Sobel Edge Detection: Gradient Approximation

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Sobel Edge Detection Gradient Approximation
Note anisotropy of edge finding
Horizontal diff.
Vertical diff.
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Sobel

• These can then be combined together to find the
  absolute magnitude of the gradient at each point
  and the orientation of that gradient. The
  gradient magnitude is given by
• an approximate magnitude is computed using
• which is much faster to compute.
• The angle of orientation of the edge (relative to
  the pixel grid) giving rise to the spatial
  gradient is given by
• In this case, orientation 0 is taken to mean that
  the direction of maximum contrast from black to
  white runs from left to right on the image, and
  other angles are measured anti-clockwise from
  this.

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Derivative of Gaussian
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Smoothing and Differentiation

• Issue noise
• smooth before differentiation
• two convolutions to smooth, then differentiate?
• actually, no - we can use a derivative of
  Gaussian filter
• because differentiation is convolution, and
  convolution is associative

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The Laplacian of Gaussian

• Another way to detect an extremal first
  derivative is to look for a zero second
  derivative
• the Laplacian

• Bad idea to apply a Laplacian without smoothing
• smooth with Gaussian, apply Laplacian
• this is the same as filtering with a Laplacian of
  Gaussian filter
• Now mark the zero points where there is a
  sufficiently large (first) derivative, and enough
  contrast

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Marr-Hildreth operator

• The Laplacian is linear and rotationally
  symmetric. Thus, we search for the zero crossings
  of the image that is first smoothed with a
  Gaussian mask and then the second derivative is
  calculated or we can convolve the image with the
  Laplacian of the Gaussian, also known as the LoG
  operator
• This defines the Marr-Hildreth operator.
• One can also get a shape similar to G'' by taking
  the difference of two Gaussians having different
  standard deviations. A ratio of standard
  deviations of 11.6 will give a close
  approximation to .This is known as the
  DoG operator (Difference of Gaussians), or the
  Mexican Hat Operator.
• Still sensitive to noise.

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Step edge detection 2nd-Derivative Operators

• Method 2nd derivative is 0 for 1st-derivative
  extrema, so find zero-crossings
• Laplacian
• Isotropic (finds edges regardless of orientation.

Three commonly used discrete approximations to
the Laplacian filter. (Note, we have defined the
Laplacian using a negative peak because this is
more common, however, it is equally valid to use
the opposite sign convention.) Source
http//www.cee.hw.ac.uk/hipr/html/log.html
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Laplacian of Gaussian

• Matlab fspecial(log,)

Below Discrete approximation to LoG function
with Gaussian 1.4
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Sobel vs. LoG Edge DetectionMatlab Automatic
Thresholds
Sobel
LoG
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1 2
There are three major issues 1) The gradient
magnitude at different scales is different which
should we choose? 2) The gradient
magnitude is large along thick trail (for 3rd
fig) how do we identify the significant
points? 3) How do we link the relevant points
up into curves?
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We wish to mark points along the curve where the
magnitude is biggest. We can do this by looking
for a maximum along a slice normal to the
curve (non-maximum suppression). These points
should form a curve. There are then two
algorithmic issues at which point is the
maximum, and where is the next one?
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Non-maximum suppression
At q, we have a maximum if the value is larger
than those at both p and at r. Interpolate to get
these values.
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Predicting the next edge point
Assume the marked point is an edge point. Then
we construct the tangent (along) to the edge
curve (which is normal to the gradient at that
point) and use this to predict the next points
(here either r or s).
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Remaining issues

• Check that maximum value of gradient value is
  sufficiently large
• drop-outs? use hysteresis
• use a high threshold to start edge curves and a
  low threshold to continue them.

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(No Transcript)
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fine scale high Threshold (be strict
in Accepting Edge points)
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coarse scale, high threshold
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coarse scale low threshold
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Canny Edge Detection

• Steps
• Apply derivative of Gaussian (not Laplacian!)
• Non-maximum suppression
• Thin multi-pixel wide ridges down to single
  pixel
• Thresholding
• Low, high edge-strength thresholds
• Accept all edges over low threshold that are
  connected to edge over high threshold (in the
  stage of predicting next edge point)
• Matlab edge(I, canny)

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Edge Smearing
Input
Result
from Forsyth Ponce
Sobel filter example Yields 2-pixel wide edge
band
We want to localize the edge to within 1 pixel
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Non-Maximum Suppression Steps

1. Consider 9-pixel neighborhood around each edge
   candidate (i.e., already over a threshold)
2. Interpolate edge strengths E at neighborhood
   boundaries in negative positive gradient
   directions from the center pixel
3. If the pixel under consideration is not greater
   than these two values (i.e. not a maximum), it is
   suppressed

Interpolating the E value E(r) (1 a)E(x, y)
aE(x 1, y)
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Example Non-Maximum Suppression
courtesy of G. Loy
Non-maxima suppressed
Original image
Gradient magnitude
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Edge Streaking

• Can predict next pixel in edge orthogonal to
  gradient to make edge chain
• Can also just use 8-connectedness to define
  chains
• Streaking Gaps in edge chain due to edge
  strength dipping below threshold

courtesy of G. Loy
Original image
Strong edges
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Edge Hysteresis

• Hysteresis A lag or momentum factor
• Idea Maintain two thresholds khigh and klow
• Use khigh to find strong edges to start edge
  chain
• Use klow to find weak edges which continue edge
  chain
• Usual ratio of thresholds is roughly
• khigh / klow 2 or 3

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Example Canny Edge Detection
Strong connected weak edges
Original image
Strong edges only
Weak edges
courtesy of G. Loy
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Example Canny Edge Detection
(Matlab automatically set thresholds)
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Image Pyramids

• Observation Fine-grained template matching
  expensive over a full image
• Idea Represent image at smaller
  scales, allowing efficient coarse-
  to-fine search
• Downsampling Cut width, height in
• half at each iteration

from Forsyth Ponce
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Gaussian Pyramid

• Let the base (the finest resolution) of an
  n-level Gaussian pyramid be defined as P0 I.
  Then the ith level is reduced from the level
  below it by
• Upsampling S"(I) Double size of image,
  interpolate missing pixels
  

courtesy of Wolfram
Gaussian pyramid
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Reconstruction
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Example fromhttp//sepwww.stanford.edu/morgan/t
exturematch/paper_html/node3.html
?decompose
Reconstruct ?
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Laplacian Pyramids

• The tip (the coarsest resolution) of an n-level
  Laplacian pyramid is the same as the Gaussian
  pyramid at that level Ln(I) Pn(I)
• The ith level is obtained from the level above
  according to Li(I) Pi(I) S"(Pi1(I))
• Synthesizing the original image Get I back by
  summing upsampled Laplacian pyramid levels

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Laplacian Pyramid

• The differences of images at successive levels of
  the Gaussian pyramid define the Laplacian
  pyramid. To calculate a difference, the image at
  a higher level in the pyramid must be increased
  in size by a factor of four prior to subtraction.
  This computes the pyramid.
• The original image may be reconstructed from the
  Laplacian pyramid by reversing the previous
  steps. This interpolates and adds the images at
  successive levels of the pyramid beginning with
  the coarsest level.
• Laplacian is largely uncorrelated, and so may be
  represented pixel by pixel with many fewer bits
  than Gaussian.

courtesy of Wolfram
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Splining

• Build Laplacian pyramids LA and LB for A B
  images
• Build a Gaussian pyramid GR from selected region
  R
• Form a combined pyramid LS from LA and LB using
  nodes of GR as weights
• LS(I,j) GR(I,j)LA(I,j)(1-GR(I,j))LB(I,j)
• Collapse the LS pyramid to get the final blended
  image

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Splining (Blending)

• Splining two images simply requires 1)
  generating a Laplacian pyramid for each image, 2)
  generating a Gaussian pyramid for the bitmask
  indicating how the two images should be merged,
  3) merging each Laplacian level of the two images
  using the bitmask from the corresponding Gaussian
  level, and 4) collapsing the resulting Laplacian
  pyramid.
• i.e. GS Gaussian pyramid of bitmask LA
  Laplacian pyramid of image "A" LB Laplacian
  pyramid of image "B" therefore, "Lout (GS)LA
  (1-GS)LB"

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Example images from GTech
Image-1 bit-mask
image-2 Direct addition splining
bad bit-mask choice
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Outline

• Corner detection
• RANSAC

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Matching with Invariant Features

• Darya Frolova, Denis Simakov
• The Weizmann Institute of Science
• March 2004

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Example Build a Panorama
M. Brown and D. G. Lowe. Recognising Panoramas.
ICCV 2003
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How do we build panorama?

• We need to match (align) images

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Matching with Features

• Detect feature points in both images

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Matching with Features

• Detect feature points in both images
• Find corresponding pairs

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Matching with Features

• Detect feature points in both images
• Find corresponding pairs
• Use these pairs to align images

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Matching with Features

• Problem 1
• Detect the same point independently in both images

no chance to match!
We need a repeatable detector
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Matching with Features

• Problem 2
• For each point correctly recognize the
  corresponding one

?
We need a reliable and distinctive descriptor
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More motivation

• Feature points are used also for
• Image alignment (homography, fundamental matrix)
• 3D reconstruction
• Motion tracking
• Object recognition
• Indexing and database retrieval
• Robot navigation
• other