Computer Vision Optical Flow Some slides from K. H. Shafique [ and T. Darrell.

Computer Vision

Optical Flow

Some slides from K. H. Shafique [http://www.cs.ucf.edu/courses/cap6411/cap5415/] and T. Darrell

Bahadir K. Gunturk EE 7730 - Image Analysis II 2

Correspondence

Which pixel went where?

Time: t Time: t + dt


Motion Field vs. Optical Flow

Scene flow: 3D velocities of scene points. Motion field: 2D projection of scene flow. Optical flow: Approximation of motion field derived

from apparent motion of brightness patterns in image.


Motion Field vs. Optical Flow

Consider perfectly uniform sphere rotating in front of camera.

Motion field follows surface points. Optical flow is zero since motion is not visible.

Now consider stationary sphere with moving light source.

Motion field is zero. But optical flow results from changing shading.

But, in general, optical flow is a reliable indicator of motion field.


Applications

Object tracking Video compression Structure from motion Segmentation Correct for camera jitter (stabilization) Combining overlapping images (panoramic image

construction) …


Optical Flow Problem

How to estimate pixel motion from one image to another?

H I


Computing Optical Flow

Assumption 1: Brightness is constant.

Assumption 2: Motion is small.

( , ) ( , )H x y I x u y v

(from Taylor series expansion)



Combine

tI

0 t x yI I u I v

In the limit as u and v goes to zero, the equation becomes exact

(optical flow equation)



At each pixel, we have one equation, two unknowns.

This means that only the flow component in the gradient direction can be determined.

0 t x yI I u I v (optical flow equation)

The motion is parallel to the edge, and it cannot be determined.

This is called the aperture problem.



We need more constraints. The most commonly used assumption is that optical

flow changes smoothly locally.



One method: The (u,v) is constant within a small neighborhood of a pixel.

0 t x yI I u I v Optical flow equation:

Use a 5x5 window:

Two unknowns, 25 equation !



Find minimum least squares solution

Lucas – Kanade method



Lucas – Kanade

When is This Solvable?• ATA should be invertible • ATA should not be too small. Other wise noise will be

amplified when inverted.)


Computing Optical Flow What are the potential causes of errors in this procedure?

Brightness constancy is not satisfied The motion is not small A point does not move like its neighbors

window size is too large


This is a polynomial root finding problem Can solve using Newton’s method

Also known as Newton-Raphson method

Lucas-Kanade method does one iteration of Newton’s method Better results are obtained via more iterations

Improving accuracy

Recall our small motion assumption

This is not exact To do better, we need to add higher order terms back in:


Iterative Refinement Iterative Lucas-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


Revisiting the small motion assumption

What can we do when the motion is not small?


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 optical flow estimation



image Iimage H

Coarse-to-fine optical flow estimation

run iterative L-K

run iterative L-K

warp & upsample

.

.

.


Multi-resolution Lucas-Kanade Algorithm


Block-Based Motion Estimation





image Iimage H


image Iimage H

Hierarchical search


Parametric (Global) Motion Sometimes few parameters are enough to represent the motion of whole image

translation

scale

rotation


Parametric (Global) Motion

Affine Flow

1 2 1

3 4 2

( , )

( , )

u x y a x a y b

v x y a x a y b

u xA B

v y

2

1

43

21

b

b

y

x

aa

aa

yy

xx

v

u



Affine Flow

1 2 1

3 4 2

( , )

( , )

u x y a x a y b

v x y a x a y b

2

4

3

1

2

1

1000

0001

b

a

a

b

a

a

yx

yx

v

u

2

1

43

21

b

b

y

x

aa

aa

yy

xx

v

u



Affine Flow – Approach

1 1 1 1

1 1 1 2

2 2 2 1

2 2 2 3

43 3 3

23 33

1 0 0 0

0 0 0 1

1 0 0 0

0 0 0 1

1 0 0 0

0 0 0 1

u x y a

v x y a

u x y b

v x y a

au x y

bx yv


Iterative Refinement Iterative Estimation

1. Estimate parameters by solving the linear system.

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

3. Repeat until convergence or a fixed number of iterations


image Iimage J


image Iimage H

Coarse-to-fine global flow estimation

Compute Flow Iteratively

Compute Flow Iteratively

warp & upsample

.

.

.


Global Flow

Application: Mosaic construction

Find featuresMatch featuresFit parametric model


Image Warping

yx, yx ,

2

1

43

21

b

b

y

x

aa

aa

yy

xx

v

u

2

1

43

21

1

1

b

b

y

x

aa

aa

y

x

2

1

1

43

21

1

1

b

b

y

x

aa

aa

y

x


Image Warping

yx,

yx ,

Interpolate to find the intensity at (x’,y’)

Pixel values are known in this image

Pixel values are to found in this image


Other Parametric Motion Models

Perspective

Pseudo-Perspective: Approximation to perspective. (Planar scene + Perspective projection)

1 2 3

7 8

'1

a x a y ax

a x a y

4 5 6

7 8

'1

a x a y ay

a x a y

'

'

u x x

v y y

and

Computer Vision Optical Flow Some slides from K. H. Shafique [ and T. Darrell.

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