Since edge focusing is obtained at a finer resolution, some edges (i.e., the blurred ones, such as shadows) present a juggling effect at small scales. This is due to the splitting of a coarse edge into several finer edges, and tends to give rise to broken, discontinuous edges.

Lacroix’s method

- Idea: avoids the problem of splitting edges by tracking edges from a fine-to-coarse resolution
- Start with Canny method
- then considers three scales
- The smallest scale is the detection scale
- The largest scale is the coarsest scale, at which the edgel still remains
- An edgel is validated and then tracked if: 1) it is the local maximum of a Gaussian gradient and 2) the two regions it separates are significantly different from one another.

Lacroix’s method - problems

- problem of localization error as it is the coarsest resolution that is used to determine the location of the edges.
- No explanation as to how to decide which scales are to be used and under what conditions.

Williams-Shah method

- Idea: Starts with Canny’s method and after thinning gradient maxima points, they linked based on four measures:
- 1) noisiness; 2) curvature; 3) contour length; and 4) gradient magnitude.
- The set of points having the highest average weight is chosen.
- Then, repeatedly, the next smaller scale is used, and the regions around the end points of the contours are examined to determine if there are possible edge points at the smaller scale having similar directions to the end points of the contours.

Williams-Shah method - problems

- They did not suggest the best way to choose the value of scales and under what conditions.

Goshtasby’s method

- Idea: modified scale-space representation
- Instead of the zero-crossings, the signs of pixels after filtering with LOG operator are recorded.

Figures from:

Goshtasby’s method

Algorithm B(1,2)

- Assuming the sign images obtained at scales 1 and 2 are I1 and I2, respectively, then:
- If a region in I1 falls on more than two regions of the same sign in I2 then make a convolution in scale to determine the sign image at scale , where =(1+ 2)/2. Then make a recursive calls to B. Otherwise, exit B.

Figures from:

Goshtasby’s method - problems

- The major problem with Goshtasby’s edge focusing algorithm is the need for a considerable amount memory to store the three-dimensional (3-D) edge images.

Deng-Cahill method

- Idea: adapting the variance of the Gaussian filter to the noise characteristics and the local variance of the image data
- They proposed that the variance of a 1-D Gaussian filter at location is

Deng-Cahill method - problem

- The major drawback of this algorithm is that it assumes the noise is Gaussian with known variance. In practical situations, however, the noise variance has to be estimated.
- The algorithm is also very computationally intensive.

Bennamoun’s method

- present a hybrid detector (GoG+LoG) that divides the tasks of edge localization and noise suppression between two subdetectors.

Figures from:

Bennamoun’s method – Scale & threshold

- The work is extended to automatically determine the optimal scale and threshold by:
- 1) finding the probability of detecting an edge for a signal with noise P(A)
- 2) finding the probability of detecting an edge in noise only P(B)
- Maximizing below cost function

Bennamoun’s method - problem

- As the authors’ results show, their technique is still susceptible to false edge-detection, especially in the presence of high noise levels.

Qian-Huang method

- A new edge detection scheme that detects two-dimensional (2-D) edges by a curve-segment-based detection functional guided by the zero-crossing contours of the Laplacian-of-Gaussian (LOG) to approach the true edge locations.
- Algorithm:
- convolving an image with the LOG operator and finding the zero-crossing contours.
- contours are then segmented at points with large curvatures.
- 2-D edge detection functional.
- Adaptive thresholding based on the global noise estimation
- Edge segments are combined from different scales using a fine-to-coarse strategy.

Qian-Huang method – problems

- They used seven scales between 2.5 and 6.7; However, this may not be the ideal range for computational methods.
- In addition, the range may also change depending on the type of image and the amount of noise it contains.

Lindeberg’s method

- Idea: suggested a framework for automatic scale selection based on maximization of two specific measures of edge strength
- First one is the simplest measure of edge strength.
- Second one originates from the sign condition in the edge definition.
- The parameter Gamma makes the scale selection method dependent on the diffuseness of the edge, i.e., fine scale is selected for sharp edges and coarse scale is selected to deal with diffused (blurred) edges. However, the authors choose it as 1 in all their experiments.

Lindeberg’s method - problems

- It still requires the user to specify a scale range.
- A major drawback of this approach is the need to compute high-order derivatives, which are known to contribute toward computational difficulties.
- One does not see any significant advantage in the use of such high-order derivatives from theoretical or experimental results.

Elder-Zucker method

- Idea: A local method for scale selection
- Making the scale a function of the second moment of sensor noise (available information)
- the authors introduce the idea of a minimum reliable scale at which and at larger scales, the possibility of detecting edges due to sensor noise is below a specified tolerance

Elder-Zucker method - problem

- the process of detecting and identifying important edges cannot be avoided.

Perona-Malik method

- Idea: space variant blurring
- Consider:
- This one parameter family of derived images may equivalently be viewed as the solution of the heat conduction, or diffusion, equation
- Anisotropic heat equation (diffusion equation):

Formulas from:

Perona-Malik method

- making the diffusion coefficient in the heat equation a function of space and scale. The goal is to smooth within a region and keep the boundaries sharp.
- Two function used in experiments

Non-linear diffusion results

adapted from: ICASSP-2000 presentation, by G. Gilboa, Y.Y. Zeevi, N. Sochen

Perona-Malik method - problem

- Large number of iteration
- Convergence problems

Fontaine-Basu method

- Idea: use of wavelets to solve the anistropic diffusion equation.
- Compact representations of images with regions of low contrast separated by high-contrast edges
- No new features are introduced in the derived images (i.e., in the scale-space representation of the original image) in passing from fine to coarse scale

Fontaine-Basu method - problem

- The drawback of this approach is that the discretization scheme for the diffusion equation proposed in this paper cannot be directly expressed in the wavelet transform domain. This requires an iterative procedure of going back and forth between the spatial and the wavelet domains of representation and adds to the numerical complexity of the algorithm.

Aurich-weule method

- Idea: modification of the way the solution of the heat equation is obtained. The method uses a nonlinear modification of Gaussian filters
- To preserve edges:

Formulas from:

Aurich-weule method

- 1) How an edge is preserved: Consider a non-edge pixel p.
- Case I: I(p)-I(q) is small for all q in the neighborhood of p.
- Case II: I(p)-I(q) is small for all q in the neighborhood of p except at one pixel .
- 2) How an edge is enhanced: Consider an edge pixel p. After weighted averaging is done, It is obvious that, the new pixel value of p will be more than its previous value.

Aurich-weule method - problems

- Although pixel value is increasing due to filtering, the overall effect may not produce enhancement.
- The slope of the edge is a critical factor here. Enhancement is achieved if the edge is steep.
- The possibility of the appearance of new features in the image has not been explored mathematically or experimentally.

Summery

- The Gaussian filter has several desirable features.
- However, Linear methods presented in this paper suffer from problems associated with Gaussian filtering, namely, edge displacement, vanishing edges, and false edges.
- The introduction of multiscale analysis further complicates the issue by creating two major problems: 1) how to choose the size of the filters and 2) how to combine edge information from different scales.

Summery

- Nonlinear approaches show significant improvement in edge-detection and localization.
- However, problems of computational speed, convergence, and difficulties associated with multiscale analysis remain.
- As it currently stands, use of the Gaussian filter requires making compromises when developing algorithms to give the best overall edge-detection performance.

Conclusion

- For detecting the edges of buildings
- We should note that An edge is not a line...
- We have to choose some of these methods based on our own environment. Fortunately, the scale of our desirable lines are different from the scale of most of edges in the environment.
- We will face with broken lines. Look at this figure:

Conclusion

How can we detect lines ?

I think we can use the line detection methods and empower them with the techniques of the edge detection methods so that we can cope with detecting broken lines.

Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

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