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.[1]
The scale-invariant feature operator (SFOP) is based on two theoretical concepts:
Desired properties of keypoint detectors:
scale-invariant corner/circle detector.
Maximize the weight w {\displaystyle w} = 1/variance of a point p {\displaystyle p}
w ( p , α , τ , σ ) = ( N ( σ ) − 2 ) λ m i n ( M ( p , α , τ , σ ) ) Ω ( p , α , τ , σ ) {\displaystyle 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[2]
Ω ( p , α , τ , σ ) = ∑ n = 1 N ( σ ) [ ( q n − p ) T R α ∇ T g ( q n ) ] 2 G σ ( q n − p ) = N ( σ ) t r { R α ∇ τ ∇ τ T R α T ∗ p p T G σ ( p ) } {\displaystyle {\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}}}
2. the smaller eigenvalue of the structure tensor M ( p , α , τ , σ ) ⏟ structure tensor = G σ ( p ) ⏟ weighted summation ∗ ( R σ ∇ τ ∇ τ T R σ T ) ⏟ squared rotated gradients {\displaystyle \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}}}
Reduce the 5-dimensional search space by