CSCE 643 Computer Vision: Lucas-Kanade Registration

transcript

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 separatelyThen will consider motion of entire image

Problem Definition: Optical Flow

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

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

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?

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:

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:

H(x,y) - I(x+u,v+y) = 0

Optical Flow EquationCombining 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?

Intuitively, what does this constraint mean?

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

Ambiguity

Stripes moved upwards 6 pixels

Stripes moved left 5 pixels

Ambiguity

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

Solution: solve least squares problem

Lukas-Kanade Flow

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 1 and 2 of ATA should not be too small• ATA should be well-conditioned

– 1/ 2 should not be too large (1 = larger eigenvalue)

Lukas-Kanade Flow

Look familiar?

Lukas-Kanade Flow

Look familiar? Harris Corner detection criterion!

Bad for motion estimation

- large1, small 2

Low Texture Region

Bad for motion estimation: - gradients have small magnitude

- small1, small 2

High Textured Region

Good for motion estimation: - gradients are different, large magnitudes

- large1, large 2

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

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.

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?

Iterative RefinementIterative Lukas-Kanade Algorithm

1. Estimate velocity at each pixel by solving Lucas-Kanade equations

2. Warp H towards I using the estimated flow field- use image warping techniques

3. Repeat until convergence

Idea I: Iterative RefinementIterative Lukas-Kanade Algorithm

1. Estimate velocity at each pixel by solving Lucas-Kanade equations

2. Warp H towards I using the estimated flow field- use image warping techniques

3. Repeat until convergence

Idea II: Reduce the Resolution!

image Iimage H

Gaussian pyramid of image H Gaussian pyramid of image I

image Iimage H u=10 pixels

u=5 pixels

u=2.5 pixels

u=1.25 pixels

Coarse-to-fine Motion Estimation

image Iimage J

Gaussian pyramid of image H Gaussian pyramid of image I

image Iimage H

Coarse-to-fine Optical Flow Estimation

run iterative L-K

Upsample & warp

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:

);();('

pxwpxwx

Transformed ImageInput Image

Geometric Image Warping

w(x;p) describes the geometric relationship between two images:

);();('

pxwpxwx

Transformed ImageInput Image

(x’)

Warping parameters

Warping Functions

Translation:

Affine:

Perspective:

1);(py

pypxpypxp

321);(pypxp

pypxppxw

Image Registration

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

Image Registration

Again, we can formulate this as an optimization problem:

pxHpxwI 2)());((minarg

Image Registration

The above problem can be solved by many gradient-based optimization algorithms:

- Steepest descent

- Gauss-newton

- Levenberg-marquardt, etc.

Image Registration

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

xHppxwI 2)());((minarg Taylor series expansion

Similar to optical flow:

Image Registration

xHppxwI 2)());((minarg

WIpxwI

)());((minarg

Taylor series expansion

Image Registration

WIpxwI

)());((minarg

Image gradient

Image Registration

xHpxwI 2)());((minarg

WIpxwI

)());((minarg

Image gradient

translation

affine

……

Jacobian matrix

Image Registration

WIpxwI

)());((minarg

- Intuition?

Image Registration

WIpxwI

)());((minarg

- Intuition: a delta change of p results in how much change of pixel values at pixel w(x;p)!

Image Registration

WIpxwI

)());((minarg

- 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

WIpxwI

)());((minarg

pxwIxHpp

)));(()((minarg

Rearrange

WIpxwI

)());((minarg

pxwIxHpp

)));(()((minarg

Rearrange

WIpxwI

)());((minarg

pxwIxHpp

)));(()((minarg

)));(()((()( 1 pxwIxHp

Rearrange

WIpxwI

)());((minarg

(ATA)-1

Lucas-Kanade Registration

Initialize p=p0:

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

));(()(()( 1 pxwIxHp

Initialize p=p0:

2. Compute the error image

));(()(()( 1 pxwIxHp

Initialize p=p0:

3. Warp the gradient with w(x;p)

));(()(()( 1 pxwIxHp

Initialize p=p0:

4. Evaluate the Jacobian at (x;p)

));(()(()( 1 pxwIxHp

Initialize p=p0:

5. Compute the using linear system solvers

));(()(()( 1 pxwIxHp

Initialize p=p0:

5. Compute the using linear system solvers

6. Update the parameters

));(()(()( 1 pxwIxHp

Lucas-Kanade Algorithm

Iteration 1: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?

Issue: corresponding pixels do not have consistent values due to illumination changes?

- linear models

- can model gain and bias (H1=H0, H2= const., other zeros)

iii xHxHxH

10 )()()(

cxHxH )()( 0

Linear ModelCan 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

iii xHxHxH

10 )()()(

- Images under unknown illuminations can be represented as a weighted combination of precomputed illumination image templates

iii xHxHxH

10 )()()(

Mean Eigen vectors

- Images under unknown illuminations can be represented as a weighted combination of precomputed illumination image template

iii xHxHxH

10 )()()(

Mean Eigen vectors

Unknown weights/parameters

mean face

lighting variation

Image Registration

Similarly, we can formulate this as an optimization problem:

pxHpxwApxwA

,)());(());((minarg

Image Registration

pxHpxwApxwA

,)());(());((minarg

Geometric warping Illumination variations

Image Registration

pxHpxwApxwA

,)());(());((minarg

pxHppxwAppxwA

,)());(()());((minarg

For iterative registration, we have

Image Registration

pxHpxwApxwA

,)());(());((minarg

pxHppxwAppxwA

,)());(()());((minarg

orderhighxHpp

WAxApxwAp

WApxwA

,)())(());(());((minarg

For iterative registration, we have

Gauss-newton optimization

WApxwApxwAp

WApxwA

,)()));((());(());((minarg

WApxwApxwAp

WApxwA

,)()));((());(());((minarg

pxHpxwApxwAp

WApxwAp

,)());(());(());((minarg

WApxwApxwAp

WApxwA

,)()));((());(());((minarg

pxHpxwApxwAp

WApxwAp

,)());(());(());((minarg

let similarly

WApxwApxwAp

WApxwA

,)()));((());(());((minarg

pxHpxwApxwAp

WApxwAp

,)());(());(());((minarg

,minarg

let similarly

WApxwApxwAp

WIpxwA

,)()));((());(());((minarg

pxHpxwApxwAp

WApxwAp

)());(());(());((minarg

,minarg

);(()),...,;((,) 11

0 pxwApxwAp

WAAJ B

let similarly

Jacobian matrix Error image

iii pxwApxwAxHb

10 ));(());(()(

WApxwApxwAp

WIpxwA

,)()));((());(());((minarg

pxHpxwApxwAp

WApxwAp

)());(());(());((minarg

,minarg

);(()),...,;((,) 11

0 pxwApxwAp

WAAJ B

let similarly

Jacobian matrix Error image

iii pxwApxwAxHb

10 ));(());(()(

bJJJq TT 1)( 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

http://www2.imm.dtu.dk/~aam/datasets/datasets.html

• 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

A long vector stacking positions of vertices

…,sn

Mean shape

…,sn

Eigen vectors

…,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

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

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)

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

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)

CSCE 643 Computer Vision: Lucas-Kanade Registration

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