edge detection & image segmentation dr. md. altab hossain associate professor dept. of computer...

Edge Detection &

Image SegmentationDr. Md. Altab Hossain

Associate ProfessorDept. of Computer Science & Engineering, RU

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Edge Detection

PreprocessImage acquisition, restoration, and enhancement

Intermediate processImage segmentation and feature extraction

High level processImage interpretation and recognition

Element of Image Analysis

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1. Similarity properties of pixels inside the object are used to grouppixels into the same set.

2. Discontinuity of pixel properties at the boundary between objectand background is used to distinguish between pixels belonging to the object and those of background.

Discontinuity:Intensity change

at boundaryPoint, Line, Edge

Similarity:Internal

pixels sharethe same intensity

Image Attributes for Image Segmentation

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Point Detection We can use Laplacian masksfor point detection.

Laplacian masks have the largest coefficient at the center of the maskwhile neighbor pixels have anopposite sign.

This mask will give the high response to the object that has the similar shape as the mask such as isolated points.

Notice that sum of all coefficients of the mask is equal to zero. This is due to the need that the response of the filter must be zero inside a constant intensity area

-1 -1

-1

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-1

-1

-1

-1

-1

-1 0

0

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-1

-1

0

-1

0

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Point Detection

X-ray image of the turbine blade with

porosity

Laplacian image After thresholding

Location of porosity

Point detection can be done by applying the thresholding function:

otherwise 0

),( 1),(

Tyxfyxg

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

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Line Detection Similar to point detection, line detection can be performedusing the mask the has the shape look similar to a part of a line

There are several directions that the line in a digital image can be.

For a simple line detection, 4 directions that are mostly used areHorizontal, +45 degree, vertical and –45 degree.

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Line detection masks 7

Line Detection Example Binary wirebond mask

image

Absolute valueof result after

processing with-45 line detector

Result after thresholding

(Images from Rafael C. Gonzalez and Richard E. Wood, Digital Image Processing, 2nd Edition.

Notice that –45 degreelines are most sensitive

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Definition of Edges

Edges are significant local changes of intensity in an image.

Edges typically occur on the boundary between two different regions in an image.

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What Causes Intensity Changes?

Geometric events surface orientation (boundary) discontinuities depth discontinuities color and texture discontinuities

Non-geometric events illumination changes specularities shadows inter-reflections

depth discontinuity

color discontinuity

illumination discontinuity

surface normal discontinuity

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Why is Edge Detection Useful?

Important features can be extracted from the edges of an image (e.g., corners, lines, curves).

These features are used by higher-level computer vision algorithms (e.g., recognition).

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Where are the edges?

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Edge normal: unit vector in the direction of maximum intensity change.

Edge direction: unit vector to perpendicular to the edge normal. Edge position or center: the image position at which the edge is

located. Edge strength: related to the local image contrast along the

normal.

Edge Descriptors

Modeling Intensity Changes

Step edge: the image intensity abruptly changes from one value on one side of the discontinuity to a different value on the opposite side.

Ramp edge: a step edge where the intensity change is not instantaneous but occur over a finite distance.

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Modeling Intensity Changes (cont’d) Ridge edge: the image intensity abruptly changes value but

then returns to the starting value within some short distance (i.e., usually generated by lines).

Roof edge: a ridge edge where the intensity change is not instantaneous but occur over a finite distance (i.e., usually generated by the intersection of two surfaces).

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Main Steps in Edge Detection

(1) Smoothing: suppress as much noise as possible, without destroying true edges.

(2) Enhancement: apply differentiation to enhance the quality of edges (i.e., sharpening).

(3) Thresholding: determine which edge pixels should be discarded as noise and which should be retained (i.e., threshold edge magnitude).

(4) Localization: determine the exact edge location.

sub-pixel resolution might be required for some applications to estimate the location of an edge to better than the spacing between pixels.

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Edge Detection Using Derivatives

Often, points that lie on an edge

are detected by:

(1) Detecting the local maxima or

minima of the first derivative.

(2) Detecting the zero-crossings

of the second derivative.

2nd derivative

1st derivative

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Gray level profile

The 1st derivative

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-0.20

0.20.40.60.8

11.2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-0.06-0.04-0.02

00.020.040.06

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-5-4-3-2-1012345

x 10 -3

Edge Edge

Minimum point

Maximumpoint

+ +- -

Zero crossing

Inte

nsit

y

Smoothed Step Edge and Its Derivatives

The 2nd derivative

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Image Derivatives

How can we differentiate a digital image?

Option 1: reconstruct a continuous image, f(x,y), then compute the derivative.

Option 2: take discrete derivative (i.e., finite differences)

Consider this case first!

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Edge Detection Using First Derivative

(upward) step edge

(centered at x)

1D functions(not centered at x)

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Edge Detection Using Second Derivative

Approximate finding maxima/minima of gradient magnitude by finding places where:

Can’t always find discrete pixels where the second derivative is zero – look for zero-crossing instead.

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Replace x+1 with x (i.e., centered at x):

1D functions:

(centered at x+1)

Edge Detection Using Second Derivative (cont’d)

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Edge Detection Using Second Derivative (cont’d)

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Edge Detection Using Second Derivative (cont’d)

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Noise Effect on Images Edges

First column: images and gray-level profiles of a ramp edge corrupted by random Gaussian noise of mean 0 and = 0.0, 0.1, 1.0 and 10.0, respectively.

Second column: first-derivative images and gray-level profiles.

Third column : second-derivative images and gray-level profiles.

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Edge Detection Using First Derivative (Gradient)

The first derivate of an image can be computed using the gradient:

2D functions:

f

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The gradient is a vector which has magnitude and direction:

Magnitude: indicates edge strength.

Direction: indicates edge direction.i.e., perpendicular to edge direction

or

(approximation)

Gradient Representation

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Approximate Gradient

Approximate gradient using finite differences:

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Approximate Gradient (cont’d)

Cartesian vs pixel-coordinates:

- j corresponds to x direction

- i to -y direction

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Approximating Gradient (cont’d)

We can implement and using the following masks:

(x+1/2,y)

(x,y+1/2)*

*

good approximationat (x+1/2,y)

good approximationat (x,y+1/2)

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Approximating Gradient (cont’d)

A different approximation of the gradient:

3 x 3 neighborhood:

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Approximating Gradient (cont’d)

and can be implemented using the following masks:

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Another Approximation

Consider the arrangement of pixels about the pixel (i, j):

The partial derivatives can be computed by:

The constant c implies the emphasis given to pixels closer to the center of the mask.

3 x 3 neighborhood:

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Prewitt Operator

Setting c = 1, we get the Prewitt operator:

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Sobel Operator

Setting c = 2, we get the Sobel operator:

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Edge Detection Steps Using Gradient

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An Example

Idx

d

Idy

d

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An Example (cont’d)

22d d

I Idx dy

100Threshold

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Comparison of Gradient based Operator

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The Canny Edge Detector

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The Canny Edge Detector Canny – smoothing and derivatives:

xf

yf

f

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The Canny Edge Detector Canny – gradient magnitude:

image gradient magnitude

Practical Issues Choice of threshold.

gradient magnitude

low threshold high threshold

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Practical Issues (cont’d)Edge thinning and linking.

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Criteria for Optimal Edge Detection

(1) Good detection Minimize the probability of false positives (i.e., spurious edges). Minimize the probability of false negatives (i.e., missing real edges).

(2) Good localization Detected edges must be as close as possible to the true edges.

(3) Single response Minimize the number of local maxima around the true edge.

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Edge Contour Extraction

Two main categories of methods: local methods (extend edges by seeking the most "compatible" candidate

edge in a neighborhood). global methods (more computationally expensive - domain knowledge can

be incorporated in their cost function).

Edge detectors typically produce short, disjoint edge segments.

These segments are generally of little use until they are aggregated into extended edges.

We assume that edge thinning has already be done (e.g., non-maxima suppression).

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Edge Contour Extraction

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Local Processing Methods

Edge Linking using neighbor

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Local Processing Methods

Contour extraction using heuristic search

A more comprehensive approach to contour extraction is based on graph searching.

Graph representation of edge points: (1) Edge points at position pi correspond to graph nodes.

(2) The nodes are connected to each other if local edge linking rules are satisfied.

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Local Processing Methods Contour extraction using

heuristic search – cont.

The generation of a contour (if any) from pixel pA to pixel pB the generation of a minimum-cost path in the directed graph.

A cost function for a path connecting nodes p1 = pA to pN = pB could be defined as follows:

Finding a minimum-cost path is not trivial in terms of computation.

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Local Processing Methods Contour extraction using dynamic programming

Dynamic programming

It is an optimization method that searches for optima of functions in which not all the variables are simultaneously interrelated.

It subdivides a problem recursively into smaller sub-problems that may need to be solved in the future, solving each sub-problem – proceeding from the smaller ones to the larger ones – and storing the solutions in a table that can be looked up when need arises.

Principle of optimality (applied to the case of graph searching): the optimal path between two nodes pA and pB can be split into two optimal sub-paths pApi and pipB for any pi lying on the optimal path pApB.

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Global Processing Methods If the gaps between pixels are very large, local processing

methods are not effective. Global methods are more effective in this case !!

Hough Transform can be used to determine whether points lie on a curve of a specified shape (model-based method).

Deformable Models (Snakes) can be used to extract the boundaries of objects having arbitrary shapes.

Grouping can be used to decide which groups of features are likely to be part of the same object.

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Image segmentation

Definition of Segmentation

Segmentation: Image segmentation refers to the partition of an image into a set of regions that have a strong correlation with objects or areas of the real world. Complete segmentation (High-level): Region(s) correspond(s)

uniquely with image object(s); not an easy task; requires domain specific knowledge; context is known and a

priori knowledge is available and used for segmentation. Partial segmentation (Low-level): Regions do not always

correspond directly with image objects. Image is divided into separate regions that are homogenous with respect to a chosen property such as brightness, color, texture, motion, etc.

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Example-Complete Segmentation

The image is segmented into two regions: face and background. We know that the human face is of some certain color. Several other heuristics using the knowledge of human face can

be used. Small holes and isolated regions are eliminated.

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Example-Partial Segmentation

Regions do not directly correspond to objects, but to regionsthat are similar in color Needs further processing

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Segmentation Criteria

Segmentation of an image I into a set of regions S should satisfy:1. Si = S Partition covers the whole

image.

2. Si Sj = , i j No regions intersect.

3. Si, P(Si) = true Homogeneity predicate is satisfied by each

region.

4. P(Si Sj) = false, Union of adjacent regionsi j, Si adjacent Sj does not satisfy it.

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Image segmentation

So, all we have to do is to define and implement the similarity predicate. But, what do we want to be similar in each region? Is there any property that will cause the regions to

be meaningful objects?

Example approaches: Histogram-based Clustering-based Region growing Split-and-merge Graph-based

Histogram-based segmentation

Global Thresholding = Choose threshold T that separates object from background.

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Simple thresholding is not always possible: Many objects at different gray

levels. Variations in background gray

level. Noise in image.

Histogram-based segmentation

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Local Thresholding-4 Thresholds Divide image in to regions. Perform thresholding independently

in each region.

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Adaptive Thresholding

Every pixel in image is thresholded according to the histogram of the pixel neighborhood.

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Any feature vector that can be associated with a pixel can be used to group pixels into clusters. Once pixels have been grouped into clusters, it is easy to find connected regions using connected components labeling.

Clustering-based segmentation

K-means algorithm can be used for clustering 63

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Clustering-based segmentation

K-means clustering of color.

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Clustering-based segmentation Clustering can also be used with other features (e.g., texture)

in addition to color.

Original Images Color Regions Texture Regions

Region Growing

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Region growing

Usually a statistical test is used to decide which pixels can be added to a region. Region is a population with similar statistics. Use statistical test to see if neighbor on border fits

into the region population.

Let R be the N pixel region so far and p be a neighboring pixel with gray level.

Define the mean X and scatter S2 (sample variance) by

Rc)(r,

c)I(r,N

1X 2

Rc)(r,

2 X-c)I(r,N1

S

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Region growing

The T for a pixel p statistic is defined by

It has a TN-1 distribution if all the pixels in R and the test pixel p are independent and identically distributed Gaussians (i.i.d. assumption).

1/2

22 /S)X(p1)(N

1)N(NT

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Region growing

For the T distribution, statistical tables give us the probability Pr(T ≤ t) for a given degrees of freedom and a confidence level. From this, pick a suitable threshold t.

If the computed T ≤ t for desired confidence level, add p to region R and update the mean and scatter using p.

If T is too high, the value p is not likely to have arisen from the population of pixels in R. Start a new region.

Many other statistical and distance-based methods have also been proposed for region growing.

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Region growing

image

segmentation

Region growing

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Split and Merge Segmentation

2 Stage Algorithm:

Stage 1: Split Split image into regions using a Quad Tree representation.

Stage 2: Merge Merge "leaves" of the Quad Tree which are neighboring

and "similar".

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Quad Tree Representation

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Split and Merge Example

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Graph Based Segmentation Segmentation can be viewed as a graph partitioning problem

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Forming Graph

Each pixel is treated as a node in a graph.

Each pixel has an edge to its neighbours. (e.g. 8 adjacent neighbours in 2D).

Edge weight w(i,j) is a function of the similarity between nodes i and j.

high weight vertices are probably part of the same element.

Low weight nodes are probably part of different element.

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Images as graphs

Nodes for every pixel

Edge between every pair of pixel (or every pair of “sufficiently close” pixels)

Each edge is weighted by the affinity or similarity of the two nodes

wij

i

j

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Constructing a Graph from an Image

Two kinds of vertices

Two kinds of edges

Cut-Segmentation

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What is a “cut”?

A graph G = (V,E) can be partitioned into two disjoint sets,

by simply removing edges connecting the two parts.

The degree of dissimilarity between these two pieces can be computed as total weight of the edges that have been removed. In graph theoretic language it is called the cut:

BvAu

vuwBAcut,

,,

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Example cut

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Graph Cut

Each cut corresponds to some cost (cut ): sum of the weights for the edges that have been removed.

BvAu

vuwBAcut,

,,

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Graph Based Algorithms

Graph Cut-Wu and Leahy 1993

Segmentation by normalized-cut

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Interactive Image Segmentation

Magic Wand (198?)

Intelligent Scissors [Mortensen and Barrett, 1995]

GrabCut[Rother et al., 2004]

Graph Cuts [Boykov and Jolly, 2001]

LazySnapping[Li et al., 2004]

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Thank You

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