Motion estimation
Digital Visual Effects Yung-Yu Chuang
with slides by Michael Black and P. Anandan
Motion estimation
• Parametric motion (image alignment)
• Tracking
• Optical flow
Parametric motion
direct method for image stitching
Tracking
Optical flow
Three assumptions
• Brightness consistency
• Spatial coherence
• Temporal persistence
Brightness consistency
Image measurement (e.g. brightness) in a small region remain the same although their location may change.
Spatial coherence
• Neighboring points in the scene typically belong to the same surface and hence typically have similar motions.
• Since they also project to nearby pixels in the image, we expect spatial coherence in image flow.
Temporal persistence
The image motion of a surface patch changes gradually over time.
Image registration
Goal: register a template image T(x) and an input image I(x), where x=(x,y)T. (warp I so that it
matches T)
Image alignment: I(x) and T(x) are two images
Tracking: T(x) is a small patch around a point p in the image at t. I(x) is the image at time t+1.
Optical flow: T(x) and I(x) are patches of images at t and t+1.
T fixed
I warp
Simple approach (for translation)
• Minimize brightness difference
y x
y x T v
y u x
I v
u E
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Simple SSD algorithm
For each offset (u, v) compute E(u,v);
Choose (u, v) which minimizes E(u,v);
Problems:
• Not efficient
• No sub-pixel accuracy
Lucas-Kanade algorithm
Newton’s method
• Root finding for f(x)=0
• March x and test signs
• Determine Δx (small slow; large miss)→ →
Newton’s method
• Root finding for f(x)=0
Newton’s method
• Root finding for f(x)=0 Taylor’s expansion:
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Newton’s method
• Root finding for f(x)=0
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Newton’s method
pick up x=x0 iterate
compute
update x by x+Δx until converge
Finding root is useful for optimization because Minimize g(x) find root for → f(x)=g’(x)=0
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x f
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Lucas-Kanade algorithm
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Lucas-Kanade algorithm
iterate
shift I(x,y) with (u,v)
compute gradient image Ix, Iy
compute error image T(x,y)-I(x,y) compute Hessian matrix
solve the linear system (u,v)=(u,v)+(∆u,∆v)
until converge
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Parametric model
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x
x p)
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translation W(x;
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affine W(x;
Our goal is to find p to minimize E(p) for all x in T’s domain
Parametric model
x
x Δp)
p
W(x; ) ( ) 2
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minimize I
with respect to Δp
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W(x;
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minimize I
Parametric model
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p) W W(x;
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image gradient
Jacobian of the warp warped image
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n x x
x
y x
p W p
W p
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Jacobian matrix
• The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
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Parametric model
x
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image gradient
Jacobian of the warp warped image
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Jacobian of the warp
For example, for affine
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Parametric model
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2
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I
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p) W p W(x;
W ( ) ( )
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T
x
p) W(x;
p x H W
Δp 1 I T( ) I( )
T
x p
W p
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(Approximated) Hessian
min
Δparg
Lucas-Kanade algorithm
iterate
1) warp I with W(x;p)
2) compute error image T(x,y)-I(W(x,p)) 3) compute gradient image with W(x,p) 4) evaluate Jacobian at (x;p)
5) compute
6) compute Hessian 7) compute
8) solve
9) update p by p+
until converge
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p) W(x;
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Coarse-to-fine strategy
J warp Jw refine I
ain
a
+
J warp Jw refine I
a
a
+ J
pyramid construction
J warp Jw refine I
a
+
I
pyramid construction
a
outApplication of image alignment
Direct vs feature-based
• Direct methods use all information and can be very accurate, but they depend on the fragile
“brightness constancy” assumption.
• Iterative approaches require initialization.
• Not robust to illumination change and noise images.
• In early days, direct method is better.
• Feature based methods are now more robust and potentially faster.
• Even better, it can recognize panorama without initialization.
Tracking
Tracking
I(x,y,t) (u, v) I(x+u,y+v,t+1)
Tracking
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I optical flow constraint equation brightness constancy
Optical flow constraint equation
Multiple constraints
Area-based method
• Assume spatial smoothness
Area-based method
• Assume spatial smoothness
y x
t y
xu I v I
I v
u E
,
) 2
, (
Area-based method
must be invertible
Area-based method
• The eigenvalues tell us about the local image structure.
• They also tell us how well we can estimate the flow in both directions.
• Link to Harris corner detector.
Textured area
Edge
Homogenous area
KLT tracking
• Select features by
• Monitor features by measuring dissimilarity
, ) ( 1 2 min
Aperture problem
Aperture problem
Aperture problem
Demo for aperture problem
• http://www.sandlotscience.com/Distortions/Br eathing_Square.htm
• http://www.sandlotscience.com/Ambiguous/Ba rberpole_Illusion.htm
Aperture problem
• Larger window reduces ambiguity, but easily violates spatial smoothness assumption
KLT tracking
http://www.ces.clemson.edu/~stb/klt/
KLT tracking
http://www.ces.clemson.edu/~stb/klt/
SIFT tracking (matching actually)
Frame 0 Frame 10
SIFT tracking
Frame 0 Frame 100
SIFT tracking
Frame 0 Frame 200
KLT vs SIFT tracking
• KLT has larger accumulating error; partly
because our KLT implementation doesn’t have affine transformation?
• SIFT is surprisingly robust
• Combination of SIFT and KLT (example)
http://www.frc.ri.cmu.edu/projects/buzzard/smalls/
Rotoscoping (Max Fleischer 1914)
1937
Tracking for rotoscoping
Tracking for rotoscoping
Waking life (2001)
A Scanner Darkly (2006)
• Rotoshop, a proprietary software. Each minute of animation required 500 hours of work.
Optical flow
Single-motion assumption
Violated by
• Motion discontinuity
• Shadows
• Transparency
• Specular reflection
• …
Multiple motion
Multiple motion
Simple problem: fit a line
Least-square fit
Least-square fit
Robust statistics
• Recover the best fit for the majority of the data
• Detect and reject outliers
Approach
Robust weighting
Truncated quadratic
Robust weighting
Geman & McClure
Robust estimation
Fragmented occlusion
Regularization and dense optical flow
• Neighboring points in the scene typically belong to the same surface and hence typically have similar motions.
• Since they also project to nearby pixels in the image, we expect spatial coherence in image flow.
Input image Horizontal motion
Vertical motion
Application of optical flow
video
matching
Input for the NPR algorithm
Brushes
Edge clipping
Gradient
Smooth gradient
Textured brush
Edge clipping
Temporal artifacts
Frame-by-frame application of the NPR algorithm
Temporal coherence
References
• B.D. Lucas and T. Kanade, An Iterative Image Registration Technique with an Application to Stereo Vision, Proceedings of the 1981 DARPA Image
Understanding Workshop, 1981, pp121-130.
• Bergen, J. R. and Anandan, P. and Hanna, K. J. and Hingorani, R., Hierarchical Model-Based Motion Estimation, ECCV 1992, pp237-252.
• J. Shi and C. Tomasi, Good Features to Track, CVPR 1994, pp593-600.
• Michael Black and P. Anandan, The Robust Estimation of Multiple Motions:
Parametric and Piecewise-Smooth Flow Fields, Computer Vision and Image Understanding 1996, pp75-104.
• S. Baker and I. Matthews, Lucas-Kanade 20 Years On: A Unifying
Framework, International Journal of Computer Vision, 56(3), 2004, pp221 - 255.
• Peter Litwinowicz, Processing Images and Video for An Impressionist Effects, SIGGRAPH 1997.
• Aseem Agarwala, Aaron Hertzman, David Salesin and Steven Seitz,
Keyframe-Based Tracking for Rotoscoping and Animation, SIGGRAPH 2004, pp584-591.
