Zernike Features

Since the Zernike moments are defined over the unit circle, two steps were required to convert a rectangular region of each image to a unit circle for calculation of Zernike moments. First, the `center of fluorescence' (analogous to the center of mass) for each image was calculated and used to define the center of the pixel coordinate system. Second, the x and y coordinates were divided by 150 (this corresponds to the size of an average cell at the magnification used in these experiments). Only pixels within the unit circle of the resulting normalized image, f(x,y), were used for subsequent calculations. The Zernike moments, Z_nl, for an image were then calculated using

$$Z_{nl} = \frac{n+1}{\pi}\sum_{x}^{}\sum_{y}^{}V_{nl}^{*}(x,y)f(x,y)$$ (2.4)

where $x^{2}+y^{2} \leq 1$, $0 \leq l \leq n$, n-l is even, f(x,y)describes the intensity values of the normalized image and V_nl^*is the complex conjugate of a Zernike polynomial of degree n and angular dependence l

$$V_{nl}(x,y) = \sum_{m=0}^{\frac{n-l}{2}}(-1)^m \frac{(n-m)!} ... ...frac{n-2m-l}{2}\right)!} (x^2+y^2)^{\frac{n}{2}-m}e^{il\theta}$$ (2.5)

where $0 \leq l \leq n$, n-l is even, $\theta = \tan^{-1}\left(\frac{y}{x}\right)$, and $i=\sqrt{-1}$. Figure 2.1 contains plots of the magnitude of two Zernike polynomials.

    

Figure 2.1: Plots of the magnitude of Zernike polynomials for n=2, l=2 (A) and n=2, l=12 (B).

Zernike moments through degree 12 were calculated (Z_nl such that $n\leq 12$ in Equation 2.4) using the code in Section 5.2.1 (p. ). Since the moments themselves are complex numbers and are sensitive to rotation of the image, the magnitudes of the moments were used as features (i.e. |Z_nl|) [21]. This provided 49 descriptive features for each image.