edge detection
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Edge Detection. 27 th Nov ember 2012 /. Edge Detection in Images. Goal: Automatically find the contour of objects in a scene. What For: Edges are significant descriptors, useful for classification, compression…. Edge Detection in Images. What is an object? - PowerPoint PPT PresentationTRANSCRIPT
Computer Vision Group
Edge Detection in Images
Goal: Automatically find the contour of objects in a scene.
What For: Edges are significant descriptors, useful for classification, compression…
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Edge Detection in Images
What is an object?It is one of the goals of computer vision to identify objects in scenes.
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Edges May Have Different Sources
Object Boundaries Occlusions Textures Shading
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What is an Edge
Lets define an edge to be a discontinuity in image intensity function.
Several Models• Step Edge• Ramp Edge• Roof Edge• Spike Edge
They can bethus detected asdiscontinuitiesof image Derivatives
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Differentiation and convolution
Recall
Now this is linear and shift invariant. Therefore, in discrete domain, it will be represented as a convolution
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Differentiation and convolution
Recall
Now this is linear and shift invariant. Therefore, in discrete domain, it will be represented as 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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Finite Difference in 2D
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Discrete Approximation
Horizontal
Convolution Kernels
Vertical
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A 1D Example
Take a line on a grayscale image
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A 1D Example (II)
Filter the discrete image values, convolution against [1 -1]
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Differentiating Filters
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1D Discrete derivatives
1 0 -1*ff x
2D Discrete derivatives (separable)
*ff y 1 0 -1 t
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Classical Operators : Prewitt
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Smooth Differentiate
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Horizontal
Vertical
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Classical Operators: Sobel
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Smooth Differentiate
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Horizontal
Vertical
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Another famous test image - cameraman
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Horizontal Derivatives using Sobel
1 * ty y h
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Vertical Derivatives using Sobel
2 * ty y h
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Gradient Magnitude and edge detectors
2 2| | * *ty y h y h
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Finite differences
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The Gradient Orientation
Like for continuous function, the gradient at each pixel points at the steepest intensity growth direction.
The gradient norm indicates the inclination of the intensity growth.
Matlab…..
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Finite differences responding to noise
Increasing noise ->(this is zero mean additive Gaussian noise)
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Finite differences responding to noise
Increasing noise ->(this is zero mean additive Gaussian noise)
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Finite differences responding to noise
Increasing noise ->(this is zero mean additive Gaussian noise)
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Smoothing reduces noise
Generally expect pixels to “be like” their neighbors• surfaces turn slowly• relatively few reflectance changes
Generally expect noise processes to be independent from pixel to pixel and with zero mean
Implies that smoothing suppresses noise, (i.i.d. noise!)
Gaussian Filtering• 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 somehow suppressed
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Smoothing reduces noise
Generally expect pixels to “be like” their neighbors• surfaces turn slowly• relatively few reflectance changes
Generally expect noise processes to be independent from pixel to pixel and with zero mean
Implies that smoothing suppresses noise, (i.i.d. noise!)
Gaussian Filtering• 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 somehow suppressed
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The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realisations of an image of gaussian noise.
Low - Pass
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Gradient Magnitude and edge detectors
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Gradient Magnitute is not a binary image -> “shows edges” but “does not allow to identify them” yet
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Sobel Edge Detector
Detecting Edges in Image
Image I
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ThresholdEdges
any alternative ?
Discrete Derivatives Gradient Norms Threshold
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Canny Edge Detector Criteria
Good Detection: The optimal detector must minimize the probability of false positives as well as false negatives.
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. similar to good detection but requires an ad-hoc formulation to get rid of multiple responses to a single edge
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Poor robustness to noise Poor localization Too many responsesTrue Edge
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Canny Edge Detector
Basically 3 steps• Convolution with derivative of Gaussian• Non-maximum Suppression• Hysteresis Thresholding
J. Canny “A Computational Approach to Edge Detection” IEEE PAMI vol 8, no. 6, Nov. 1986http://perso.limsi.fr/Individu/vezien/PAPIERS_ACS/canny1986.pdf
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Canny Edge Detector
Smooth by Gaussian
Compute x and y derivatives
Compute gradient magnitude and orientation
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Canny Edge Operator
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Canny Edge Detector
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An Overview on Threshold
According to the way the Threshold T is used/determined they are divided into• Global Threshold• Local (or) Adaptive Threshold
According to the output they can be classified in • Binary Threshold• Hard Threshold• Soft Threshold
Matlab…
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Canny Edge Detector
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22yx SSS
25 ThresholdS
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Non-Maximum Suppression: The Idea
We wish to determine the points along the curve where the gradient magnitude is largest.
Non-maximum suppression: we look for a maximum along a slice orthogonal to the curve. These points form a 1D curve.
There are two issues:• which point is the maximum, • and where is the next one?
06/12/2011Original Image Gradient Magnitude Segment orthogonal
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Non-Maximum Suppression: Quantize Gradient Directions
For each pixel compute gradient direction and quantize it in 4 main direction, each covering 45° (orientation is not considered)
For each pixel buld up a segment following the quantized directions 0,1,2,3
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Non-Maximum Suppression – Idea (II)
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The intensity profile along the segment
A segment orthogonal to the gradient directionin a pixel
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Non-Maximum Suppression - Threshold
Suppress the pixels in ‘Gradient Magnitude Image’ which are not local maximum
and are the
neighbors of in
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Non-Maximum Suppression
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25ThresholdM06/12/2011
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Hysteresis Thresholding
Use of two different threshold High and Low for • For new edge starting point• For continuing edges
In such a way the edges continuity is preserved
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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’.
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Hysteresis Thresholding
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LowHigh
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Examples
an image its detected edges
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In matlab..
Canny edge detector is implemented with theedge.m command
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A brief overview on Morphological Operators in Image Processing
Giacomo Boracchi24/11/2010
home.dei.polimi.it/boracchi/teaching/IAS.htm
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An overview on morphological operations
Erosion, Dilation
Open, Closure
We assume the image being processed is binary, as these operators are typically meant for refining “mask” images.
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AND operator in Binary images
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OR in Binary Images
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Erosion
General definition: Nonlinear Filtering procedure that replace to each pixel valuethe minimum on a given neighbor
As a consequence on binary images
E(x)=1 iff the image in the neighbor is constantly 1
This operation reduces thus the boundaries of binary images
It can be interpreted as an AND operation of the image and the neighbor overlapped at each pixel
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Erosion Example
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Erosion Example
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Dilation
General definition: Nonlinear Filtering procedure that replace to each pixel valuethe maximum on a given neighbor
As a consequence on binary images
E(x)=1 iff at least a pixel in the neighbor is 1
This operation grows fat the boundaries of binary images
It can be interpreted as an OR operation of the image and the neighbor overlapped at each pixel
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Dilation Example
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Dilation Example
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In matlab
They are performed using the
bwmorph.m script passing the name of the operation as a parameter
Examples…
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Open and Closure
Open Erosion followed by a Dilation
Closure Dilation followed by an Erosion
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Open
Open Erosion followed by a Dilation• Smoothes the contours of an object• Typically eliminate thin protrusions
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Open
Figure to Open,
Structuring element a Disk
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Open
First Erode,
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Open
Then Dilate,
Open results, the bridge has been eliminate from the first erosion and cannot be replaced by the dilation.
Edges are smoothed
Corners are rounded
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Closure
Closure Dilation followed by an Erosion• Smoothes the contours of an object, typically creates• Generally fuses narrow breaks
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Open
Figure to Open,
Structuring element a Disk
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Close
First Dilate,
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Close
Then Erode,
Close results, the bridge has been preserved
Edges are smoothed
Corners are rounded
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There are several other Non Linear Filters
Ordered Statistic based• Median Filter • Weight Ordered Statistic Filter• Trimmed Mean• Hybrid Median
Thresholding
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Fitting – A brief Introduction
We assume that the images are generated from a “straight lines” word.
By performing edge detection we are only able to select those pixels that belongs to an edge
We are not able to determine if such an edge is a straight line, and in case it is, which are the parameters of such line
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Fitting – A brief Introduction
Let now assume we are interested in detecting lines only, and we want to determine the parameters of such a lines
Line Fitting can be performed via • Hough Transform• Error minimization• Maximum Likelihood
But we have to determine first how the process generating data is:i.e. • Observation are noisy, how is noise affecting straight lines?
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There are different noise models 1. Noise displaces line pixels orthogonally to the straight line and it
is gaussian distributed2. Noise displaces line pixels only along the y coordinate and it is
gaussian distributed
Then, in both case there exist a closed form solution
Or it can be performed via numerical minimization … matlab
Fitting – A brief Introduction
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Fitting and Convolution
One can prove that the least square fit of polynomial of 0-th order (i.e constant) is given by
where
and thus
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Frequency Response of Differential Kernel
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Transform
xf ufFFourierTransform
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Canny Edge Detector
Difficult to find closed-form solutions.
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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)
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Finite differences and noise
Finite difference filters respond strongly to noise• obvious reason: image noise results in pixels that look
very different from their neighbors
The more relevant is the noise, the stronger the response
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Finite differences and noise
What is to be done?• intuitively, most pixels in images look quite a lot like their
neighbors.• 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 neighbors (=noise pixels?) to look more like neighbors.
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Gaussian Filter
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