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**CSCE 643 Computer Vision: Lucas-Kanade Registration**

Jinxiang Chai

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**Appearance-based Tracking**

Slide from Robert Collins

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**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.

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**Pixel-based Registration: Optical flow**

Will start by estimating motion of each pixel separately Then will consider motion of entire image

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**Problem Definition: Optical Flow**

How to estimate pixel motion from image H to image I?

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**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

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**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

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**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)

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**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

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**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:

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**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:

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Optical Flow Equation Combining these two equations

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Optical Flow Equation Combining these two equations

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Optical Flow Equation Combining these two equations

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Optical Flow Equation Combining these two equations

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**Optical Flow Equation Combining these two equations**

In the limit as u and v go to zero, this becomes exact

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**Optical Flow Equation How many unknowns and equations per pixel?**

A: u and v are unknown, 1 equation

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**Optical Flow Equation How many unknowns and equations per pixel?**

Intuitively, what does this constraint mean? A: u and v are unknown, 1 equation

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**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

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**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

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Ambiguity

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Ambiguity Stripes moved upwards 6 pixels Stripes moved left 5 pixels

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

Stripes moved left 5 pixels How to address this problem?

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**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!

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**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!

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

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

Solution: solve least squares problem

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**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:

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

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**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)

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**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?

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**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!

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Edge Bad for motion estimation - large l1, small l2

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**Low Texture Region Bad for motion estimation:**

- gradients have small magnitude - small l1, small l2

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**High Textured Region Good for motion estimation:**

- gradients are different, large magnitudes - large l1, large l2

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**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

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**Errors in Lucas-Kanade**

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

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**Errors in Lucas-Kanade**

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.

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**Errors in Lucas-Kanade**

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.

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**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?

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**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

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**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

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**Idea II: Reduce the Resolution!**

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**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

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**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

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**Errors in Lucas-Kanade**

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.

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

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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)

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**Beyond Translation So far, our patch can only translate in (u,v)**

What about other motion models? rotation, affine, perspective

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Warping Review Figure from Szeliski book

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**Geometric Image Warping**

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

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**Geometric Image Warping**

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

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Warping Functions Translation: Affine: Perspective:

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

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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:

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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.

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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

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

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

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

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

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

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

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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)!

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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.

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**Gauss-newton Optimization**

Rearrange

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**Gauss-newton Optimization**

Rearrange

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**Gauss-newton Optimization**

Rearrange A b

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**Gauss-newton Optimization**

Rearrange A b (ATA)-1 ATb

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**Lucas-Kanade Registration**

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

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**Lucas-Kanade Registration**

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

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**Lucas-Kanade Registration**

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)

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**Lucas-Kanade Registration**

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)

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**Lucas-Kanade Registration**

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

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**Lucas-Kanade Registration**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**Lucas-Kanade Algorithm**

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

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**How to Break Assumptions**

Small motion Constant optical flow in the window Color constancy

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**Break the Color Constancy**

How to deal with illumination change?

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**Break the Color Constancy**

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

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**Break the Color Constancy**

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

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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

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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

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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

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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

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

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

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

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

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**Gauss-newton optimization**

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**Gauss-newton optimization**

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**Gauss-newton optimization**

let similarly

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**Gauss-newton optimization**

let similarly

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**Gauss-newton optimization**

let similarly Jacobian matrix Error image

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**Gauss-newton optimization**

let similarly Jacobian matrix Error image Update equation:

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**Results with Illumination Changes**

[Hagar and Belhumeur 98]

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**Applications: 2D Face Registration**

Face registration using active appearance models

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**AAM for Image registration**

Goal: automatic detection of facial features from a single image

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**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

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**AAM: database construction**

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

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

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

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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

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Shape Variation

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**Shape Variation modeling**

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

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**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

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**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

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**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

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**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

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**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.

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**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)!

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Model Instantiations A new image can be generated via AAM

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**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

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**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

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Iterative Approach

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**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)

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**Lucas-Kanade for Image Alignment**

Pros: All pixels get used in matching Can get sub-pixel accuracy (important for good mosaicing!) Relatively fast and simple Applicable to optical flow estimation, parametric motion tracking, and AAM registration Cons: Prone to local minima Relative small movement

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**Beyond 2D Tracking/Registration**

So far, we focus on registration between 2D images The same idea can be used in registration between 3D and 2D (model-based tracking) We will go back to this when we talk about model-based 3D tracking (e.g., head, human body and hand gesture)

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