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Motion estimation

- Introduction to Computer VisionCS223B, Winter

2005Richard Szeliski

Why Visual Motion?

- Visual Motion can be annoying
- Camera instabilities, jitter
- Measure it. Remove it.
- Visual Motion indicates dynamics in the scene
- Moving objects, behavior
- Track objects and analyze trajectories
- Visual Motion reveals spatial layout of the scene
- Motion parallax

Todays lecture

- Motion estimation
- background image pyramids, image warping
- application image morphing
- parametric motion (review)
- optic flow
- layered motion models

Image Pyramids

Image Pyramids

Pyramid Creation

filter mask

Gaussian Pyramid

- Laplacian Pyramid
- Created from Gaussianpyramid by subtractionLl

Gl expand(Gl1)

Octaves in the Spatial Domain

Lowpass Images

- Bandpass Images

Pyramids

- Advantages of pyramids
- Faster than Fourier transform
- Avoids ringing artifacts
- Many applications
- small images faster to process
- good for multiresolution processing
- compression
- progressive transmission
- Known as mip-maps in graphics community
- Precursor to wavelets
- Wavelets also have these advantages

Laplacian level 0

left pyramid

right pyramid

blended pyramid

Pyramid Blending

Image Warping

Image Warping

- image filtering change range of image
- g(x) h(f(x))
- image warping change domain of image
- g(x) f(h(x))

Image Warping

- image filtering change range of image
- g(x) h(f(x))
- image warping change domain of image
- g(x) f(h(x))

f

g

f

g

Parametric (global) warping

- Examples of parametric warps

aspect

rotation

translation

perspective

cylindrical

affine

2D coordinate transformations

- translation x x t x (x,y)
- rotation x R x t
- similarity x s R x t
- affine x A x t
- perspective x ? H x x (x,y,1) (x is a

homogeneous coordinate) - These all form a nested group (closed w/ inv.)

Image Warping

- Given a coordinate transform x h(x) and a

source image f(x), how do we compute a

transformed image g(x) f(h(x))?

h(x)

x

x

f(x)

g(x)

Forward Warping

- Send each pixel f(x) to its corresponding

location x h(x) in g(x)

- What if pixel lands between two pixels?

h(x)

x

x

f(x)

g(x)

Forward Warping

- Send each pixel f(x) to its corresponding

location x h(x) in g(x)

- What if pixel lands between two pixels?

- Answer add contribution to several pixels,

normalize later (splatting)

h(x)

x

x

f(x)

g(x)

Inverse Warping

- Get each pixel g(x) from its corresponding

location x h-1(x) in f(x)

- What if pixel comes from between two pixels?

h-1(x)

x

x

f(x)

g(x)

Inverse Warping

- Get each pixel g(x) from its corresponding

location x h-1(x) in f(x)

- What if pixel comes from between two pixels?

- Answer resample color value from interpolated

(prefiltered) source image

x

x

f(x)

g(x)

Interpolation

- Possible interpolation filters
- nearest neighbor
- bilinear
- bicubic (interpolating)
- sinc / FIR
- Needed to prevent jaggies and texture crawl

(see demo)

Prefiltering

- Essential for downsampling (decimation) to

prevent aliasing - MIP-mapping Williams83
- build pyramid (but what decimation filter?)
- block averaging
- Burt Adelson (5-tap binomial)
- 7-tap wavelet-based filter (better)
- trilinear interpolation
- bilinear within each 2 adjacent levels
- linear blend between levels (determined by pixel

size)

Prefiltering

- Essential for downsampling (decimation) to

prevent aliasing - Other possibilities
- summed area tables
- elliptically weighted Gaussians (EWA)

Heckbert86

Image Warping non-parametric

- Specify more detailed warp function
- Examples
- splines
- triangles
- optical flow (per-pixel motion)

Image Warping non-parametric

- Move control points to specify spline warp

Image Morphing

Image Morphing

- How can we in-between two images?
- Cross-dissolve(all examples from Gomes

et al.99)

Image Morphing

- How can we in-between two images?
- Warp then cross-dissolve morph

Warp specification

- How can we specify the warp?
- Specify corresponding points
- interpolate to a complete warping

function - Nielson, Scattered Data Modeling, IEEE CGA93

Warp specification

- How can we specify the warp?
- Specify corresponding vectors
- interpolate to a complete warping function

Warp specification

- How can we specify the warp?
- Specify corresponding vectors
- interpolate Beier Neely, SIGGRAPH92

Warp specification

- How can we specify the warp?
- Specify corresponding spline control points
- interpolate to a complete warping function

Final Morph Result

Motion estimation

Classes of Techniques

- Feature-based methods
- Extract salient visual features (corners,

textured areas) and track them over multiple

frames - Analyze the global pattern of motion vectors of

these features - Sparse motion fields, but possibly robust

tracking - Suitable especially when image motion is large

(10-s of pixels) - Direct-methods
- Directly recover image motion from

spatio-temporal image brightness variations - Global motion parameters directly recovered

without an intermediate feature motion

calculation - Dense motion fields, but more sensitive to

appearance variations - Suitable for video and when image motion is small

(lt 10 pixels)

The Brightness Constraint

- Brightness Constancy Equation

Or, better still, Minimize

Linearizing (assuming small (u,v))

Gradient Constraint (or the Optical Flow

Constraint)

Local Patch Analysis

Patch Translation Lucas-Kanade

Assume a single velocity for all pixels within an

image patch

Minimizing

LHS sum of the 2x2 outer product tensor of the

gradient vector

The Aperture Problem

and

Let

- Algorithm At each pixel compute by

solving - M is singular if all gradient vectors point in

the same direction - e.g., along an edge
- of course, trivially singular if the summation

is over a single pixel or there is no texture - i.e., only normal flow is available (aperture

problem) - Corners and textured areas are OK

Aperture Problem and Normal Flow

Local Patch Analysis

Iterative Refinement

- Estimate velocity at each pixel using one

iteration of Lucas and Kanade estimation - Warp one image toward the other using the

estimated flow field - (easier said than done)
- Refine estimate by repeating the process

Optical Flow Iterative Estimation

x

x0

Optical Flow Iterative Estimation

Optical Flow Iterative Estimation

Optical Flow Iterative Estimation

x

x0

Optical Flow Iterative Estimation

- Some Implementation Issues
- warping is not easy (make sure that errors in

interpolation and warping are not bigger than the

estimate refinement) - warp one image, take derivatives of the other so

you dont need to re-compute the gradient after

each iteration. - often useful to low-pass filter the images before

motion estimation (for better derivative

estimation, and somewhat better linear

approximations to image intensity)

Optical Flow Iterative Estimation

- Some Implementation Issues
- warping is not easy (make sure that errors in

interpolation and warping are not bigger than the

estimate refinement) - warp one image, take derivatives of the other so

you dont need to re-compute the gradient after

each iteration. - often useful to low-pass filter the images before

motion estimation (for better derivative

estimation, and somewhat better linear

approximations to image intensity)

Optical Flow Aliasing

Temporal aliasing causes ambiguities in optical

flow because images can have many pixels with the

same intensity. I.e., how do we know which

correspondence is correct?

nearest match is correct (no aliasing)

nearest match is incorrect (aliasing)

To overcome aliasing coarse-to-fine estimation.

Iterative refinement

BUT!!

Limits of the gradient method

- Fails when intensity structure in window is poor
- Fails when the displacement is large (typical

operating range is motion of 1 pixel) - Linearization of brightness is suitable only for

small displacements - Also, brightness is not strictly constant in

images - actually less problematic than it appears, since

we can pre-filter images to make them look similar

Coarse-to-Fine Estimation

Coarse-to-Fine Estimation

I

J

J

Jw

I

refine

warp

J

I

Jw

pyramid construction

pyramid construction

refine

warp

J

I

Jw

refine

warp

Global Motion Models

- 2D Models
- Affine
- Quadratic
- Planar projective transform (Homography)
- 3D Models
- Instantaneous camera motion models
- Homographyepipole
- PlaneParallax

Example Affine Motion

- Substituting into the B.C. Equation

Each pixel provides 1 linear constraint in 6

global unknowns

Other 2D Motion Models

3D Motion Models

Correlation and SSD

- For larger displacements, do template matching
- Define a small area around a pixel as the

template - Match the template against each pixel within a

search area in next image. - Use a match measure such as correlation,

normalized correlation, or sum-of-squares

difference - Choose the maximum (or minimum) as the match
- Sub-pixel interpolation also possible

SSD Surface Textured area

SSD Surface -- Edge

SSD homogeneous area

Discrete Search vs. Gradient Based Estimation

Consider image I translated by

The discrete search method simply searches for

the best estimate. The gradient method linearizes

the intensity function and solves for the estimate

Uncertainty in Local Estimation

Consider image I translated by

Now,

This assumes uniform priors on the velocity field

Quadratic Approximation

When

are small

After some fiddling around, we can show

Posterior uncertainty

At edges is singular, but just take

pseudo-inverse

Note that the error is always convex, since

is positive semi-definite i.e., even for

occluded points and other false matches, this is

the case seems a bit odd!

Match plus confidence

- Numerically compute error for various
- Search for the peak
- Numerically fit a qudratic to around the

peak - Find sub-pixel estimate for and covariance
- If the matrix is negative, it is false match
- Or even better, if you can afford it, simply

maintain a discrete sampling of and

Quadratic Approximation and Covariance Estimation

When

where

Correlation Window Size

- Small windows lead to more false matches
- Large windows are better this way, but
- Neighboring flow vectors will be more correlated

(since the template windows have more in common) - Flow resolution also lower (same reason)
- More expensive to compute
- Another way to look at this
- Small windows are good for local search but more

precise and less smooth - Large windows good for global search but less

precise and more smooth method

Optical Flow Robust Estimation

Issue 7 Noise distributions are often

non-Gaussian, having much heavier tails. Noise

samples from the tails are called outliers.

- Sources of outliers (multiple motions)
- specularities / highlights
- jpeg artifacts / interlacing / motion blur
- multiple motions (occlusion boundaries,

transparency)

Robust Estimation

- Noise distributions are often non-Gaussian,

having much heavier tails. Noise samples from

the tails are called outliers. - Sources of outliers (multiple motions)
- specularities / highlights
- jpeg artifacts / interlacing / motion blur
- multiple motions (occlusion boundaries,

transparency)

Robust Estimation

Standard Least Squares Estimation allows too much

influence for outlying points

Robust Estimation

Robust gradient constraint

Robust SSD

Robust Estimation

Layered Motion Models

Layered models provide a 2.5 representation, like

cardboard cutouts.

Key players

- intensity (appearance)
- alpha map (opacity)
- warp maps (motion)

Layered Scene Representations

Motion representations

- How can we describe this scene?

Block-based motion prediction

- Break image up into square blocks
- Estimate translation for each block
- Use this to predict next frame, code difference

(MPEG-2)

Layered motion

- Break image sequence up into layers
- ?
- Describe each layers motion

Outline

- Why layers?
- 2-D layers Wang Adelson 94 Weiss 97
- 3-D layers Baker et al. 98
- Layered Depth Images Shade et al. 98
- Transparency Szeliski et al. 00

Layered motion

- Advantages
- can represent occlusions / disocclusions
- each layers motion can be smooth
- video segmentation for semantic processing
- Difficulties
- how do we determine the correct number?
- how do we assign pixels?
- how do we model the motion?

Layers for video summarization

Background modeling (MPEG-4)

- Convert masked images into a background sprite

for layered video coding

What are layers?

- Wang Adelson, 1994
- intensities
- alphas
- velocities

How do we composite them?

How do we form them?

How do we form them?

How do we estimate the layers?

- compute coarse-to-fine flow
- estimate affine motion in blocks (regression)
- cluster with k-means
- assign pixels to best fitting affine region
- re-estimate affine motions in each region

Layer synthesis

- For each layer
- stabilize the sequence with the affine motion
- compute median value at each pixel
- Determine occlusion relationships

Results

What if the motion is not affine?

- Use a regularized (smooth) motion field
- Weiss, CVPR97

A Layered Approach To Stereo Reconstruction

- Simon Baker, Richard Szeliski and P. Anandan
- CVPR98

Layered Stereo

- Assign pixel to different layers (objects,

sprites) - already covered in Stereo Lecture 2

Layer extraction from multiple images containing

reflections and transparency

- Richard Szeliski
- Shai Avidan
- P. Anandan
- CVPR2000
- extra bonus material

Transparent motion

- Photograph (Lee) and reflection (Michael)

Previous work

- Physics-based vision and polarizationShafer et

al. Wolff Nayar et al. - Perception of transparency Adelson
- Transparent motion estimationShizawa Mase

Bergen et al. Irani et al. Darrell

Simoncelli - 3-frame layer recovery Bergen et al.

Problem formulation

MotionX,i( )

X

Y

MotionY,i( )

Image formation model

- Pure additive mixing of positive signals
- mk(x) ?l Wkl ? fl(x)
- or
- mk ?l Wkl fl
- Assume motion is planar (perspective transform,

aka homography)

Two processing stages

- Estimate the motions and initial layer estimates
- Compute optimal layer estimates (for known motion)

Dominant motion estimation

- Stabilize sequence by dominant motionrobus

t affine Bergen et al. 92 Szeliski Shum

Dominant layer estimate

- How do we form composite (estimate)?

Average?

Median?

- Hint all layers are non-negative

Min-composite

- Smallest value is over-estimate of layer

Difference sequence

- Subtract min-composite from original image

?

original - min composite

difference image

Min composite

Intensity

Time

(overestimate of background layer)

Difference sequence

(underestimate of foreground layer)

Stabilizing secondary motion

Max-composite

Largest value is under-estimate of layer

Min-max alternation

- Subtract secondary layer (under-estimate) from

original sequence - Re-compute dominant motion and better

min-composite - Iterate
- Does this process converge?

Min-max alternation

- Does this process converge?
- in theory yes
- each iteration reduces number of mis-estimated

pixels (tightens the bounds) proof in paper

Min-max alternation

- Does this process converge?
- in practice no
- resampling errors and noise both lead to

divergence discussion in paperresampling

error noisy

Two processing stages

- Estimate the motions and initial layer estimates
- Compute optimal layer estimates (for known motion)

Optimal estimation

- Recall additive mixing of positive signals
- mk ?l Wkl fl
- Use constrained least squares(quadratic

programming) - min ?k ?l Wkl fl mk 2 s.t. fl ? 0

Least squares example

background foreground

Uniqueness of solution

- If any layer does not have a black region,

i.e., if fl ? c, then can add this offset to

another layer (and subtract it from fl)

background foreground

Degeneracies in solution

- If motion is degenerate (e.g., horizontal),

regions (scanlines) decouple (w/o MRF)

mixed

Noise sensitivity

- In general, low-frequency components hard to

recover for small motions

? recovered ?

mixed

? scaled errors ?

Three-layer example

- 3 layers with general motion works well

Complete algorithm

- Dominant motion with min-composites
- Difference (residual) images
- Non-dominant motion on differences
- Improve the motion estimates
- Unconstrained least-squares problem
- Constrained least-squares problem

Complete example

original

stabilized

Complete example

difference

stabilized

Final Results

Another example

- original stabilized min-comp. resid.? 2

Results Anne and books

original background foreground (photo)

Transparent layer recovery

- Pure (additive) mixing of intensities
- simple constrained least squares problem
- degeneracies for simple or small motions
- Processing stages
- dominant motion estimation
- min- and max-composites to initialize
- optimization of motion and layers

Future work

- Mitigating degeneracies (regularization)
- Opaque layers (? estimation)
- Non-planar geometry (parallax)

Bibliography

- L. Williams. Pyramidal parametrics.
- Computer Graphics, 17(3)1--11, July 1983.
- L. G. Brown. A survey of image registration

techniques. - Computing Surveys, 24(4)325--376, December 1992.
- C. D. Kuglin and D. C. Hines. The phase

correlation image alignment method. - In IEEE 1975 Conference on Cybernetics and

Society, pages - 163--165, New York, September 1975.
- J. Gomes, L. Darsa, B. Costa, and L. Velho.

Warping and Morphing of Graphical Objects. - Morgan Kaufmann Publishers, San Francisco Altos,

California, 1999. - G. M. Nielson. Scattered data modeling.
- IEEE Computer Graphics and Applications,

13(1)60--70, January 1993. - T. Beier and S. Neely. Feature-based image

metamorphosis. - Computer Graphics (SIGGRAPH'92), 26(2)35--42,

July 1992.

Bibliography

- J. R. Bergen, P. Anandan, K. J. Hanna, and R.

Hingorani. Hierarchical model-based motion

estimation. In ECCV92, pp. 237252, Italy, May

1992. - M. J. Black and P. Anandan. The robust estimation

of multiple motions Parametric and

piecewise-smooth flow fields. Comp. Vis. Image

Understanding, 63(1)75104, 1996. - H. S. Sawhney and S. Ayer. Compact representation

of videos through dominant multiple motion

estimation. IEEE Trans. Patt. Anal. Mach. Intel.,

18(8)814830, Aug. 1996. - Y. Weiss. Smoothness in layers Motion

segmentation using nonparametric mixture

estimation. In CVPR97, pp. 520526, June 1997.

Bibliography

- J. Y. A. Wang and E. H. Adelson. Representing

moving images with layers. IEEE Transactions on

Image Processing, 3(5)625--638, September 1994.

- Y. Weiss and E. H. Adelson. A unified mixture

framework for motion segmentation Incorporating

spatial coherence and estimating the number of

models. In IEEE Computer Society Conference on

Computer Vision and Pattern Recognition

(CVPR'96), pages 321--326, San Francisco,

California, June 1996. - Y. Weiss. Smoothness in layers Motion

segmentation using nonparametric mixture

estimation. In IEEE Computer Society Conference

on Computer Vision and Pattern Recognition

(CVPR'97), pages 520--526, San Juan, Puerto Rico,

June 1997. - P. R. Hsu, P. Anandan, and S. Peleg. Accurate

computation of optical flow by using layered

motion representations. In Twelfth International

Conference on Pattern Recognition (ICPR'94),

pages 743--746, Jerusalem, Israel, October 1994.

IEEE Computer Society Press

Bibliography

- T. Darrell and A. Pentland. Cooperative robust

estimation using layers of support. IEEE

Transactions on Pattern Analysis and Machine

Intelligence, 17(5)474--487, May 1995. - S. X. Ju, M. J. Black, and A. D. Jepson. Skin

and bones Multi-layer, locally affine, optical

flow and regularization with transparency. In

IEEE Computer Society Conference on Computer

Vision and Pattern Recognition (CVPR'96), pages

307--314, San Francisco, California, June 1996. - M. Irani, B. Rousso, and S. Peleg. Computing

occluding and transparent motions. International

Journal of Computer Vision, 12(1)5--16, January

1994. - H. S. Sawhney and S. Ayer. Compact

representation of videos through dominant

multiple motion estimation. IEEE Transactions on

Pattern Analysis and Machine Intelligence,

18(8)814--830, August 1996. - M.-C. Lee et al. A layered video object coding

system using sprite and affine motion model.

IEEE Transactions on Circuits and Systems for

Video Technology, 7(1)130--145, February 1997.

Bibliography

- S. Baker, R. Szeliski, and P. Anandan. A layered

approach to stereo reconstruction. In IEEE

CVPR'98, pages 434--441, Santa Barbara, June

1998. - R. Szeliski, S. Avidan, and P. Anandan. Layer

extraction from multiple images containing

reflections and transparency. In IEEE CVPR'2000,

volume 1, pages 246--253, Hilton Head Island,

June 2000. - J. Shade, S. Gortler, L.-W. He, and R. Szeliski.

Layered depth images. In Computer Graphics

(SIGGRAPH'98) Proceedings, pages 231--242,

Orlando, July 1998. ACM SIGGRAPH. - S. Laveau and O. D. Faugeras. 3-d scene

representation as a collection of images. In

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