computer vision : cisc 4/689 gradients and edges points of sharp change in an image are interesting:...

Computer Vision : CISC 4/689

Gradients and edges

• Points of sharp change in an image are interesting:– change in reflectance

– change in object

– change in illumination

– noise

• Sometimes called edge points

• General strategy– determine image gradient

– now mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).


The Gradient and Edges

• Consider image intensities as a 2-D height function I(x, y). Then the image gradient is the vector field defined by:

• Definition of an edge– Line segment separating regions of contrasting intensity– Location: Where gradient magnitude is high – Direction: Orthogonal to the gradient


Edge Causes

• Depth discontinuity

• Surface orientation discontinuity

• Reflectance discontinuity (i.e., change in surface material properties)

• Illumination discontinuity (e.g., shadow)


Edge Detection• An edge point can be regarded as a point in an image

where a discontinuity (in gradient) occurs across some line. A discontinuity may be classified as one of five types

• Searching for Edges:– Filter: Smooth image– Enhance: Apply numerical derivative approximation– Detect: Threshold to find strong edges– Localize/analyze: Reject spurious edges, include

weak but justified edges

Gradient Discontinuity -- where the gradient of the pixel values changes across a line. This type of discontinuity can be classed as roof edges ramp edges convex edges concave edges by noting the sign of the component of the gradient perpendicular to the edge on either side of the edge. Ramp edges have the same signs in the gradient components on either side of the discontinuity, while roof edges have opposite signs in the gradient components.

A Jump or Step Discontinuity -- where pixel values themselves change suddenly across some line. A Bar Discontinuity -- where pixel values rapidly increase then decrease again (or vice versa) across some line.

Source: http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MARSHALL/node28.html

http://homepages.inf.ed.ac.uk/rbf/CVonline/


Step edge detection: First Derivative Operators

• Method: Differentiate and find extrema

• Examples– Sobel operator (Matlab: edge(I, ‘sobel’))

– Prewitt, Roberts cross

– Derivative of Gaussian

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Sobel Edge Filtering Example

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Sobel Edge Filtering Example: Result

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(pad with zeroes again, the boundary)and then we threshold…


Sobel Edge Detection: Gradient Approximation

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Note anisotropy of edge finding


Sobel

• These can then be combined together to find the absolute magnitude of the gradient at each point and the orientation of that gradient. The gradient magnitude is given by:

• an approximate magnitude is computed using:

which is much faster to compute.

• The angle of orientation of the edge (relative to the pixel grid) giving rise to the spatial gradient is given by:

In this case, orientation 0 is taken to mean that the direction of maximum contrast from black to white runs from left to right on the image, and other angles are measured anti-clockwise from this.


Derivative of Gaussian


Smoothing and Differentiation

• Issue: noise– smooth before differentiation

– two convolutions: to smooth, then differentiate?

– actually, no - we can use a derivative of Gaussian filter

• because differentiation is convolution, and convolution is associative


The Laplacian of Gaussian

• Another way to detect an extremal first derivative is to look for a zero second derivative– the Laplacian

• Bad idea to apply a Laplacian without smoothing– smooth with Gaussian, apply

Laplacian

– this is the same as filtering with a Laplacian of Gaussian filter

• Now mark the zero points where there is a sufficiently large (first) derivative, and enough contrast


Marr-Hildreth operator

• The Laplacian is linear and rotationally symmetric. Thus, we search for the zero crossings of the image that is first smoothed with a Gaussian mask and then the second derivative is calculated; or we can convolve the image with the Laplacian of the Gaussian, also known as the LoG operator;

• This defines the Marr-Hildreth operator.

• One can also get a shape similar to G'' by taking the difference of two Gaussians having different standard deviations. A ratio of standard deviations of 1:1.6 will give a close approximation to .This is known as the DoG operator (Difference of Gaussians), or the Mexican Hat Operator.

• Still sensitive to noise.


Step edge detection: 2nd-Derivative Operators

• Method: 2nd derivative is 0 for 1st-derivative extrema, so find “zero-crossings”– Laplacian

Isotropic (finds edges regardless of orientation.

Three commonly used discrete approximations to the Laplacian filter. (Note, we have defined the Laplacian using a negative peak because this is more common, however, it is equally valid to use the opposite sign convention.) Source: http://www.cee.hw.ac.uk/hipr/html/log.html

http://www.cee.hw.ac.uk/hipr/html/log.html


Laplacian of Gaussian

• Matlab: fspecial(‘log’,…)

Below: Discrete approximation to LoG function with Gaussian 1.4


Sobel vs. LoG Edge Detection:Matlab Automatic Thresholds

Sobel LoG


There are three major issues: 1) The gradient magnitude at different scales is different; which should we choose? 2) The gradient magnitude is large along thick trail (for 3rd fig); how do we identify the significant points? 3) How do we link the relevant points up into curves?

= 1 = 2


We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?


Non-maximumsuppression

At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.


Predictingthe nextedge point

Assume the marked point is an edge point. Then we construct the tangent (along) to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).


Remaining issues

• Check that maximum value of gradient value is sufficiently large– drop-outs? use hysteresis

• use a high threshold to start edge curves and a low threshold to continue them.


fine scalehigh Threshold(be strict inAcceptingEdge points)


coarse scale,high threshold


coarsescalelowthreshold


Canny Edge Detection

• Steps1. Apply derivative of Gaussian (not Laplacian!)

2. Non-maximum suppression

• Thin multi-pixel wide “ridges” down to single pixel

3. Thresholding

• Low, high edge-strength thresholds

• Accept all edges over low threshold that are connected to edge over high threshold (in the stage of predicting next edge point)

• Matlab: edge(I, ‘canny’)


Edge “Smearing”

from Forsyth & Ponce

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We want to localize the edge to within 1 pixel

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Non-Maximum Suppression: Steps

1. Consider 9-pixel neighborhood around each edge candidate (i.e., already over a threshold)

2. Interpolate edge strengths E at neighborhood boundaries in negative & positive gradient directions from the center pixel

3. If the pixel under consideration is not greater than these two values (i.e. not a maximum), it is suppressed

Interpolating the E value:

E(r) = (1 ¡ a)E(x, y) + aE(x + 1, y)

a 1 ¡ a

(x, y) (x + 1, y)r


Example: Non-Maximum Suppression

courtesy of G. Loy

Original image Gradient magnitudeNon-maxima suppressed


Edge “Streaking”

• Can predict next pixel in edge orthogonal to gradient to make edge chain– Can also just use 8-connectedness to define chains

• Streaking: Gaps in edge chain due to edge strength dipping below threshold

courtesy of G. Loy

Original image Strong edges

gap


Edge Hysteresis

• Hysteresis: A lag or momentum factor

• Idea: Maintain two thresholds khigh and klow

– Use khigh to find strong edges to start edge chain

– Use klow to find weak edges which continue edge chain

• Usual ratio of thresholds is roughly

khigh / klow = 2 or 3


Example: Canny Edge Detection

courtesy of G. Loy

gap is gone

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Example: Canny Edge Detection

(Matlab automatically set thresholds)


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


Reconstruction


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 obtained 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 coarsest level.

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

courtesy of Wolfram


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




•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


Contents

• Harris Corner Detector

– Description

– Analysis

• Detectors

– Rotation invariant

– Scale invariant

– Affine invariant

• Descriptors

– Rotation invariant

– Scale invariant

– Affine invariant


Harris Detector: Mathematics

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( , ) ( , ) ( , ) ( , )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:F(x+dx,y+dy) = f(x,y)+fx(x,y)dx+fy(x,y)dy+…http://mathworld.wolfram.com.TaylorSeries.html



( , ) ,u

E u v u v Mv

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

2

2,

( , ) x x y

x y x y y

I I IM w x y

I I I

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

direction of the slowest change

direction of the fastest change

(max)-1/2

(min)-1/2

Ellipse E(u,v) = const



1

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

1 2

1 2

det

trace

M

M

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



1

2 “Corner”

“Edge”

“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 < 0

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

computer vision : cisc 4/689 gradients and edges points of sharp change in an image are interesting:...

Documents

gradient magnitude

gradient components

spatial gradient

gradient perpendicular

step discontinuity

edge pointsgeneral

edge relative

bar discontinuity