Lecture VI: Corner and Blob Detection

Lecture VI: Corner and Blob Detection
CS 558 Computer Vision Lecture VI: Corner and Blob Detection Slides adapted from S. Lazebnik

Outline Corner detection Blob detection Why detecting features?
Finding corners: basic idea and mathematics Steps of Harris corner detector Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region

Feature extraction: Corners
9300 Harris Corners Pkwy, Charlotte, NC

Why extract features? Motivation: panorama stitching
We have two images – how do we combine them?

Why extract features? Step 1: extract features Step 2: match features
Motivation: panorama stitching We have two images – how do we combine them? Step 2: match features Step 1: extract features

Why extract features? Step 1: extract features Step 2: match features
Motivation: panorama stitching We have two images – how do we combine them? Step 1: extract features Step 2: match features Step 3: align images

Characteristics of good features
Repeatability The same feature can be found in several images despite geometric and photometric transformations Saliency Each feature is distinctive Compactness and efficiency Many fewer features than image pixels Locality A feature occupies a relatively small area of the image; robust to clutter and occlusion

Feature points are used for:
Applications Feature points are used for: Image alignment 3D reconstruction Motion tracking Robot navigation Indexing and database retrieval Object recognition

Finding Corners Key property: in the region around a corner, image gradient has two or more dominant directions Corners are repeatable and distinctive C.Harris and M.Stephens. "A Combined Corner and Edge Detector.“ Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.

Corner Detection: Basic Idea
We should easily recognize the point by looking through a small window Shifting a window in any direction should give a large change in intensity “flat” region: no change in all directions “edge”: no change along the edge direction “corner”: significant change in all directions Source: A. Efros

Corner Detection: Mathematics
Change in appearance of window w(x,y) for the shift [u,v]: I(x, y) E(u, v) E(3,2) w(x, y)

Change in appearance of window w(x,y) for the shift [u,v]: I(x, y) E(u, v) E(0,0) w(x, y)

Change in appearance of window w(x,y) for the shift [u,v]: Intensity Window function Shifted intensity or Window function w(x,y) = Gaussian 1 in window, 0 outside Source: R. Szeliski

Change in appearance of window w(x,y) for the shift [u,v]: We want to find out how this function behaves for small shifts E(u, v)

Change in appearance of window w(x,y) for the shift [u,v]: We want to find out how this function behaves for small shifts Local quadratic approximation of E(u,v) in the neighborhood of (0,0) is given by the second-order Taylor expansion:

Second-order Taylor expansion of E(u,v) about (0,0):

The quadratic approximation simplifies to where M is a second moment matrix computed from image derivatives: M

Interpreting the second moment matrix
The surface E(u,v) is locally approximated by a quadratic form. Let’s try to understand its shape.

First, consider the axis-aligned case (gradients are either horizontal or vertical) If either λ is close to 0, then this is not a corner, so look for locations where both are large.

Consider a horizontal “slice” of E(u, v): This is the equation of an ellipse.

Consider a horizontal “slice” of E(u, v): This is the equation of an ellipse. Diagonalization of M: The axis lengths of the ellipse are determined by the eigenvalues and the orientation is determined by R direction of the slowest change direction of the fastest change (max)-1/2 (min)-1/2

Visualization of second moment matrices

Interpreting the eigenvalues
Classification of image points using eigenvalues of M: 2 “Edge” 2 >> 1 “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 “Flat” region 1

Corner response function
α: constant (0.04 to 0.06) “Edge” R < 0 “Corner” R > 0 |R| small “Edge” R < 0 “Flat” region

Harris detector: Steps
Compute Gaussian derivatives at each pixel Compute second moment matrix M in a Gaussian window around each pixel Compute corner response function R Threshold R Find local maxima of response function (nonmaximum suppression) C.Harris and M.Stephens. “A Combined Corner and Edge Detector.” Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.

Harris Detector: Steps

Compute corner response R

Find points with large corner response: R>threshold

Take only the points of local maxima of R

Invariance and covariance
We want corner locations to be invariant to photometric transformations and covariant to geometric transformations Invariance: image is transformed and corner locations do not change Covariance: if we have two transformed versions of the same image, features should be detected in corresponding locations

Affine intensity change
I  a I + b Only derivatives are used => invariance to intensity shift I  I + b Intensity scaling: I  a I R x (image coordinate) threshold Partially invariant to affine intensity change

Corner location is covariant w.r.t. translation
Image translation Derivatives and window function are shift-invariant Corner location is covariant w.r.t. translation

Corner location is covariant w.r.t. rotation
Image rotation Second moment ellipse rotates but its shape (i.e. eigenvalues) remains the same Corner location is covariant w.r.t. rotation

Corner location is not covariant to scaling!
All points will be classified as edges Corner location is not covariant to scaling!

Blob detection

Achieving scale covariance
Goal: independently detect corresponding regions in scaled versions of the same image Need scale selection mechanism for finding characteristic region size that is covariant with the image transformation

Recall: Edge detection
f Derivative of Gaussian Edge = maximum of derivative Source: S. Seitz

Edge detection, Take 2 f Edge
Second derivative of Gaussian (Laplacian) An edge filtered with the Laplacian of Gaussian is the difference between the blurry edge and the original edge. Edge = zero crossing of second derivative Source: S. Seitz

From edges to blobs Edge = ripple Blob = superposition of two ripples maximum Spatial selection: the magnitude of the Laplacian response will achieve a maximum at the center of the blob, provided the scale of the Laplacian is “matched” to the scale of the blob

original signal (radius=8)
Scale selection We want to find the characteristic scale of the blob by convolving it with Laplacians at several scales and looking for the maximum response However, Laplacian response decays as scale increases: increasing σ original signal (radius=8) Why does this happen?

Scale normalization The response of a derivative of Gaussian filter to a perfect step edge decreases as σ increases The area under the first derivative of Gaussian from –infinity to 0 is equal to the value of the Gaussian at zero

Scale normalization The response of a derivative of Gaussian filter to a perfect step edge decreases as σ increases To keep response the same (scale-invariant), must multiply Gaussian derivative by σ Laplacian is the second Gaussian derivative, so it must be multiplied by σ2

Effect of scale normalization
Original signal Unnormalized Laplacian response Scale-normalized Laplacian response maximum

Blob detection in 2D Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D

Blob detection in 2D Scale-normalized:
Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D Scale-normalized:

Scale selection At what scale does the Laplacian achieve a maximum response to a binary circle of radius r? r image Laplacian

Scale selection At what scale does the Laplacian achieve a maximum response to a binary circle of radius r? To get maximum response, the zeros of the Laplacian have to be aligned with the circle The Laplacian is given by: Therefore, the maximum response occurs at circle Laplacian r image

Characteristic scale characteristic scale
We define the characteristic scale of a blob as the scale that produces peak of Laplacian response in the blob center characteristic scale T. Lindeberg (1998). "Feature detection with automatic scale selection." International Journal of Computer Vision 30 (2): pp

Scale-space blob detector
Convolve image with scale-normalized Laplacian at several scales

Scale-space blob detector: Example

Scale-space blob detector
Convolve image with scale-normalized Laplacian at several scales Find maxima of squared Laplacian response in scale-space

Scale-space blob detector: Example

Efficient implementation
Approximating the Laplacian with a difference of Gaussians: (Laplacian) (Difference of Gaussians)

Efficient implementation
David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp , 2004.

Invariance and covariance properties
Laplacian (blob) response is invariant w.r.t. rotation and scaling Blob location and scale is covariant w.r.t. rotation and scaling What about intensity change?

Achieving affine covariance
Affine transformation approximates viewpoint changes for roughly planar objects and roughly orthographic cameras

Achieving affine covariance
Consider the second moment matrix of the window containing the blob: direction of the slowest change direction of the fastest change (max)-1/2 (min)-1/2 Recall: This ellipse visualizes the “characteristic shape” of the window

Affine adaptation example
Scale-invariant regions (blobs)

Affine adaptation example
Affine-adapted blobs

From covariant detection to invariant description
Geometrically transformed versions of the same neighborhood will give rise to regions that are related by the same transformation What to do if we want to compare the appearance of these image regions? Normalization: transform these regions into same-size circles

Affine normalization Problem: There is no unique transformation from an ellipse to a unit circle We can rotate or flip a unit circle, and it still stays a unit circle

Eliminating rotation ambiguity
To assign a unique orientation to circular image windows: Create histogram of local gradient directions in the patch Assign canonical orientation at peak of smoothed histogram 2 p

From covariant regions to invariant features
Eliminate rotational ambiguity Compute appearance descriptors Extract affine regions Normalize regions SIFT (Lowe ’04)

Invariance vs. covariance
features(transform(image)) = features(image) Covariance: features(transform(image)) = transform(features(image)) Covariant detection => invariant description

Lecture VI: Corner and Blob Detection

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