Representation, Description and Matching CS485/685 Computer Vision Dr. George Bebis.

Representation, Description and Matching

CS485/685 Computer Vision

Dr. George Bebis

Matching

• Given covariant region descriptors, what information should we use for matching corresponding regions in different images?

?

Simplest approach: correlation

• Directly compare intensities using “sum of squared differences” or “normalized cross-correlation”

Simplest approach: correlation (cont’d)

• Works satisfactorily when we matching corresponding regions related mostly by translation.– e.g., stereo pairs, video sequence assuming small camera

motion

Simplest approach: correlation (cont’d)

• Sensitive to small variations with respect to:– Location

– Pose

– Scale

– Intra-class variability

• Poorly distinctive!

• We will discuss a powerful descriptor called SIFT

Region Detection Steps: Review

Extract affine regions Normalize regionsEliminate rotational

ambiguityCompute appearance

descriptors

SIFT (Lowe ’04)

FeatureExtraction

• Remember how we resolved the orientation ambiguity?

Scale Invariant Feature Transform (SIFT)

Extract affine regions Normalize regionsEliminate rotational

ambiguityCompute appearance

descriptors

SIFT (Lowe ’04)?

• Find dominant gradient direction using the histograms of gradient direction.


Dominant direction of gradient

0 2

(36 bins)

• Same theory, except that we use 16 histograms (8 bins each).


16 histograms x 8 orientations = 128 features

1. Take a 16 x16 window around detected interest point

2. Divide into a 4x4 grid of cells

3. Compute histogram in each cell

Main idea:

Properties of SIFT

• Highly distinctive!– A single feature can be correctly matched with high probability against a large

database of features from many images.

• Scale and rotation invariant.

• Partially invariant to 3D camera viewpoint– Can tolerate up to about 60 degree out of plane rotation

• Can be computed fast and efficiently

Properties of SIFT (cont’d)

• Partially invariant to changes in illumination

http://people.csail.mit.edu/albert/ladypack/wiki/index.php/Known_implementations_of_SIFT

SIFT – Main Steps

(1) Scale-space extrema detection– Extract scale and rotation invariant interest points (i.e., keypoints).

(2) Keypoint localization– Determine location and scale for each interest point.

(3) Orientation assignment– Assign one or more orientations to each keypoint.

(4) Keypoint descriptor– Use local image gradients at the selected scale.

D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints”, International Journal of Computer Vision, 60(2):91-110, 2004.

Cited 9589 times (as of 3/7/2011)

1. Scale-space Extrema Detection

• Harris-Laplacian

• Find local maxima of:– Harris detector in space – LoG in scale

scale

x

y

Harris

L

oG

• SIFT

Find local maxima of:– DoG in space – DoG in scale

scale

x

y

Hessian

D

oG

σn =knσ0

(k=2)

1. Scale-space Extrema Detection (cont’d)

• DoG images are grouped by octaves-An octave corresponds to doubling the value of σ

• Fixed number of scales (i.e., levels) per octave

σ0

2σ0

22σ0

2 2

( , , ) ( , , )

( 1)

G x y k G x y

k G

( , , )

( , , )* ( , )

L x y

G x y I x y

( , , )

( , , ) ( , , )

D x y

L x y k L x y

Down-sample


σ0

2σ0

kσ0

k2σ0

… • Images separated by a constant factor kk

• If each octave is divided in ss intervals, we have s+1 DoG images/octave where:

kkss=2=2 or k=2 or k=21/s1/s

Note: need (s+1)+2 = s+3 blurred images

2 2( , , ) ( , , ) ( 1)G x y k G x y k G (ks=2)

Choosing SIFT parameters

• Experimentally using a matching task:

- 32 real images (outdoor, faces, aerial etc.)

- Images subjected to a wide range of transformations (i.e., rotation, scaling, shear, change in brightness, noise).

- Keypoints are detected in each image.

- Parameters are chosen based on keypoint repeatability, localization, and matching accuracy.


• How many scales sampled per octave?

3 scales # of keypoints increases but they are not stable!


• Smoothing is applied to the first level of each octave.• How to choose σ? (i.e., integration scale)

σ =1.6


σ

2σ

kσ

k2σ

…• Pre-smoothing discards high frequencies.

• Double the size of the input image(i.e., using linear interpolation) prior tobuilding the first level of the DoG pyramid.

• Increases the number of stable keypointsby a factor of 4.


• Extract local extrema (i.e., minima or maxima) in DoG pyramid.-Compare each point to its 8 neighbors at the same level, 9 neighbors in the level above, and 9 neighbors in the level below (i.e., 26 total).

2. Keypoint Localization

• Determine the location and scale of keypoints to sub-pixel and sub-scale accuracy by fitting a 3D quadratic function at each keypoint.

• Substantial improvement to matching and stability!

( , , )i i i i iX x y X X


• Use Taylor expansion of D(x,y,σ) (i.e., DoG function) around the sample point :

where

is the offset from this point.

2

2

( ) ( )1( ) ( )

2

TTi i

i

D X D XD X D X

( , , )i i iX x x y y

( , , )i i i iX x y


• To find the extrema of D(ΔX):

• ΔX can be computed by solving a 3x3 linear system:

2

2

( ) ( )i iD X D XX

X X

2 2 2

2

2 2 2

2

2 2 2

2

D D D Dy x

D D D Dy

y y yx yx

DD D Dxx yx x

4

)()(

1

2

2

,11

,11

,11

,11

2

,1

,,1

2

2

,1

,1

jik

jik

jik

jik

jik

jik

jik

jik

jik

DDDD

y

D

DDDD

DDD

use finite

differences:

2 1

2

( ) ( )i iD X D XX

X X

2

2

( ) ( )0i iD X D X

XX X

2. Keypoint Localization (cont’d)

If in any dimension, repeat.0.5X

i iX X X

• Sub-pixel, sub-scale interpolated estimate:


• Reject keypoints having low contrast.– i.e., sensitive to noise

If reject keypoint– i.e., assumes that image values have been normalized in [0,1]

| ( ) | 0.03iD X X


• Reject points lying on edges (or being close to edges)

• Harris uses the 2nd order moment matrix:

2

2,

( , ) x x yW

x W y W x y y

f f fA x y

f f f

R(AW) = det(AW) – α trace2(AW)

or R(AW) = λ1 λ2- α (λ1+ λ2)2


• SIFT uses the Hessian matrix for efficiency.– i.e., encodes principal curvatures

α: largest eigenvalue (λmax)β: smallest eigenvalue (λmin)(proportional to principal curvatures)

(SIFT uses r = 10)

(r = α/β)


(a) 233x189 image

(b) 832 DoG extrema

(c) 729 left after low contrast threshold

(d) 536 left after testing ratio based on Hessian

3. Orientation Assignment

• Create histogram of gradient directions, within a region around the keypoint, at selected scale (i.e., scale invariance):

2 2( , ) ( ( 1, ) ( 1, )) ( ( , 1) ( , 1))

( , ) tan 2(( ( , 1) ( , 1)) / ( ( 1, ) ( 1, )))

m x y L x y L x y L x y L x y

x y a L x y L x y L x y L x y

36 bins (i.e., 10o per bin)

• Histogram entries are weighted by (i) gradient magnitude and (ii) aGaussian function with σ equal to 1.5 times the scale of the keypoint.

0 2

( , , ) ( , , )* ( , )L x y G x y I x y

3. Orientation Assignment (cont’d)

• Assign canonical orientation at peak of smoothed histogram (fit parabola to better localize peak).

• In case of peaks within 80% of highest peak, multiple orientations assigned to keypoints. – About 15% of keypoints has multiple orientations assigned.

– Significantly improves stability of matching.

0 2

3. Orientation Assignment (cont’d)

• Stability of location, scale, and orientation (within 15 degrees) under noise.

• Have achieved invariance to location, scale, and orientation.

• Next, tolerate illumination and viewpoint changes.

4. Keypoint Descriptor

8 bins

Orientation histogram of gradient magnitudesOrientation histogram of gradient magnitudes

4. Keypoint Descriptor (cont’d)

16 histograms x 8 orientations = 128 features

1. Take a 16 x16 window around detected interest point.

2. Divide into a 4x4 grid of cells.

3. Compute histogram in each cell.

(8 bins)


• Each histogram entry is weighted by (i) gradient magnitude and (ii) a Gaussian function with σ equal to 0.5 times the width of the descriptor window.


• Partial Voting: distribute histogram entries into adjacent bins (i.e., additional robustness to shifts)– Each entry is added to all bins, multiplied by a weight of 1-d,

where d is the distance from the bin it belongs.


128 features

• Descriptor depends on two parameters:(1) number of orientations r(2) n x n array of orientation histograms

•

SIFT: r=8, n=4

rn2 features

4. Keypoint Descriptor (cont’d)• Invariance to affine (linear) illumination changes:

– Normalization to unit length is sufficient.

128 features

4. Keypoint Descriptor (cont’d)• Non-linear illumination changes :

– Saturation affects gradient magnitudes more than orientations

– Threshold gradient magnitudes to be no larger than 0.2 and renormalize to unit length

(i.e., emphasizes gradient orientations than magnitudes)

128 features

Robustness to viewpoint changes

• Match features after random change in image scale and orientation, with 2% image noise, and affine distortion.

• Find nearest neighbor in database of 30,000 features.

Additional robustness canbe achieved usingaffine invariantregion detectors.

Distinctiveness• Vary size of database of features, with 30 degree affine

change, 2% image noise.

• Measure % correct for single nearest neighbor match.

Matching SIFT features

• Given a feature in I1, how to find the best match in I2?

1. Define distance function that compares two descriptors.

2. Test all the features in I2, find the one with min distance.

I1I2

Matching SIFT features (cont’d)

• How to define the distance between two features f1, f2?

– Simple approach: SSD(f1, f2) (i.e., sum of squared differences)

– Can give good scores to very ambiguous (bad) matches

I1 I2

f1 f2


SSD(f1,f2) < t How to set t?

Matching SIFT features (cont’d)• How to define the difference between two features f1, f2?

– Better approach: SSD(f1, f2) / SSD(f1, f2’)

• f2 is best SSD match to f1 in I2

• f2’ is 2nd best SSD match to f1 in I2

I1 I2

f1 f2f2'


• Accept a match if SSD(f1, f2) / SSD(f1, f2’) < t

• t=0.8 has given good results in object recognition.– Eliminated 90% of false matches.

– Discarded less than 5% of correct matches


• How to evaluate the performance of a feature matcher?

5075

200


• True positives (TP) = # of detected matches that are correct

• False positives (FP) = # of detected matches that are incorrect

5075

200false match

true match

• Threshold t affects # of correct/false matches

Matching SIFT features(cont’d)

10.7

0 1FP rate

TPrate

0.1

• ROC Curve

- Generated by computing (FP, TP) for different thresholds.

- Need to maximize area under the curve (AUC)

http://en.wikipedia.org/wiki/Receiver_operating_characteristic

Applications of SIFT

• Object recognition• Object categorization• Location recognition• Robot localization• Image retrieval• Image panoramas

Object Recognition

Object Models

Object Categorization

Location recognition

Robot Localization

Map continuously built over time

Image retrieval – Example 1

…> 5000images

change in viewing angle

Matches

22 correct matches

Image retrieval – Example 2

…> 5000images

change in viewing angle

+ scale change

Matches

33 correct matches

Image panoramas from an unordered image set

• Steps 1-3 are the same; Step 4 is modified.

• Take a 41 x 41 patch at the given scale, centered at the keypoint, and normalized to a canonical direction.

PCA-SIFT

Yan Ke and Rahul Sukthankar, “PCA-SIFT: A More Distinctive Representation for Local Image Descriptors”, Computer Vision and Pattern Recognition, 2004

• Instead of using weighted histograms, concatenate the horizontal and vertical gradients into a long vector.

• Normalize vector to unit length.

PCA-SIFT

Yan Ke and Rahul Sukthankar, “PCA-SIFT: A More Distinctive Representation for Local Image Descriptors”, Computer Vision and Pattern Recognition, 2004

2 x 39 x 39 = 3042 vector

PCA-SIFT

PCA

N x 1 K x 1

'11 KxNxKxN IIA

• Reduce the dimensionality of the vector using Principal Component Analysis (PCA)– e.g., from 3042 to 36

• Proved to be less discriminatory than SIFT.

SURF: Speeded Up Robust Features

• Fast implementation of a variation of the SIFT descriptor.

• Key idea: fast approximation of (i) Hessian matrix and (ii) descriptor using “integral images”.

• What is an “integral image”?

Herbert Bay, Tinne Tuytelaars, and Luc Van Gool, “SURF: Speeded Up Robust Features”, European Computer Vision Conference (ECCV), 2006.

Integral Image

• The integral image IΣ(x,y) of an image I(x, y) represents the sum of all pixels in I(x,y) of a rectangular region formed by (0,0) and (x,y).

• .Using integral images, it takes only four array references to calculate the sum of pixels over a rectangular region of any size.

0 0

( , ) ( , )j yi x

i j

I x y I i j

SURF: Speeded Up Robust Features (cont’d)

• Approximate Lxx, Lyy, and Lxy using box filters.

• Can be computed very fast using integral images!

(box filters shown are 9 x 9 – good approximations for a Gaussian with σ=1.2)

derivative approximation approximationderivative


• In SIFT, images are repeatedly smoothed with a Gaussian and subsequently sub-sampled in order to achieve a higher level of the pyramid.


• Alternatively, we can use filters of larger size on the original image.

• Due to using integral images, filters of any size can be applied at exactly the same speed!

(see paper for details on how the filter size and octaves are computed)


• Approximation of H:

:

ˆ ˆ:

ˆ ˆ

xx xySIFTapprox

yx yy

xx xySURFapprox

yx yy

D DSIFT H

D D

L LSURF H

L L

Using DoG

Using box filters


• Instead of using a different measure for selecting the location and scale of interest points (like in SIFT(, SURF uses the determinant of to find both.

• Determinant elements must be weighted to obtain a good approximation of det(H):

SURFapproxH

2ˆ ˆ ˆdet( ) (0.9 )SURFapprox xx yy xyH L L L


• Once interest points have been localized both in space and scale, the next steps are:

(1) Orientation assignment

(2) Keypoint descriptor


• Orientation assignment

Circular neighborhood of radius 6σ around the interest point(σ = the scale at which the point was detected)

Haar wavelets (responses weighted with Gaussian)Side length = 4σ

x response y response

Can be computed very fast using integral images!

600

angle

( , )dx dy


• Sum the response over each sub-region for dx and dy separately.

• To bring in information about the polarity of the intensity changes, extract the sum of absolute value of the responses too.

Feature vector size:

4 x 16 = 64

• Keypoint descriptor (square region of size 20σ)

4 x 4grid


• SURF-128– The sum of dx and

|dx| are computed separately for points where dy < 0 and dy >0

– Similarly for the sum of dy and |dy|

– More discriminatory!

SURF: Speeded Up Robust Features

• Has been reported to be 3 times faster than SIFT.

• Less robust to illumination and viewpoint changes compared to SIFT.

K. Mikolajczyk and C. Schmid,"A Performance Evaluation of Local Descriptors", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 10, pp. 1615-1630, 2005.

Gradient location-orientation histogram (GLOH)

• Compute SIFT using a log-polar location grid:– 3 bins in radial direction (i.e., radius 6, 11, and 15)

– 8 bins in angular direction

• Gradient orientation quantized in 16 bins.

• Total: (2x8+1)*16=272 bins PCA.

K. Mikolajczyk and C. Schmid,"A Performance Evaluation of Local Descriptors", IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 27, no. 10, pp. 1615-1630, 2005.

Shape Context

• A 3D histogram of edge point locations and orientations. – Edges are extracted by the Canny edge detector.

– Location is quantized into 9 bins (using a log-polar coordinate system).

– Orientation is quantized in 4 bins (i.e., (horizontal, vertical, and two diagonals).

• Total number of features: 4 x 9 = 36


Spin image

• A histogram of quantized pixel locations and intensity values.– A normalized histogram is computed for each of five rings

centered on the region.

– The intensity of a normalized patch is quantized into 10 bins.

• Total number of features: 5 x 10 = 50


Differential Invariants

Example: some Gaussian derivatives up to fourth order

• “Local jets” of derivatives obtained by convolving the image with Gaussian derivates.• Derivates are computed at different orientations by rotating the image patches. ( , ) ( )

( , ) ( )

( , )* ( )

( , )* ( )

( , )* ( )

( , )* ( )

x

y

xx

xy

yy

I x y G

I x y G

I x y G

I x y G

I x y G

I x y G

invariants

Bank of Filters

(e.g., Gabor filters)

Moment Invariants

• Moments are computed for derivatives of an image patch using:

where p and q is the order, a is the degree, and Id is the image gradient in direction d.

• Derivatives are computed in x and y directions.

Bank of Filters: Steerable Filters

Representation, Description and Matching CS485/685 Computer Vision Dr. George Bebis.

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