Image Analysis Boundary Detection Dr. Ziya Telatar

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Image Analysis Boundary Detection
Dr. Ziya Telatar
Digital Image Processing
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Image Analysis
IMAGE ANALYSIS TECHNIQUES

• 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

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Image Analysis Preview

• Feature extraxtion is the first and important
  step for succesfully analysis
• Segmentation is to subdivide an image into its
  constituent regions or objects.
• Segmentation should stop when the objects of
  interest in an application have been isolated.
• Classification is closely related to segmentation

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Principal approaches

• Segmentation based feature extraction algorithms
  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 set of
  predefined criteria.

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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

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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.

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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

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Example
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Line Detection

• Horizontal mask will result with max response
  when a line passed through the middle row of the
  mask with a constant background.
• the similar idea is used with other masks.
• note the preferred direction of each mask is
  weighted with a larger coefficient (i.e.,2) than
  other possible directions.

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Line Detection

• Apply every masks on the image
• let R1, R2, R3, R4 denotes the response of the
  horizontal, 45 degree, vertical and -45 degree
  masks, respectively.
• if, at a certain point in the image
• Ri gt Rj,
• for all j?i, that point is said to be more likely
  associated with a line in the direction of mask
  i.

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Line Detection

• Alternatively, if we are interested in detecting
  all lines in an image in the direction defined by
  a given mask, we simply run the mask through the
  image and threshold the absolute value of the
  result.
• The points that are left are the strongest
  responses, which, for lines one pixel thick,
  correspond closest to the direction defined by
  the mask.

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Example
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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.

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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.

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Ideal and Ramp Edges
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Thick edge

• The slope of the ramp is inversely proportional
  to the degree 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 the ramp, and an edge would then be a set of
  such points that are connected.
• The thickness is determined by the length of the
  ramp.
• The length is determined by the slope, which is
  in turn determined by the degree of blurring.
• Blurred edges tend to be thick and sharp edges
  tend to be thin

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First and Second derivatives
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Second derivatives

• produces 2 values for every edge in an image (an
  undesirable feature)
• an imaginary straight line joining the extreme
  positive and negative values of the second
  derivative would cross zero near the midpoint of
  the edge. (zero-crossing property)
• quite useful for locating the centers of thick
  edges

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Noisy Images

• 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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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 is likely to be present.

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Edge point

• to determine a point as an edge point
• the transition in grey level associated with the
  point 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 specified
  threshold.

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Gradient Operator

• first derivatives are implemented using the
  magnitude of the gradient.

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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

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Gradient Masks
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Diagonal edges with Prewitt and Sobel masks
Sobel masks have slightly superior
noise-suppression characteristics which is an
important issue when dealing with derivatives.
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Example
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Example
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Example
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Laplacian
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Laplacian of Gaussian

• Laplacian combined with smoothing as a precursor
  to find edges via zero-crossing.

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Mexican hat
positive central term surrounded by an adjacent
negative region (a function of distance) zero
outer region
the coefficient must be sum to zero
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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

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Example
a). Original image b). Sobel Gradient c). Spatial
Gaussian smoothing function d). Laplacian
mask e). LoG f). Threshold LoG g). Zero crossing
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Zero crossing LoG

• Approximate the zero crossing from LoG image
• to threshold the LoG image by setting all its
  positive values to white and all negative values
  to black.
• the zero crossing occur between positive and
  negative values of the thresholded LoG.

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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.

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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.

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Parametric Grouping Grouping Points into Lines

• Basic Facts about Lines

(a,b)

• (x,y) is on line if (x,y).(a,b) c
• ax by c
• Distance from (x,y) to line is
• (a,b).(x,y) ax by

c
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Line Grouping Problem
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This is difficult because of

• Extraneous data Clutter
• Missing data
• Noise

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RANSAC Random Sample Consensus

• Generate a bunch of reasonable hypotheses.
• Test to see which is the best.

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RANSAC for Lines

• Generate Lines using Pairs of Points
• How many samples?
• Suppose p is fraction of points from line.
• n points needed to define hypothesis (2 for
  lines)
• k samples chosen.
• Probability one sample correct is

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RANSAC for Lines Continued

• Decide how good a line is
• Count number of points within e of line.
• Parameter e measures the amount of noise
  expected.
• Other possibilities. For example, for these
  points, also look at how far they are.
• Pick the best line.

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Some comparisons

• Complexity of RANSAC nnn
• Complexity of Hough nd
• 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 improve some of
  Houghs problems.

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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

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Neighbors of Pixels

• 4- Neighborhood
• (x1,y) (x-1,y) (x,y1) (x,y-1)
• 8- Neighbor with diagonal pixels
• (x1,y1) (x1,y-1) (x-1,y1) (x-1,y-1)

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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

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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

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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.

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Boundary ExtractionLocal Processing

• Criteria
• the strength of the response of the gradient
  operator used to produce the edge pixel
• an edge pixel with coordinates (x0,y0) in a
  predefined neighborhood of (x,y) is similar in
  magnitude to the pixel at (x,y) if
• ?f(x,y) - ?f (x0,y0) ? E

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Boundary ExtractionLocal Processing

• Criteria
• 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) lt A

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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 the
  image
• a record must be kept
• simply by assigning a different gray level to
  each set of linked edge pixels.

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Example
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Boundary Extraction Hough Transformation (Line)
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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 ap results in solution bq then 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 aixbj

b - axi yi
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??-plane

• ?90? measured with respect to x-axis

• problem of using equation y ax b is that
  value of a is infinite for a vertical line.
• To avoid the problem, use equation x cos ? y sin
  ? ? to represent a line instead.
• vertical line has ? 90? with ? equals to the
  positive y-intercept or ? -90? with ? equals to
  the negative y-intercept

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where D is the distance between corners in the
image
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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

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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 c1 and c2 , solve of c3 that satisfies
  the equation
• update the accumulator corresponding to the cell
  associated with triplet (c1, c2, c3)

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Edge-linking based on Hough Transformation

1. Compute the gradient of an image and threshold it
   to obtain a binary image.
2. Specify subdivisions in the ??-plane.
3. Examine the counts of the accumulator cells for
   high pixel concentrations.
4. Examine the relationship (principally for
   continuity) between pixels in a chosen cell.

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The Hough Transform for Lines

• A line is the set of points (x, y) such that
• Different choices of q, dgt0 give different lines
• For any (x, y) there is a one parameter family of
  lines through this point. Just let (x,y) be
  constants and q, 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

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Mechanics of the Hough transform

• Construct an array representing q, d
• For each point, render the curve (q, d) into this
  array, adding one at each cell
• Difficulties
• how big should the cells be? (too big, and we
  cannot 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 which line?
• tag the votes
• Can modify voting, peak finding to reflect noise.
• Big problem if noise in Hough space different
  from noise in image space.

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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 closet neighbor
  exceeds a certain threshold.

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link criteria 1). the pixels belonged to one of
the set of pixels linked according to the highest
count 2). no gaps were longer than 5 pixels
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Boundary Representation

• Proper representation of object boundaries is
  important 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-aided
  design (CAD) of parts and assemblies, image
  simulation applications such as video games,
  cartoon movies, environmental modeling of
  aircraft-landing testing and training, etc.

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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 a predetermined
  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, and recognition
  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.

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Example Boundary superimposed
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Region Representation

• The shape of an object may be directly
  represented by the region it occupies
• Run-length codes
• Quad-trees
• Projections

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Moment Representation

• The theory of moments provides an interesting and
  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

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Structure

• In many computer vision applications, the objects
  in a scene can be characterized satisfactorily by
  structures composed of line or arc patterns like
  handwritten or printed characters, circuit
  diagrams and engineering drawings, etc.
• Transformations useful for analysis of structure
  of patterns
• Medial Axes Transforms
• Skeleton Algorithms
• Thinning Algorithms
• Morphological Transforms

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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

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Shape Features

• Shape Representation

Regenerative 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
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Texture

• Texture is observed in the structural patterns of
  objects such as wood, grain, sand, grass, and
  cloth
• A texture contains several pixels, whose
  placement could 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.

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Texture

• Classification of Texture

Statistical - ACF (Autocorrelation Function) -
Transforms - Edge-ness - Concurrence Matrix -
TextureTransforms - Random Field Models
Other
Structural
Random - Edge Density - Extreme Density - Run
Lengths
Periodic - Primitives - Gray Levels - Shape -
Homogeneity - Placement Rules - Period -
Adjacency - Closest Distances
Mosaic Models
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Scene Matching and Detection

• A problem of much significance in image analysis
  is the detection of change or presence of 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