# The Active Segmentation Platform for Microscopic Image Classification and Segmentation
## 2 Theory
### Scale Space Theory  
In the discrete domain, smoothing leads to loss of resolution and, therefore, of some information. However, the information loss can be limited if one uses multiple smoothing scales. A systematic way to treat such information loss is offered by the scale space theory [23; 24]. The axiomatic linear scale space theory was formulated in series of works by [PERSON] and [PERSON] [25; 26]. In its original version, the theory depends on several properties of the Gaussian filters as solutions of the diffusion equation in the scale-space generated by the image. That is, the generic smoothing kernel \(G\) is identified with a radially-symmetric Gaussian kernel of scale \(s=\sigma^{2}\in\mathbb{R}\)  
\[G(r)=\frac{e^{-r^{2}/2s}}{2\pi s} \tag{4}\]  
The Gaussian kernels provide several advantages: (i) they are rotationally invariant (ii) they do not produce artificial extrema in the resulting image (iii) successive convolutions with different kernels can be combined. Mathematically, this imposes a very useful semi-group structure, equivalent to the heat/diffusion equation. In this sense, the image structures diffuse or \"melt-down\", so that the rate of this diffusion indicates the \"robustness\" of the structure. In its typical presentation, the scale space theory applies only smoothing steps. Later, the theory was extended to include also differentiation and thus account for the differential structure of the images [27].  
Linear diffusion scale-spaces are well-posed and have a solid axiomatic foundation. On the other hand, for some applications, they have the undesirable property that they do not permit contrast enhancement and that they may blur and delocalize structures. Non-linear scale spaces try to overcome some of these limitations. Such scale spaces arise in non-linear partial-differential equation framework. The formal properties of some types of scale spaces have been established by [PERSON] et al. [28] and encompass anisotropic diffusion, among other approaches.  
In particular, the Laplacian operator can be decomposed into two mutually orthogonal components: one along the direction of the gradient and another one along the direction of the isophote ([29], Chapter 1; see Appendix B). In a similar way, the Laplacian operator can be iterated to give rise to the Power-of-Laplacian (PoL) operator [21]:  
\[\Delta_{G}^{n}I=(\underbrace{\Delta_{G}\dots}_{n\text{ times}})I \tag{5}\]  
This operator enhances high-frequency features of an images given the scale cut-off. This can be seen easily from the frequency response of the LoG filter \(\mathcal{F}_{k}\Delta_{G}=-k^{2}s^{2}\) where \(k\) is the wave vector and \(\mathcal{F}_{k}\) is the symbol of Fourier transform. The PoL operator is immediately generalizable from its Fourier space representation to the positive real domain of the exponent. This can be achieved in terms of the \(\alpha\)-scale space theory [30], which offers interesting possibilities of a broader class of smoothing filters, which are derivable from the Riesz Laplacian operator. The PoL operators are implemented by 2 filters--BoG and LoGN for speed reasons (see Table 3).