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CAP 5415 Computer Vision Fall 2004. Dr. Alper Yilmaz Univ. of Central Florida www.cs.ucf.edu/courses/cap5415/fall2004 Office: CSB 250. Recap Circle Fitting. Circle equation Change x 0 , and y 0 compute r Change only r and compute ( x 0 , y 0 ) from image gradient angle 

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cap 5415 computer vision fall 2004

CAP 5415 Computer VisionFall 2004

Dr. Alper Yilmaz

Univ. of Central Florida

www.cs.ucf.edu/courses/cap5415/fall2004

Office: CSB 250

Alper Yilmaz, Fall 2004 UCF

recap circle fitting
RecapCircle Fitting
  • Circle equation
    • Change x0, and y0 compute r
    • Change only r and compute (x0, y0) from image gradient angle 
  • Increment (x0, y0 , r) in accumulator array
  • Find the local maxima

Alper Yilmaz, Fall 2004 UCF

recap generalized hough transform
RecapGeneralized Hough Transform
  • For shapes with no analytical expression
  • Requires learning of r-table
    • Distance vector and its angle for each boundary pixel to object centroid
  • Finding a shape in an image
    • For input edge-map compute angle (can be from image gradient)
    • For each distance vector corresponding to edge angle increment a 2D accumulator array corresponding to (x0, y0)

Alper Yilmaz, Fall 2004 UCF

recap medial axis transform
RecapMedial Axis Transform
  • Represents object shape: skeleton of an object
  • Computed using an iterative algorithm,
  • Inverse transform exist

Alper Yilmaz, Fall 2004 UCF

recap interest points
RecapInterest Points
  • High texture variation in neighborhood of a pixel
  • Movarec’s operator
    • Compute directional intensity variations in 4x4 neighborhood
    • Pick points with high intensity variations in a neighborhood
  • Harris corner detector
    • Compute a moment matrix M from gradients in a neighborhood
    • Min eigenvalue of M higher than a threshold indicates corner

Alper Yilmaz, Fall 2004 UCF

object motion

Object Motion

Alper Yilmaz, Fall 2004 UCF

motion
Motion
  • Projection of object motion in real-world (3D)results in motion in image plane (2D)
  • 2D motion is defined over a sequence of frames
  • Computed by using brightness constancy constraint
    • Intensity of a moving pixel does not change over-time

Alper Yilmaz, Fall 2004 UCF

what is its use
What is its use?
  • Lots of uses
    • Motion Detection
    • Track object behavior
    • Correct for camera jitter (stabilization)
    • Align images (mosaics)
    • 3D shape reconstruction
    • Video Compression

Alper Yilmaz, Fall 2004 UCF

measurement of motion at every pixel
Measurement of motion at every pixel

Alper Yilmaz, Fall 2004 UCF

measurement of motion at every pixel1
Measurement of motion at every pixel

Alper Yilmaz, Fall 2004 UCF

visual mosaics
Visual Mosaics

Alper Yilmaz, Fall 2004 UCF

visual mosaics1
Visual Mosaics

Alper Yilmaz, Fall 2004 UCF

geo registration
Geo Registration

Alper Yilmaz, Fall 2004 UCF

video segmentation
Video Segmentation

Alper Yilmaz, Fall 2004 UCF

structure from motion
Structure From Motion

Alper Yilmaz, Fall 2004 UCF

optical flow

Optical Flow

Alper Yilmaz, Fall 2004 UCF

optical flow1
Optical Flow
  • Flow vector in image space (2D)
  • Taylor series expansion of right side around t

Alper Yilmaz, Fall 2004 UCF

optical flow2
Optical Flow

Brightness constancy equation

Equation of a line in (u,v) space

Alper Yilmaz, Fall 2004 UCF

optical flow3
Optical Flow
  • We know Ix, Iy, and It from images
  • For every point these values provide one equation
  • We have 2 unknowns: u, v
  • Solution lie any where on the line

v

u

Alper Yilmaz, Fall 2004 UCF

optical flow4

d

.

Optical Flow

v

p

  • Let (u’, v’) be true flow
  • True flow has two components
    • Normal flow: d
    • Parallel flow: p
  • Normal flow can be computed
  • Parallel flow cannot

u

Alper Yilmaz, Fall 2004 UCF

computing true flow
Computing True Flow
  • Horn & Schunck
  • Schunk
  • Lukas & Kanade

Alper Yilmaz, Fall 2004 UCF

horn schunck
Horn & Schunck
  • Define an energy function and minimize
  • Differentiate w.r.t. unknowns u and v

laplacian of u

laplacian of v

Alper Yilmaz, Fall 2004 UCF

horn schunck1
Horn & Schunck
  • Laplacian controls smoothness of optical flow
    • A particular choice can be 2u=u-uavg, 2v=v-vavg.
  • Rearranging equations
  • 2 equations 2 unknowns
  • Write v in terms of u
  • Plug it in the other equation

Alper Yilmaz, Fall 2004 UCF

horn schunck2
Horn & Schunck
  • Iteratively compute u and v
    • Assume initially u and v are 0
    • Compute uavg and vavg in a neighborhood

Alper Yilmaz, Fall 2004 UCF

schunck
Schunck
  • If two neighboring pixels move with same velocity
    • Corresponding flow equations intersect at a point in (u,v) space
    • Find the intersection point of lines
    • If more than 1 intersection points find clusters
    • Biggest cluster is true flow

v

u

Alper Yilmaz, Fall 2004 UCF

lucas kanade
Lucas & Kanade
  • Similar to line fitting we have seen
    • Define an energy functional
      • Take derivatives equate it to 0
      • Rearrange and construct an observation matrix

Alper Yilmaz, Fall 2004 UCF

lucas kanade 1
Lucas & Kanade 1

Alper Yilmaz, Fall 2004 UCF

lucas kanade 2
Lucas & Kanade 2

Alper Yilmaz, Fall 2004 UCF

discussion
Discussion
  • Horn-Schunck and Lucas-Kanade optical method works only for small motion.
  • If object moves faster, the brightness changes rapidly, derivative masks fail to estimate spatiotemporal derivatives.
  • Pyramids can be used to compute large optical flow vectors.

Alper Yilmaz, Fall 2004 UCF

coarse to fine optical flow estimation

u=1.25 pixels

u=2.5 pixels

u=5 pixels

u=10 pixels

image It

image It+1

Gaussian pyramid of image It

Gaussian pyramid of image It+1

Coarse to Fine Optical Flow Estimation

Alper Yilmaz, Fall 2004 UCF