Computer Vision : CISC 4/689 Example: Canny Edge Detection (Matlab automatically set thresholds)

Computer Vision : CISC 4/689

Example: Canny Edge Detection

(Matlab automatically set thresholds)

More: facts and figures

• The convolution of two Gaussians with variances {1}2 and {2}2 is {1}2+{2}2. This is same as consecutive smoothing with the two corresponding SD’s.

• Thus, generic formula is: i{i}2

• Problem: A discrete appx. to a 1D Gaussian can be obtained by sampling g(x). In practice, samples are taken uniformly until the truncated values at the tails of the distribution are less than 1/1000 of the peak value.

a) For =1, show that the filter is 7 pixels wide.

Answer..

• Lets pick (n+1) pixels from the center of kernel(including center). This way, total kernel size is 2n+1, n pixels on either side of origin.

Exp(-{(n+1)2}/{22}) < 1/1000So, n > 3.7 -1n must be the nearest integer to 3.7 -0.5

For =1, n=3, 2n+1=7.Filter coefficients can be obtained as {-3,-2,-1,0,1,2,3}

Choice of

• The choice of depends on the scale at which the image is to be displayed.

• Small values bring out edges at a fine scale, vice-versa.

• Noise is another factor to look into the selection, along with computational cost

Some comparisons

Zero-crossings easy to find than threshold

• Many implementations of the Canny edge detector approximate this process by first convolving the image with a Gaussian to smooth the signal, and then looking for maxima in the first partial derivatives of the resulting signal (using masks similar to the Sobel masks).

• Thus we can convolve the image with 4 masks, looking for horizontal, vertical and diagonal edges. The direction producing the largest result at each pixel point is marked.

• Record the convolution result and the direction of the edge at each pixel.

Marr-Hildreth vs. Canny

• Laplacian is isotropic, computationally efficient: single convolution, look for zero-crossing. (one way to explain zero-crossing is, if first derivative can be looked at as a function, its maximum will be its derivative=0).

• Canny being a directional operator (derivative in 4 or 3 directions), more costly, esp. due to hysterisis.

• Two derivatives -> more sensitive to noise

Image Pyramids

• Observation: Fine-grained template matching expensive over a full image – Idea: Represent image at smaller

scales, allowing efficient coarse- to-fine search

• Downsampling: Cut width, height in half at each iteration:

from Forsyth & Ponce

Gaussian Pyramid

• Let the base (the finest resolution) of an n-level Gaussian pyramid be defined

as P0 = I. Then the ith level is reduced from the level below it by:

• Upsampling S"(I): Double size of image, interpolate missing pixels

courtesy of Wolfram

Gaussian pyramid

Laplacian Pyramids

• The tip (the coarsest resolution) of an n-level Laplacian pyramid is the same as the Gaussian pyramid at that level: Ln(I) = Pn(I)

• The ith level is expanded from the level above according to Li(I) = Pi(I) ¡ S"(Pi+1(I))

• Synthesizing the original image: Get I back by summing upsampled Laplacian pyramid levels

Laplacian Pyramid

• The differences of images at successive levels of the Gaussian pyramid define the Laplacian pyramid. To calculate a difference, the image at a higher level in the pyramid must be increased in size by a factor of four prior to subtraction. This computes the pyramid.

• The original image may be reconstructed from the Laplacian pyramid by reversing the previous steps. This interpolates and adds the images at successive levels of the pyramid beginning with the lowest level.

• Laplacian is largely uncorrelated, and so may be represented pixel by pixel with many fewer bits than Gaussian.

courtesy of Wolfram

Reconstruction

Splining

• Build Laplacian pyramids LA and LB for A & B images

• Build a Gaussian pyramid GR from selected region R

• Form a combined pyramid LS from LA and LB using nodes of GR as weights:

LS(I,j) = GR(I,j)*LA(I,j)+(1-GR(I,j))*LB(I,j)

Collapse the LS pyramid to get the final blended image

Splining (Blending)

• Splining two images simply requires: 1) generating a Laplacian pyramid for each image, 2) generating a Gaussian pyramid for the bitmask indicating how the two images should be merged, 3) merging each Laplacian level of the two images using the bitmask from the corresponding Gaussian level, and 4) collapsing the resulting Laplacian pyramid.

• i.e. GS = Gaussian pyramid of bitmask LA = Laplacian pyramid of image "A" LB = Laplacian pyramid of image "B" therefore, "Lout = (GS)LA + (1-GS)LB"

Example images from GTech

Image-1 bit-mask image-2

Direct addition splining bad bit-mask choice

Outline

• Corner detection

• RANSAC

Matching with Invariant Features

Darya Frolova, Denis Simakov

The Weizmann Institute of Science

March 2004

Example: Build a Panorama

M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003

How do we build panorama?

• We need to match (align) images

Matching with Features

•Detect feature points in both images

•Find corresponding pairs

•Use these pairs to align images

• Problem 1:– Detect the same point independently in both images

no chance to match!

We need a repeatable detector

• Problem 2:– For each point correctly recognize the corresponding one

We need a reliable and distinctive descriptor

More motivation…

• Feature points are used also for:– Image alignment (homography, fundamental matrix)

– 3D reconstruction

– Motion tracking

– Object recognition

– Indexing and database retrieval

– Robot navigation

– … other

Corner Detection

• Basic idea: Find points where two edges meet—i.e., high gradient in two directions

• “Cornerness” is undefined at a single pixel, because there’s only one gradient per point– Look at the gradient behavior over a small window

• Categories image windows based on gradient statistics– Constant: Little or no brightness change– Edge: Strong brightness change in single direction– Flow: Parallel stripes– Corner/spot: Strong brightness changes in orthogonal directions

Corner Detection: Analyzing Gradient Covariance

• Intuitively, in corner windows both Ix and Iy should be high– Can’t just set a threshold on them directly, because we want rotational invariance

• Analyze distribution of gradient components over a window to differentiate between types from previous slide:

• The two eigenvectors and eigenvalues ¸1, ¸2 of C (Matlab: eig(C)) encode the predominant directions and magnitudes of the gradient, respectively, within the window

• Corners are thus where min(¸1, ¸2) is over a threshold courtesy of Wolfram

Covariance is 1/n sum(1 to n) (xi-x’)(xi-x’)^t

Contents

• Harris Corner Detector

– Description

– Analysis

• Detectors

– Rotation invariant

– Scale invariant

– Affine invariant

• Descriptors

– Scale invariant

Harris Detector: Mathematics

( , ) ( , ) ( , ) ( , )x y

E u v w x y I x u y v I x y

Change of intensity for the shift [u,v]:

IntensityShifted intensity

Window function

orWindow function w(x,y) =

Gaussian1 in window, 0 outside

Taylor series:I(x+u,y+v) = I(x,y)+Ix(x,y)u+Iy(x,y)v+…http://mathworld.wolfram.com.TaylorSeries.html

( , ) ,u

E u v u v Mv

For small shifts [u,v] we have a bilinear approximation:

( , ) x x y

x y x y y

I I IM w x y

where M is a 22 matrix computed from image derivatives:

( , ) ,u

E u v u v Mv

Intensity change in shifting window: eigenvalue analysis

1, 2 – eigenvalues of M

Ellipse E(u,v) = const

If we try every possible orientation n,the max. change in intensity is 2

“Corner”1 and 2 are large,

1 ~ 2;

E increases in all directions

1 and 2 are small;

E is almost constant in all directions

“Edge” 1 >> 2

“Edge” 2 >> 1

“Flat” region

Classification of image points using eigenvalues of M:

Measure of corner response:

2det traceR M k M

(k – empirical constant, k = 0.04-0.06)

2 “Corner”

“Edge”

“Flat”

• R depends only on eigenvalues of M

• R is large for a corner

• R is negative with large magnitude for an edge

• |R| is small for a flat region

R < 0|R| small

Harris Detector

• The Algorithm:– Find points with large corner response function R (R >

threshold)

– Take the points of local maxima of R

Harris Detector: Workflow

Compute corner response R

Find points with large corner response: R>threshold

Take only the points of local maxima of R

Example: Gradient Covariances

Full imageDetail of image with gradient covar-

iance ellipses for 3 x 3 windows

from Forsyth & Ponce

Corners are where both eigenvalues are big

Example: Corner Detection (for camera calibration)

courtesy of B. Wilburn

Example: Corner Detection

courtesy of S. Smith

SUSAN corners

Harris Detector: Summary

• Average intensity change in direction [u,v] can be expressed as a bilinear form:

• Describe a point in terms of eigenvalues of M:measure of corner response

• A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive

( , ) ,u

E u v u v Mv

1 2 1 2R k

Contents

– Description

– Analysis

• Detectors

– Scale invariant

• Descriptors

– Scale invariant

Tracking: compression of video information

• Harris response (uses criss-cross gradients)• Dinosaur tracking (using features)• Dinosaur Motion tracking (using correlation)• Final Tracking (superimposed)Courtesy: (http://www.toulouse.ca/index.php4?/CamTracker/index.php4?/CamTracker/FeatureTracking.html)

This figure displays results of feature detection over the dinosaur test sequence with the algorithm set to extract the 6 most "interesting" features at every image frame.

It is interesting to note that although no attempt to extract frame-to-frame feature correspondences was made, the algorithm still extracts the same set of features at every frame.

This will be useful very much in feature tracking.

One More..

• Office sequence

• Office Tracking

Harris Detector: Some Properties

• Rotation invariance

Ellipse rotates but its shape (i.e. eigenvalues) remains the same

Corner response R is invariant to image rotation

• Partial invariance to affine intensity change

Only derivatives are used => invariance to intensity shift I I + b

Intensity scale: I a I

x (image coordinate)

threshold

x (image coordinate)

• But: non-invariant to image scale!

All points will be classified as edges

Corner !

Harris Detector: Some Properties• Quality of Harris detector for different scale changes-- Correspondences calculated using distance (and threshold)

-- Improved Harris is proposed by Schmid et al

-- repeatability rate is defined as the number of points

repeated between two images w.r.t the total number of

detected points. Repeatability rate:

# correspondences# possible correspondences

C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000

Imp.Harris uses derivative of Gaussianinstead of standard template used byHarris et al.

Contents

– Description

– Analysis

• Detectors

– Scale invariant

• Descriptors

– Scale invariant

We want to:

detect the same interest points regardless of image changes

Models of Image Change

• Geometry– Rotation

– Similarity (rotation + uniform scale)

– Affine (scale dependent on direction)valid for: orthographic camera, locally planar object

• Photometry– Affine intensity change (I a I + b)

Contents

– Description

– Analysis

• Detectors

– Scale invariant

• Descriptors

– Scale invariant

Rotation Invariant Detection

C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000

Contents

– Description

– Analysis

• Detectors

– Scale invariant

• Descriptors

– Scale invariant

Scale Invariant Detection• Consider regions (e.g. circles) of different sizes around a point

• Regions of corresponding sizes will look the same in both images

Scale Invariant Detection• The problem: how do we choose corresponding circles independently

in each image?

Scale Invariant Detection

• Solution:

– Design a function on the region (circle), which is “scale invariant” (the same for corresponding regions, even if they are at different scales)

Example: average intensity. For corresponding regions (even of different sizes) it will be the same.

scale = 1/2

– For a point in one image, we can consider it as a function of region size (circle radius)

region size

Image 1 f

region size

Image 2

• Common approach:

scale = 1/2

region size

Image 1 f

region size

Image 2

Take a local maximum of this function

Observation: region size, for which the maximum is achieved, should be invariant to image scale.

Important: this scale invariant region size is found in each image independently!

• A “good” function for scale detection: has one stable sharp peak

region size

Good !

• For usual images: a good function would be the one with contrast (sharp local intensity change)

• Functions for determining scale

( , , )x y

G x y e

2 ( , , ) ( , , )xx yyL G x y G x y

( , , ) ( , , )DoG G x y k G x y

Kernel Imagef Kernels:

where Gaussian

Note: both kernels are invariant to scale and rotation

(Laplacian)

(Difference of Gaussians)

Scale Invariant Detectors• Harris-Laplacian1

Find local maximum of:– Harris corner detector in space

(image coordinates)– Laplacian in scale

1 K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20012 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 2004

Harris

• SIFT (Lowe)2

Find local maximum of:– Difference of Gaussians in

space and scale

Harris Laplacian

• Characteristic Scale: Given a point in an image, compute the function responses for several factors sn The characteristic scale is the local max. of the function (can be more than one).

• Easy to look for zero-crossings of 2nd derivative than maxima.

• The ratio of the scales, at which the extrema were found for corresponding points in two rescaled images, is equal to the scale factor between the images.

• Then what..?

Harris-Laplacian

• Existing methods search for maxima in the 3D representation of an image (x,y,scale). A feature point represents a local maxima in the surrounding 3D cube and its value is higher than a threshold.

• THIS (Harris-Laplacian) method uses Harris function first, then selects points for which Laplacian attains maximum over scales.

• First, prepare scale-space representation for the Harris function. At each level, detect interest points as local maxima in the image plane (of that scale) – do this by comparing 8-neighborhood. (different from plain Harris corner detection)

• Second, use Laplacian to judge if each of the candidate points found on different levels, if it forms a maximum in the scale direction. (check with n-1 and n+1)

Scale Invariant Detectors

K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 2001

• Experimental evaluation of detectors w.r.t. scale change

Repeatability rate:

# correspondences# possible correspondences

(points present)

Scale Invariant Detection: Summary

• Given: two images of the same scene with a large scale difference between them

• Goal: find the same interest points independently in each image• Solution: search for maxima of suitable functions in scale and in space

(over the image)

Methods:

1. Harris-Laplacian [Mikolajczyk, Schmid]: maximize Laplacian over scale, Harris’ measure of corner response over the image

2. SIFT [Lowe]: maximize Difference of Gaussians over scale and space

Computer Vision : CISC 4/689 Example: Canny Edge Detection (Matlab automatically set thresholds)

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