Lec-5 Image Analysis-Boundary Detection
-
Upload
balasha937617969 -
Category
Documents
-
view
220 -
download
0
Transcript of Lec-5 Image Analysis-Boundary Detection
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
1/77
1
Image Analysis
Boundary Detection
Dr. Ziya Telatar
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
2/77
Image Analysis
Feature Extraction
- Spatial Features
- Transform Features
- Edges and
Boundaries- Shape Features
- Moments
- Texture
Segmentation
- Template Matching
- Thresholding
- Boundary
Detection
- Clustering
- Quad-Trees
- Texture matching
Classification
- Clustering
- Statistical
- Decision Trees
- Similarity Measures
- Min. Spanning
Trees
IMAGE ANALYSIS TECHNIQUES
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
3/77
Image Analysis Preview
Feature extraxtion is the first and importantstep for succesfully analysis
Segmentation is to subdivide an image into
its constituent regions or objects. Segmentation should stop when the objectsof interest in an application have beenisolated.
Classification is closely related tosegmentation
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
4/77
Principal approaches
Segmentation based feature extractionalgorithms generally are based on one of 2
basis properties of intensity values discontinuity : to partition an image based on
abrupt changes in intensity (such as edges)
similarity : to partition an image into regions
that are similar according to a setof predefined criteria.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
5/77
Detection of Discontinuities
detect the three basic types of gray-level
discontinuities
points , lines , edges
the common way is to run a mask through
the image
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
6/77
Point Detection
a point has been detected at the location on
which the mark is centered if
|R| T
where
T is a nonnegative threshold
R is the sum of products of the coefficients with
the gray levels contained in the region
encompassed by the mark.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
7/77
Point Detection
Note that the mark is the same as the mask
of Laplacian Operation
The only differences that are considered of
interest are those large enough (as
determined by T) to be considered isolated
points.|R| T
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
8/77
Example
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
9/77
Line Detection
Horizontal mask will result with max response when a linepassed through the middle row of the mask with aconstant background.
the similar idea is used with other masks.
note: the preferred direction of each mask is weighted witha larger coefficient (i.e.,2) than other possible directions.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
10/77
Line Detection
Apply every masks on the image
let R1, R2, R3, R4 denotes the response of
the horizontal, +45 degree, vertical and -45degree masks, respectively.
if, at a certain point in the image
|Ri| > |Rj|, for all ji, that point is said to be more likely
associated with a line in the direction of maski.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
11/77
Line Detection
Alternatively, if we are interested in detectingall lines in an image in the direction defined
by a given mask, we simply run the maskthrough the image and threshold the absolutevalue of the result.
The points that are left are the strongestresponses, which, for lines one pixel thick,correspond closest to the direction defined bythe mask.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
12/77
Example
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
13/77
Edge Detection
the most common approach for detecting
meaningful discontinuities in gray level.
we discuss approaches for implementing
first-order derivative (Gradient operator)
second-order derivative (Laplacian operator)
Here, we will talk only about their properties
for edge detection.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
14/77
Basic Formulation
an edge is a set of connected pixels that
lie on the boundary between two regions.
an edge is a local concept whereas a
region boundary, owing to the way it is
defined, is a more global idea.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
15/77
Ideal and Ramp Edges
because of optics,sampling, imageacquisitionimperfection
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
16/77
Thick edge
The slope of the ramp is inversely proportional to thedegree of blurring in the edge.
We no longer have a thin (one pixel thick) path.
Instead, an edge point now is any point contained in theramp, and an edge would then be a set of such points thatare connected.
The thickness is determined by the length of the ramp.
The length is determined by the slope, which is in turndetermined by the degree of blurring.
Blurred edges tend to be thick and sharp edges tendto be thin
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
17/77
First and Second derivatives
the signs of the derivativeswould be reversed for an edgethat transitions from light to
dark
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
18/77
Second derivatives
produces 2 values for every edge in animage (an undesirable feature)
an imaginary straight line joining the
extreme positive and negative values ofthe second derivative would cross zeronear the midpoint of the edge. (zero-crossing property)
quite useful for locating the centers of thickedges
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
19/77
Noisy Images
First column: images andgray-level profiles of aramp edge corrupted byrandom Gaussian noiseof mean 0 and = 0.0,
0.1, 1.0 and 10.0,respectively.
Second column: first-derivative images andgray-level profiles.
Third column : second-derivative images andgray-level profiles.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
20/77
Keep in mind
fairly little noise can have such a
significant impact on the two key
derivatives used for edge detection in
images
image smoothing should be serious
consideration prior to the use of
derivatives in applications where noise islikely to be present.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
21/77
Edge point
to determine a point as an edge point
the transition in grey level associated with thepoint has to be significantly stronger than the
background at that point. use threshold to determine whether a value is
significant or not.
the points two-dimensional first-order
derivative must be greater than a specifiedthreshold.
f
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
22/77
Gradient Operator
first derivatives are implemented using
the magnitude of the gradient.
y
fx
f
G
G
y
xf
21
22
21
22 ][)f(
yf
xf
GGmagf yx
the magnitude becomes nonlinearyx GGf
commonly approx.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
23/77
Gradient direction
Let (x,y) represent the direction angle of
the vector f at (x,y)
(x,y) = tan-1(Gy/Gx)
The direction of an edge at (x,y) is
perpendicular to the direction of the
gradient vector at that point
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
24/77
Gradient Masks
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
25/77
Diagonal edges with Prewitt
and Sobel masks
Sobel masks haveslightly superiornoise-suppression
characteristics whichis an important issuewhen dealing withderivatives.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
26/77
Example
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
27/77
Example
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
28/77
Example
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
29/77
Laplacian
2
2
2
22 ),(),(
y
yxf
x
yxff
(linear operator)
Laplacian operator
)],(4)1,()1,(
),1(),1([2
yxfyxfyxf
yxfyxff
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
30/77
Laplacian of Gaussian
Laplacian combined with smoothing as a
precursor to find edges via zero-crossing.
2
2
2
)(
r
erh
where r2= x2+y2, andis the standard deviation
2
2
24
222
)(
r
er
rh
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
31/77
Mexican hat
the coefficient must be sum to zero
positive central term
surrounded by an adjacent negative region (a function of distance)zero outer region
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
32/77
Linear Operation
second derivation is a linear operation
thus, 2f is the same as convolving the
image with Gaussian smoothing function
first and then computing the Laplacian of
the result
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
33/77
Example
a). Original imageb). Sobel Gradient
c). Spatial Gaussiansmoothing functiond). Laplacian maske). LoG
f). Threshold LoGg). Zero crossing
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
34/77
Zero crossing & LoG
Approximate the zero crossing from LoGimage
to threshold the LoG image by setting all
its positive values to white and all negativevalues to black.
the zero crossing occur between positive
and negative values of the thresholdedLoG.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
35/77
Zero crossing vs. Gradient
Attractive
Zero crossing produces thinner edges
Noise reduction
Drawbacks Zero crossing creates closed loops. (spaghetti
effect)
sophisticated computation.
Gradient is more frequently used.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
36/77
Edge Linking and Grouping
Define what it means for a group to be
good.
Usually this involves simplifications
Search for the best group.
Usually this is intractable, so short-cuts are
needed.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
37/77
Parametric Grouping: Grouping
Points into Lines
Basic Facts about Lines
(a,b)
c (x,y) is on line if (x,y).(a,b) = cax + by = c
Distance from (x,y) to line is
(a,b).*(x,y) = ax + by
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
38/77
Line Grouping Problem
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
39/77
This is difficult because of:
Extraneous data: Clutter
Missing data
Noise
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
40/77
RANSAC: Random Sample
Consensus
Generate a bunch of reasonable
hypotheses.
Test to see which is the best.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
41/77
RANSAC for Lines Generate Lines using Pairs of Points
How many samples?
Supposep is fraction of points from line.
npoints needed to define hypothesis (2 for lines)
k samples chosen.
Probability one sample correct is:
kn
p)1(1
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
42/77
RANSAC for Lines: Continued
Decide how good a line is:
Count number of points within eof line.
Parameteremeasures the amount of noise
expected.
Other possibilities. For example, for these
points, also look at how far they are.
Pick the best line.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
43/77
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
44/77
Some comparisons
Complexity of RANSAC n*n*n
Complexity of Hough n*d
Error behavior: both can have problems,
RANSAC perhaps easier to understand. Clutter: RANSAC very robust, Hough falls
apart at some point.
There are endless variations that improvesome of Houghs problems.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
45/77
Boundary Extraction
Boundaries are linked edges that characterize
the shape of an object
Connectivity Boundaries can be found by tracing the connected
edges
On a rectangular grid, a pixel is said to be 4- or 8-
connected when it has the same properties as one of
its nearest 4 or 8 neighbors, respectively
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
46/77
Neighbors of Pixels
4- Neighborhood
(x+1,y) (x-1,y) (x,y+1) (x,y-1)
8- Neighbor with diagonal pixels
(x+1,y+1) (x+1,y-1) (x-1,y+1) (x-1,y-1)
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
47/77
Boundary Extraction
Contour Following
Contour-following algorithms trace boundaries by
ordering successive edge points.
Some example of algorithms are:
Edge Linking and Hueristic Graph Search Dynamic programming
Hough Transform
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
48/77
Boundary Extraction
Edge Linking:
edge detection algorithm are followed by
linking procedures to assemble edge pixels
into meaningful edges.
Basic approaches
Local Processing
Global Processing via the Hough Transform Global Processing via Graph-Theoretic
Techniques
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
49/77
Boundary Extraction
Local Processing
analyze the characteristics of pixels in a small
neighborhood (say, 3x3, 5x5) about every
edge pixels (x,y) in an image. all points that are similar according to a set of
predefined criteria are linked, forming an edge
of pixels that share those criteria.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
50/77
Boundary ExtractionLocal Processing
Criteria
1. the strength of the response of thegradient operator used to produce the
edge pixel an edge pixel with coordinates (x0,y0) in a
predefined neighborhood of (x,y) is similar inmagnitude to the pixel at (x,y) if
|f(x,y) - f (x0,y0) | E
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
51/77
Boundary ExtractionLocal Processing
Criteria
2. the direction of the gradient vector
an edge pixel with coordinates (x0,y0) in a
predefined neighborhood of (x,y) is similar in
angle to the pixel at (x,y) if
|(x,y) - (x0,y0) | < A
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
52/77
Criteria
A point in the predefined neighborhood of (x,y) is
linked to the pixel at (x,y) if both magnitude
and direction criteria are satified.
the process is repeated at every location in theimage
a record must be kept
simply by assigning a different gray level to each
set of linked edge pixels.
fi d t l h i k
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
53/77
Examplefind rectangles whose sizes makesthem suitable candidates for licenseplates
use horizontal andvertical Sobel operators
link conditions:gradient value > 25gradient direction differs < 15
eliminate isolated shortsegments
B d E t ti
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
54/77
Boundary ExtractionHough Transformation (Line)
yi=axi+ b b = - axi+ yi
ab-plane or parameter space
xy-plane
all points (xi,yi) contained on the same line must have lines in
parameter space that intersect at (a,b
)
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
55/77
Accumulator cells
(amax, amin) and (bmax, bmin) are
the expected ranges of slope
and intercept values.
all are initialized to zero
if a choice of apresults in
solution bqthen we let
A(p,q) = A(p,q)+1
at the end of the procedure,
value Q in A(i,j) corresponds to
Q points in the xy-plane lying
on the line y = aix+bj b = - axi+ yi
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
56/77
-plane
problem of using equation y = ax + b is that value of a is infinite for avertical line.
To avoid the problem, use equation x cos + y sin = to represent aline instead.
vertical line has = 90with equals to the positive y-intercept or =
-90with equals to the negative y-intercept
x cos + y sin =
-plane
= 90 measured with respect to x-axis
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
57/77
D2ofrange
where D is the distancebetween corners in the
image
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
58/77
Generalized Hough Transformation
can be used for any function of the form
g(v,c) = 0
v is a vector of coordinates
c is a vector of coefficients
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
59/77
Hough Transformation (Circle)
equation:
(x-c1)2+ (y-c2)
2= c32
three parameters (c1, c2, c3)
cube like cells accumulators of the form A(i, j, k)
increment c1and c2, solve of c3that satisfies theequation
update the accumulator corresponding to thecell associated with triplet (c1, c2, c3)
Edge linking based on Hough
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
60/77
Edge-linking based on Hough
Transformation
1. Compute the gradient of an image andthreshold it to obtain a binary image.
2. Specify subdivisions in the -plane.
3. Examine the counts of the accumulatorcells for high pixel concentrations.
4. Examine the relationship (principally for
continuity) between pixels in a chosencell.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
61/77
The Hough Transform for
Lines A line is the set of points (x, y) such that
Different choices of , d>0 give different lines
For any (x, y) there is a one parameter familyof lines through this point. Just let (x,y) be
constants and , d be unknowns.
Each point gets to vote for each line in the
family; if there is a line that has lots of votes,
that should be the line passing through the
points
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
62/77
Mechanics of the Hough
transform Construct an arrayrepresenting , d
For each point, render
the curve (, d) into this
array, adding one at
each cell
Difficulties
how big should the cells
be? (too big, and wecannot distinguish
between quite different
lines; too small, and
noise causes lines to be
missed)
How many lines?
count the peaks in the
Hough array
Who belongs to whichline?
tag the votes
Can modify voting, peak
finding to reflect noise. Big problem if noise in
Hough space different
from noise in image
space.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
63/77
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
64/77
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
65/77
Continuity
based on computing the distance between
disconnected pixels identified during
traversal of the set of pixels corresponding
to a given accumulator cell.
a gap at any point is significant if the
distance between that point and its closetneighbor exceeds a certain threshold.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
66/77
link criteria:1). the pixels belonged to one of the set of pixelslinked according to the highest count2). no gaps were longer than 5 pixels
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
67/77
Boundary Representation
Proper representation of object boundaries isimportant for analysis and synthesis of shape
Shape analysis is often required for detection
and recognition of objects in a scene.
Shape synthesis is useful in computer-aideddesign (CAD) of parts and assemblies, image
simulation applications such as video games,cartoon movies, environmental modeling ofaircraft-landing testing and training, etc.
Methods for Boundary Representations
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
68/77
Methods for Boundary Representations
Chain Codes The direction vectors between successive boundary pixels are encoded.
Fitting Line Segments Staright-line segments give simple approximation of curve boundaries.
Approximate the curve by the line segment joining its end points.
If the distance from the farthest curve point to the segment is greater than apredetermined quantitiy, join the points to line ends.
B-Spline Representation Piecewise polynomial functions that can provide local approximations of contours
of shapes using a small number of parameters.
B-Splines are used in shape synthesis and analysis, computer graphics, andrecognition parts from boundaries.
Fourier Descriptors Effect of Geometric Transformations
Boundary matching from FD
Autoregressive Models For arbitrary class of object boundaries, we have an ensemble of boundaries that
could be represented by a stochastic model.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
69/77
Example:
Boundary superimposed
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
70/77
Region Representation
The shape of an object may be directly
represented by the region it occupies
Run-length codes
Quad-trees
Projections
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
71/77
Moment Representation
The theory of moments provides an interestingand sometimes useful alternative to serious
expansions for representing shape of objects
Moment representation theorem Moment matching
Orthogonal moments
Moment invariants
Translation Scaling
Rotation and Reflection
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
72/77
Structure
In many computer vision applications, the objects in ascene can be characterized satisfactorily by structurescomposed of line or arc patterns like handwritten orprinted characters, circuit diagrams and engineeringdrawings, etc.
Transformations useful for analysis of structure ofpatterns Medial Axes Transforms
Skeleton Algorithms Thinning Algorithms
Morphological Transforms
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
73/77
Shape Features
The shape of an object refers to its profile
and physical structure
These characteristics can be represented
by the boundary, region, moment, and
structural representations
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
74/77
Shape Features
Shape RepresentationRegenerative Features
- Boundaries
- Regions
- Moments- Structural and Syntactic
Measurement Features
Geometry
- Perimetry
- Area
- Max-min Radii
and eccentricity
- Corners
- Roundness
- Bending Energy
- Holes
- Euler Numbers
- Symmetry
Moments
- Center of Mass
- Orintation
- Bounding
Rectangle
- Best-Fit Ellipse
- Eccentricity
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
75/77
Texture
Texture is observed in the structural patterns of objectssuch as wood, grain, sand, grass, and cloth
A texture contains several pixels, whose placementcould be periodic, quasi-periodic or random.
Natural textures are generally random
Artificial textures are often deterministic or periodic
Texture are classified into two main categories:Statistical and Structural.
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
76/77
TextureClassification of Texture
Statistical
- ACF (Autocorrelation
Function)
- Transforms
- Edge-ness
- Concurrence Matrix
- TextureTransforms
- Random Field
Models
Other
Random
- Edge Density
- Extreme Density- Run Lengths
Mosaic Models
Structural
Periodic
- Primitives:
- Gray Levels- Shape
- Homogeneity
- Placement Rules
- Period
- Adjacency
- Closest
Distances
-
8/10/2019 Lec-5 Image Analysis-Boundary Detection
77/77
Scene Matching and Detection
A problem of much significance in imageanalysis is the detection of change or presenceof an object in a given scene
Methods Image Subtraction
Template Matching and Area Correlation
Matched Filtering
Direct serch Two Dimensional Logarithmic search Sequential Search
Hiearchical Search