Survey of Gaussian-Based Edge-Detection Methods Mitra Basu

SURVEY OF GAUSSIAN-BASED EDGE-DETECTION METHODS

MITRA BASU

Presented by: Ali Agha23Feb2009

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?

Local Gradient

Sobel

Prewitt

Robert

Gaussian-based

Marr-Hildreth

Canny

Schunck

Edge focusing

Lacroix

William-Shah

Goshtasby

Jeong-Kim

Deng-Cahill

Bennamoun

Qian-Huang

Lindeberg

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

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

http://en.wikipedia.org/wiki/Sobel_operator

SMOOTHING FILTER

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

GAUSSIAN DERIVATIVES


LAPLACIAN OF GAUSSIAN

Laplacian of Gaussianoperator


SCALE-SPACE REPRESENTATIONFor 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.

Local Gradient

Sobel

Prewitt

Robert

Gaussian-based

Marr-Hildreth

Canny

Schunck

Edge focusing

Lacroix

William-Shah

Goshtasby

Jeong-Kim

Deng-Cahill

Bennamoun

Qian-Huang

Lindeberg

SIGNIFICANCE OF THE GAUSSIAN FILTER Babaud et 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.

Yuille et 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


Local Gradient

Sobel

Prewitt

Robert

Gaussian-based

Marr-Hildreth

Canny

Schunck

Edge focusing

Lacroix

William-Shah

Goshtasby

Jeong-Kim

Deng-Cahill

Bennamoun

Qian-Huang

Lindeberg

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.

Local Gradient

Sobel

Prewitt

Robert

Gaussian-based

Marr-Hildreth

Canny

Schunck

Edge focusing

Lacroix

William-Shah

Goshtasby

Jeong-Kim

Deng-Cahill

Bennamoun

Qian-Huang

Lindeberg

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


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

thinning(non-maximum suppression)Figures adapted from: http://www.umiacs.umd.edu/~ramani/cmsc426/

thresholding

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.


EFFECT OF (GAUSSIAN KERNEL SIZE)

Canny with Canny with original

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

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

larger

Gaussian filtered signal

first derivative peaks

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 METHODAlgorithm 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.

Local Gradient

Sobel

Prewitt

Robert

Gaussian-based

Marr-Hildreth

Canny

Schunck

Edge focusing

Lacroix

William-Shah

Goshtasby

Jeong-Kim

Deng-Cahill

Bennamoun

Qian-Huang

Lindeberg

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

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C(|R

U|)

R U

NON-LINEAR DIFFUSION RESULTS

Original With Noise

Linear diffusion Nonlinear diffusion

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.

q

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.

THANK YOU FOR YOUR ATTENTION

QUESTIONS??

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Survey of Gaussian-Based Edge-Detection Methods Mitra Basu