# An Application of Manifold Learning in Global Shape Descriptors
## 3 Method
### Scale Normalization  
For some applications, the size of an object is not a determinant factor in shape comparison and identification. Therefore, a scale-invariant shape descriptor with a solid normalization method is more desirable. For that purpose, some shape descriptors including Shape-DNA, have proposed multiple normalization methods. Most normalization methods of Shape-DNA focus on finding an appropriate scaling factor, such as the surface area, the volume, or coefficient of a fitting curve, which will be multiplied in the descriptor. Moreover, it is shown that eigenvalues with a higher order are more susceptible to noise. For that reason, the original Shape-DNA recommends cropping the spectrum and using no more than 100 eigenvalues [20]. In this section, we propose a simple and efficient normalization method that significantly reduces the effect of scale variations as well as noise. In this approach, we are interested in taking the scaling factor out from the descriptor in one step, rather than an extra step to find an appropriate neutralizing factor. Although the normalization seems simple, later in the experiments section, we show its efficiency. In general, there are two forms of scaling that can affect a shape descriptor: (1) when an arbitrary scale object is scaled in the Euclidean space with factor \(\alpha\), e.g., a Teddy Bear model and its scaled version shown in Figure 2a; and (2) when two objects within a same category are naturally in different sizes, e.g., the two glasses shown in Figure 2b. According to [PERSON]'s law [48], the eigenvalues of a scaled object (the first form of object scaling) are scaled with factor \(1/\alpha^{2}\). In the case of the second form of object scaling, the relationship between the eigenvalues of two objects is a complicated relation, which requires solving Equation (10). However, we can approximately model it with \(f(\alpha^{2})\)--a function of \(\alpha^{2}\)For Equation (6), one can use a constant value for parameter \(t\) on the entire dataset, or use a varying value for each object in proportion to the size of the object. For example, as we previously suggested in Equation (7), \(t\) can be a multiple of \(d_{max}^{2}\) for each individual object. When \(t\) is selected using the latter option, the scaling factor \(\alpha\) of the first form of object scaling is canceled out from the numerator and denominator of Equation (6). As a result, all edge weights, and consequently, all eigenvalues will be invariant of the factor \(\alpha\). However, such a parameter \(t\) cannot completely deal with the scaling differences between two different objects. In addition, when \(t\) is constant for the entire dataset, both forms of object scaling have their impact on the spectrum. Figure 2c shows the spectrum of Teddy Bear models and both glasses in two scenarios; \(t=0.5\) and \(t=d_{max}^{2}\). Note that, when \(t=d_{max}^{2}\), the spectrum of the scaled Teddy Bear model is aligned with the spectrum of the original model. Another observation from Figure 2c is that the spectrum is a function of \(t\), as well. That phenomenon can be expected as \(t\) presents in the exponential term defining edge weights. Let \(\xi_{s}=g(\alpha^{2},t)(\lambda_{1},\dots,\lambda_{d})\) be a LESI descriptor obtained from a scaled 3D model with unknown factor \(\alpha\). To normalize the descriptor and eliminate the effect of the scaling factor as well as parameter \(t\), we recommend computing:  
\[\xi_{n}(i)=\log(\xi_{s}(i))-\log(\xi_{s}(1))\;,\,\text{for}\;1\leq i\leq d \tag{12}\]  
To achieve a scale invariant shape descriptor, we first take the logarithm of the descriptor vector and then compute the difference of the new vector from its smallest element. This is equivalent to dividing the vector by its first element--as suggested by [PERSON] et al. (2018)--and taking the logarithm next.