Benutzer:Ipb.uni-bonn/Spielwiese

Motivation

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

Using:

• spiral model
• feature operator

to create a

scale-invariant corner/circle detector.

Theory

Maximize the weight

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

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

comprising:

1. the image model

• Distance d of an edge from a reference point p in a spiral feature
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${\begin{aligned}\Omega (\mathbf {p} ,\alpha ,\tau ,\sigma )&=\sum _{n=1}^{N(\sigma )}[(\mathbf {q} _{n}-\mathbf {p} )^{T}\mathbf {R} _{\alpha }\mathbf {\nabla } _{T}g(\mathbf {q} _{n})]^{2}G_{\sigma }(\mathbf {q} _{n}-\mathbf {p} )\\&=N(\sigma )\mathbf {tr} \left\{R_{\alpha }\mathbf {\nabla } _{\tau }\mathbf {\nabla } _{\tau }^{T}R_{\alpha }^{T}*\mathbf {p} \mathbf {p} ^{T}G_{\sigma }(\mathbf {p} )\right\}\end{aligned}}$		(1)

2. the smaller eigenvalue of the structure tensor

$\underbrace {M(\mathbf {p} ,\alpha ,\tau ,\sigma )} _{\text{structure tensor}}=\underbrace {G_{\sigma }(\mathbf {p} )} _{\text{weighted summation}}*\underbrace {(R_{\sigma }\nabla _{\tau }\nabla _{\tau }^{T}R_{\sigma }^{T})} _{\text{squared rotated gradients}}$		(2)

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

$\Omega (\alpha )=a-b\cos(2\alpha -2\alpha _{0})$

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

$\Omega (0^{\circ }),\Omega (60^{\circ }),\Omega (120^{\circ })\quad \rightarrow \quad a,b,\alpha _{0}\quad \rightarrow \quad {\hat {\alpha }}$

• pre-selection possible:

$\alpha =0^{\circ }\,\rightarrow \,{\mbox{junctions}},\quad \alpha =90^{\circ }\,\rightarrow \,{\mbox{circular features}}$

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
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  Sfop: junctions red, circular features cyan
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  Edge-based Regions
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  Intensity-based Regions
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  [MSER]
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  [Harris affine]
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  [Hessian affine]
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  [Lowe]

Repeatability and accuracy evaluation

Using the performance evaluation for region detectors presented in , the following results were achieved

Image Set A

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Image Set B

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References

• W. Förstner and T. Dickscheid and F. Schindler.
  Interpretable and Accurate Scale-Invariant Keypoints.
  International Conference on Computer Vision (ICCV'09), Kyoto, Japan, 2009.
• W. Förstner and T. Dickscheid and F. Schindler.
  SFOP Keypoint Detector - Project Homepage.