Scale-invariant feature operator

Template:Short description Template:Orphan Template:Feature detection (computer vision) navbox In the fields of computer vision and image analysis, the scale-invariant feature operator (or SFOP) is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009.

Algorithm

The scale-invariant feature operator (SFOP) is based on two theoretical concepts:

• spiral model
• feature operator

Desired properties of keypoint detectors:

• Invariance and repeatability for object recognition
• Accuracy to support camera calibration
• Interpretability: Especially corners and circles, should be part of the detected keypoints (see figure).
• As few control parameters as possible with clear semantics
• Complementarity to known detectors

scale-invariant corner/circle detector.

Theory

Maximize the weight

Maximize the weight $w$= 1/variance of a point $p$

$w(𝐩,\alpha ,\tau ,\sigma )=(N(\sigma )-2)\frac{{\lambda }_{min}(M(𝐩,\alpha ,\tau ,\sigma ))}{\mathrm{\Omega }(𝐩,\alpha ,\tau ,\sigma )}$  

comprising:

1. the image model

• Distance d of an edge from a reference point p in a spiral feature
• 
• 
• 

$\begin{array}{cc}\mathrm{\Omega }(𝐩,\alpha ,\tau ,\sigma )& =\sum _{n=1}^{N(\sigma )}[({𝐪}_n-𝐩{)}^T{𝐑}_{\alpha }{\mathrm{\nabla }}_{T}g({𝐪}_n){]}^2{G}_{\sigma }({𝐪}_n-𝐩)\\ & =N(\sigma )𝐭𝐫\{R_{\alpha }{\mathrm{\nabla }}_{\tau }{\mathrm{\nabla }}_{\tau }^T{R}_{\alpha }^T*𝐩{𝐩}^T{G}_{\sigma }(𝐩)\}\end{array}$

2. the smaller eigenvalue of the structure tensor $\underset{\text{structure tensor}}{\underbrace{M(𝐩,\alpha ,\tau ,\sigma )}}=\underset{\text{weighted summation}}{\underbrace{G_{\sigma }(𝐩)}}*\underset{\text{squared rotated gradients}}{\underbrace{(R_{\sigma }{\mathrm{\nabla }}_{\tau }\underset{\tau }{\overset{T}{\mathrm{\nabla }}}{R}_{\sigma }^T)}}$

Reduce the search space

Reduce the 5-dimensional search space by

• linking the differentiation scale $\tau$ to the integration scale

$\tau =\sigma /3$

• solving for the optimal $\hat{\alpha }$ using the model

$\mathrm{\Omega }(\alpha )=a-b\cos (2\alpha -2{\alpha }_0)$

• and determining the parameters from three angles, e. g.

$\mathrm{\Omega }(0^{\circ }),\mathrm{\Omega }(60^{\circ }),\mathrm{\Omega }(120^{\circ })\to a,b,{\alpha }_0\to \hat{\alpha }$

• pre-selection possible:

$\alpha =0^{\circ }\to \text{junctions},\alpha =90^{\circ }\to \text{circular features}$

Filter potential keypoints

• non-maxima suppression over scale, space and angle

• thresholding the isotropy ${\lambda }_{2(M)}$:
  eigenvalues characterize the shape of the keypoint, smallest eigenvalue has to be larger than threshold $T_{\lambda }$
  derived from noise variance $V(n)$ and significance level $S$:

$T_{\lambda }(V(n),\tau ,\sigma ,S)=\frac{N(\sigma )}{16\pi {\tau }^4}V(n){\chi }_{2,S}^2$

Algorithm

Algorithm

Results

Interpretability of SFOP keypoints

• Results of different detectors on a Siemens star
• 
  Sfop: junctions red, circular features cyan
• 
  Edge-based Regions
• 
  Intensity-based Regions
• 
  MSER
• 
  Harris affine
• 
  Hessian affine
• 
  Lowe

See also

• Corner detection
• Feature detection (computer vision)

External links

• Project website at University of Bonn