# CSCE 643 Computer Vision: Lucas-Kanade Registration

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CSCE 643 Computer Vision: Lucas-Kanade Registration
Jinxiang Chai

Appearance-based Tracking
Slide from Robert Collins

Image Registration This requires solving image registration problems
Lucas-Kanade is one of the most popular frameworks for image registration - gradient based optimization - iterative linear system solvers - applicable to a variety of scenarios, including optical flow estimation, parametric motion tracking, AAMs, etc.

Pixel-based Registration: Optical flow
Will start by estimating motion of each pixel separately Then will consider motion of entire image

Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?

Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I? Solve pixel correspondence problem given a pixel in H, look for nearby pixels of the same color in I

Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I? Solve pixel correspondence problem given a pixel in H, look for nearby pixels of the same color in I Key assumptions color constancy: a point in H looks the same in I For grayscale images, this is brightness constancy small motion: points do not move very far This is called the optical flow problem

Optical Flow Constraints
Let’s look at these constraints more closely brightness constancy: Q: what’s the equation? A: H(x,y) = I(x+u, y+v)

Optical Flow Constraints
Let’s look at these constraints more closely brightness constancy: Q: what’s the equation? A: H(x,y) = I(x+u, y+v) H(x,y) - I(x+u,v+y) = 0

Optical Flow Constraints
Let’s look at these constraints more closely brightness constancy: Q: what’s the equation? A: H(x,y) = I(x+u, y+v) H(x,y) - I(x+u,v+y) = 0 small motion: (u and v are less than 1 pixel) suppose we take the Taylor series expansion of I:

Optical Flow Constraints
Let’s look at these constraints more closely brightness constancy: Q: what’s the equation? A: H(x,y) = I(x+u, y+v) H(x,y) - I(x+u,v+y) = 0 small motion: (u and v are less than 1 pixel) suppose we take the Taylor series expansion of I:

Optical Flow Equation Combining these two equations

Optical Flow Equation Combining these two equations

Optical Flow Equation Combining these two equations

Optical Flow Equation Combining these two equations

Optical Flow Equation Combining these two equations
In the limit as u and v go to zero, this becomes exact

Optical Flow Equation How many unknowns and equations per pixel?
A: u and v are unknown, 1 equation

Optical Flow Equation How many unknowns and equations per pixel?
Intuitively, what does this constraint mean? A: u and v are unknown, 1 equation

Optical Flow Equation How many unknowns and equations per pixel?
Intuitively, what does this constraint mean? The component of the flow in the gradient direction is determined The component of the flow parallel to an edge is unknown A: u and v are unknown, 1 equation

Optical Flow Equation How many unknowns and equations per pixel?
Intuitively, what does this constraint mean? The component of the flow in the gradient direction is determined The component of the flow parallel to an edge is unknown A: u and v are unknown, 1 equation

Ambiguity

Ambiguity Stripes moved upwards 6 pixels Stripes moved left 5 pixels

Ambiguity How to address this problem? Stripes moved upwards 6 pixels
Stripes moved left 5 pixels How to address this problem?

Solving the Aperture Problem
How to get more equations for a pixel? Basic idea: impose additional constraints most common is to assume that the flow field is smooth locally one method: pretend the pixel’s neighbors have the same (u,v) If we use a 5x5 window, that gives us 25 equations per pixel!

RGB Version How to get more equations for a pixel?
Basic idea: impose additional constraints most common is to assume that the flow field is smooth locally one method: pretend the pixel’s neighbors have the same (u,v) If we use a 5x5 window, that gives us 25 equations per pixel!

Lukas-Kanade Flow Prob: we have more equations than unknowns

Lukas-Kanade Flow Prob: we have more equations than unknowns
Solution: solve least squares problem

Lukas-Kanade Flow Prob: we have more equations than unknowns
Solution: solve least squares problem minimum least squares solution given by solution (in d) of:

Lukas-Kanade Flow The summations are over all pixels in the K x K window This technique was first proposed by Lukas & Kanade (1981)

Lukas-Kanade Flow When is this Solvable? ATA should be invertible
ATA should not be too small due to noise eigenvalues l1 and l2 of ATA should not be too small ATA should be well-conditioned l1/ l2 should not be too large (l1 = larger eigenvalue)

Lukas-Kanade Flow When is this Solvable? ATA should be invertible
ATA should not be too small due to noise eigenvalues l1 and l2 of ATA should not be too small ATA should be well-conditioned l1/ l2 should not be too large (l1 = larger eigenvalue) Look familiar?

Lukas-Kanade Flow When is this Solvable? ATA should be invertible
ATA should not be too small due to noise eigenvalues l1 and l2 of ATA should not be too small ATA should be well-conditioned l1/ l2 should not be too large (l1 = larger eigenvalue) Look familiar? Harris Corner detection criterion!

Edge Bad for motion estimation - large l1, small l2

Low Texture Region Bad for motion estimation:
- gradients have small magnitude - small l1, small l2

High Textured Region Good for motion estimation:
- gradients are different, large magnitudes - large l1, large l2

Good Features to Track This is a two image problem BUT
Can measure sensitivity by just looking at one of the images! This tells us which pixels are easy to track, which are hard very useful later on when we do feature tracking... For more detail, check “Good feature to Track”, Shi and Tomasi, CVPR 1994

What are the potential causes of errors in this procedure? Suppose ATA is easily invertible Suppose there is not much noise in the image

What are the potential causes of errors in this procedure? Suppose ATA is easily invertible Suppose there is not much noise in the image When our assumptions are violated Brightness constancy is not satisfied The motion is not small A point does not move like its neighbors Optical flow in local window is not constant.

What are the potential causes of errors in this procedure? Suppose ATA is easily invertible Suppose there is not much noise in the image When our assumptions are violated Brightness constancy is not satisfied The motion is not small A point does not move like its neighbors Optical flow in local window is not constant.

Revisiting the Small Motion Assumption
Is this motion small enough? Probably not—it’s much larger than one pixel (2nd order terms dominate) How might we solve this problem?

Estimate velocity at each pixel by solving Lucas-Kanade equations Warp H towards I using the estimated flow field - use image warping techniques Repeat until convergence

Idea I: Iterative Refinement
Iterative Lukas-Kanade Algorithm Estimate velocity at each pixel by solving Lucas-Kanade equations Warp H towards I using the estimated flow field - use image warping techniques Repeat until convergence

Idea II: Reduce the Resolution!

Coarse-to-fine Motion Estimation
Gaussian pyramid of image H Gaussian pyramid of image I image I image H u=10 pixels u=5 pixels u=2.5 pixels u=1.25 pixels image H image I

Coarse-to-fine Optical Flow Estimation
Gaussian pyramid of image H Gaussian pyramid of image I image I image H run iterative L-K Upsample & warp run iterative L-K . image J image I

What are the potential causes of errors in this procedure? Suppose ATA is easily invertible Suppose there is not much noise in the image When our assumptions are violated Brightness constancy is not satisfied The motion is not small A point does not move like its neighbors Optical flow in local window is not constant.

Lucas Kanade Tracking Assumption of constant flow (pure translation) for all pixels in a larger window might be unreasonable

Lucas Kanade Tracking Assumption of constant flow (pure translation) for all pixels in a larger window might be unreasonable However, we can easily generalize Lucas-Kanade approach to other 2D parametric motion models (like affine or projective)

Beyond Translation So far, our patch can only translate in (u,v)
What about other motion models? rotation, affine, perspective

Warping Review Figure from Szeliski book

Geometric Image Warping
w(x;p) describes the geometric relationship between two images: x x’ Input Image Transformed Image

Geometric Image Warping
w(x;p) describes the geometric relationship between two images: (x) (x’) Input Image Transformed Image Warping parameters

Warping Functions Translation: Affine: Perspective:

Image Registration Find the warping parameter p that minimizes the intensity difference between template image and the warped input image

Image Registration Find the warping parameter p that minimizes the intensity difference between template image and the warped input image Again, we can formulate this as an optimization problem:

Image Registration Find the warping parameter p that minimizes the intensity difference between template image and the warped input image Again, we can formulate this as an optimization problem: The above problem can be solved by many gradient-based optimization algorithms: - Steepest descent - Gauss-newton - Levenberg-marquardt, etc.

Image Registration Find the warping parameter p that minimizes the intensity difference between template image and the warped input image Again, we can formulate this as an optimization problem: The above problem can be solved by many gradient-based optimization algorithms: - Steepest descent - Gauss-newton - Levenberg-marquardt, etc

Image Registration Mathematically, we can formulate this as an optimization problem:

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion Image gradient

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion translation affine Image gradient Jacobian matrix ……

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion - Intuition?

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion - Intuition: a delta change of p results in how much change of pixel values at pixel w(x;p)!

Image Registration Mathematically, we can formulate this as an optimization problem: Similar to optical flow: Taylor series expansion - Intuition: a delta change of p results in how much change of pixel values at pixel w(x;p)! - An optimal that minimizes color inconsistency between the images.

Gauss-newton Optimization
Rearrange

Gauss-newton Optimization
Rearrange

Gauss-newton Optimization
Rearrange A b

Gauss-newton Optimization
Rearrange A b (ATA)-1 ATb

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p))

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p)) 2. Compute the error image

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p)) 2. Compute the error image 3. Warp the gradient with w(x;p)

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p)) 2. Compute the error image 3. Warp the gradient with w(x;p) 4. Evaluate the Jacobian at (x;p)

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p)) 2. Compute the error image 3. Warp the gradient with w(x;p) 4. Evaluate the Jacobian at (x;p) 5. Compute the using linear system solvers

Initialize p=p0: Iterate: 1. Warp I with w(x;p) to compute I(w(x;p)) 2. Compute the error image 3. Warp the gradient with w(x;p) 4. Evaluate the Jacobian at (x;p) 5. Compute the using linear system solvers 6. Update the parameters

Iteration 1: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 2: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 3: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 4: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 5: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 6: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 7: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 8: H(x) I(w(x;p)) H(x)-I(w(x;p))

Iteration 9: H(x) I(w(x;p)) H(x)-I(w(x;p))

Final result: H(x) I(w(x;p)) H(x)-I(w(x;p))

How to Break Assumptions
Small motion Constant optical flow in the window Color constancy

Break the Color Constancy
How to deal with illumination change?

Break the Color Constancy
How to deal with illumination change? Issue: corresponding pixels do not have consistent values due to illumination changes?

Break the Color Constancy
How to deal with illumination change? - linear models - can model gain and bias (H1=H0, H2= const., other zeros)

Linear Model Can also model the appearance of a face under different illumination using a linear combination of base images (PCA): recording images under different illumination applying PCA to recorded images to model illumination in recorded images Images under unknown illuminations can be represented as a weighted combination of precomputed illumination image templates

Linear Model Can also model the appearance of a face under different illumination using a linear combination of base images (PCA): recording images under different illumination applying PCA to recorded images to model illumination in recorded images Images under unknown illuminations can be represented as a weighted combination of precomputed illumination image templates Mean Eigen vectors

Linear Model Can also model the appearance of a face under different illumination using a linear combination of base images (PCA): recording images under different illumination applying PCA to recorded images to model illumination in recorded images Images under unknown illuminations can be represented as a weighted combination of precomputed illumination image template Unknown weights/parameters Mean Eigen vectors

Linear Model Can also model the appearance of a face under different illumination using a linear combination of base images (PCA): mean face lighting variation

Image Registration Similarly, we can formulate this as an optimization problem:

Image Registration Similarly, we can formulate this as an optimization problem: Geometric warping Illumination variations

Image Registration Similarly, we can formulate this as an optimization problem: For iterative registration, we have

Image Registration Similarly, we can formulate this as an optimization problem: For iterative registration, we have Taylor series expansion

Gauss-newton optimization

Gauss-newton optimization

Gauss-newton optimization
let similarly

Gauss-newton optimization
let similarly

Gauss-newton optimization
let similarly Jacobian matrix Error image

Gauss-newton optimization
let similarly Jacobian matrix Error image Update equation:

Results with Illumination Changes
[Hagar and Belhumeur 98]

Applications: 2D Face Registration
Face registration using active appearance models

AAM for Image registration
Goal: automatic detection of facial features from a single image

AAM for Image registration
Goal: automatic detection of facial features from a single image Solution: register input image against a template constructed from a prerecorded facial image database

AAM: database construction
Construct a database of images (e.g., faces) with variations

AAM: Feature Labeling Label all database images by identifying key facial features

AAM: Feature Labeling Label all database images by identifying key facial features So how to build a template based on labeled database images?

Key idea Decouple image variation into shape variation and appearance variation Model each of them using PCA A combined model consists of a linear shape model and a linear appearance model

Shape Variation

Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn

Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn A long vector stacking positions of vertices

Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn Mean shape

Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn Eigen vectors

Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn Shape parameter

Appearance Variation modeling
A linear appearance model consists of a base appearance image A0 defined on the pixels inside the base mesh s0 plus a linear combination of m appearance images Ai also defined on the same set of pixels.

Appearance Variation modeling
A linear appearance model consists of a base appearance image A0 defined on the pixels inside the base mesh s0 plus a linear combination of m appearance images Ai also defined on the same set of pixels. Defined in base mesh (mean shape)!

Model Instantiations A new image can be generated via AAM

Image Registration with AAM
Analysis by synthesis: estimate the optimal parameters by minimizing the difference between input image and synthesized image 2 - min p Input image Synthesized image

Image Registration with AAM
Analysis by synthesis: estimate the optimal parameters by minimizing the difference between input image and synthesized image Solve the problem with iterative linear system solvers - for details, check “active appearance model revisited”, Iain Matthews and Simon Baker , IJCV 2004 2 - min p Input image Synthesized image

Iterative Approach

Pros and Cons of AAM Registration
It can register facial images from different peoples, different facial expressions and different illuminations The quality of results heavily depends on training datasets Gradient-based optimization is prone to local minima It often fails when face is under extreme deformation, pose, or illumination Needs to figure out a better way to measure the distance between input image and template image (e.g., gradients and edges)