[CS5670] Lecture 5: Feature Invariance

내용 출처: CS5670

 

Local features: main components

1. Detection: Identify the interest points.
2. Description: Extract vector feature descriptor surrounding each interest point.
3. Matching: Determine correspondence between descriptors in two views.

 

Harris features(in red)

 

Image transformations

• Geometric: Rotation, Scale
• Photo metric: Intensity change

 

Invariance and equivariance

We want corner location to be invariant to photometric transformations and equivariant to geometric transformations

• Invariance: image is transformed and corner locations do not change
• equivariance: if we have two transformed versions of the same image, features should be detected in corresponding locations
• (Sometimes “invariant” and “equivariant” are both referred to as “invariant”)
• (Sometimes “equivariant” is called “covariant”)

 

Invariance와 Equivariance는 서로 반대되는 개념입니다. Invariance는 불변성이라는 뜻으로, 함수의 입력 값에 따라 출력 값이 바뀌지 않는다는 듯입니다. 따라서 trainslation invariance는 입력의 위치가 변해도 출력이 변하지 않는다는 의미입니다. 예시로 max pooling의 개념이 대표적인 small translation invariance 함수입니다.

 

내용 출처: CNN의 stationarity와 locality

 

Harris detector invariance properties

 

• Image translation 

• Derivatives and window function are equivariant
• Corner location is equivariant w.r.t translation

 

• Image rotation

• Second moment ellipse rotates but its shape (i.e. eigenvalues) remanins the same
• Corner location is euqivariant w.r.t image rotation

 

• Affine intensity change

• Partially invariant to affine intensity change

 

• scailing

• Neither invariant nor equivariant to scaling

 

Scale invariant detection

• Key idea: find scale that gives local maximum of f
  • in both position and scale
  • One definition of f: the Harris operator

 

Automatic scale selection

• Normalize: rescale to fixed size

 

Implementation

• Instead of computing f for larger and larger window, we can implement using a fixed window size with a Gaussian pyramid.

 

Feature extraction: Corners and blobs

 

Another common definition of f

• The Laplacian of gaussian (LoG)

$\nabla^2g = \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 g}{\partial y^2}$

 

(very similar to a Difference of Gasussians (DoG) - i.e a Gaussian minus a slightly smaller Gaussian)

 

Laplacian of Gaussian

• “Blob” detector

• Find maxima and minima of LoG operator in sapce and scale

 

Scale selection

• At what scale does the Laplacian achieve a maximum response for a binary circle of radius $r$?

 

Characteristic scale

• We define the characteristic scale as the scale that produces peak of Laplacian response

 

Find local maxima in 3D position-sclae space

 

 

Scale-space blob detector: Example

 

 

Scale Invaraint Detection

 

• Functions for determining scale

 

f = Kernel * Image

• kernels: $\nabla^2g = \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 g}{\partial y^2}$ (Laplacian)

 

$DoG = G(x,y,k\sigma)-G(x,y,\sigma)$ (Difference of Gaussians)

 

where Gaussian

 

$G(x,y,\sigma) = \frac{1}{2\pi \sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}$

 

• Note: The LoG and DoG operators are both rotation equivariant

 

Feature descriptors

We know how to detect good points

• Next question: How to match them?
  • Answer: Come up with a descriptor for each point, find similar descriptors between the two images

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