## Abstract

We define a nonlinear filtering based on correlations on unit spheres to obtain both rotation- and scale-invariant three-dimensional (3D) object detection. Tridimensionality is expressed in terms of range images. The phase Fourier transform (PhFT) of a range image provides information about the orientations of the 3D object surfaces. When the object is sequentially rotated, the amplitudes of the different PhFTs form a unit radius sphere. On the other hand, a scale change is equivalent to a multiplication of the amplitude of the PhFT by a constant factor.
The effect of both rotation and scale changes for 3D objects means a change in the intensity of the unit radius sphere. We define a 3D filtering based on nonlinear operations between spherical correlations to achieve both scale- and rotation-invariant 3D object recognition.

© 2008 Optical Society of America

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### Equations (8)

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(1)
$$P\left(y,z\right)=\mathrm{exp}\left[ikx\left(y,z\right)\right]\text{,}$$
(2)
$$\mathrm{PhFT}\left(\theta ,\phi \right)={F}_{2D}\left\{\mathrm{exp}\left[ikx\left(y,z\right)\right]\right\}\text{,}$$
(3)
$$x\prime (y,z)=x(\frac{y}{m},\frac{z}{m})\text{,}$$
(4)
$$\mid {\mathrm{PhFT}}_{x\prime}\left(\theta ,\phi \right)\mid \cong {m}^{2}\mid {\mathrm{PhFT}}_{x}\left(\theta ,\phi \right)\mid \text{.}$$
(5)
$${C}_{fg}{\left(\alpha ,\beta ,\gamma \right)}_{\in SO\left(3\right)}=f\left(\theta ,\phi \right){*}_{SO\left(3\right)}g(\theta ,\phi )\equiv {\displaystyle \sum _{j\text{,}m\text{,}m\prime}{{d}_{m\text{,}m\prime}}^{j}\left(\beta \right){F}_{j\text{,}m\prime}{G}_{j\text{,}m}\times \mathrm{exp}\left(im\gamma \right)\mathrm{exp}\left(im\alpha \right)}\text{,}$$
(6)
$${F}_{j\text{,}m\prime}={\displaystyle \int f\left(\theta ,\phi \right){Y}_{lm}*\left(\theta ,\phi \right)\mathrm{sin}\left(\theta \right)\mathrm{d}\theta \mathrm{d}\phi}\text{,}$$
(7)
$$g\left(\theta ,\phi \right)=af\left(\theta ,\phi \right)+bw\left(\theta ,\phi \right)\text{,}$$
(8)
$${L}_{gf}\left(\alpha ,\beta ,\gamma \right)\text{\hspace{1em}}=\frac{{\left[g\left(\theta ,\phi \right){*}_{SO\left(3\right)}{f}_{o}\left(\theta ,\phi \right)\right]}^{2}}{\sqrt{N}\left[g\left(\theta ,\phi \right){*}_{SO\left(3\right)}w\left(\theta ,\phi \right)\right]-{\left[g\left(\theta ,\phi \right){*}_{SO\left(3\right)}w\left(\theta ,\phi \right)\right]}^{2}}\text{,}$$