# Analysis of Random Dynamics of Cell Segmented by a Modified Active Contour Method
## 1 Introduction  
Recently, it has become an essential theme of biology to understand the dynamics of living systems, such as cell migration [1, 2], embryogenesis [3, 4], and transport of organelles within a cell [5]. Characterization of living organisms can be performed not only by measuring their biochemical properties, but also by analyzing their dynamic properties including trajectories, velocities, and diffusion coefficients. For example, cellular migration is caused by the dynamics of cytoskeletal proteins such as actin and microfilaments within the cell, which indicate that cellular biochemical properties can be correlated to dynamic properties of the cell [1]. Sometimes, the dynamics of intracellular organelles can be explained by analysis of deformation of a migrating cell [5]. However, since cells do not have ultrasmooth surfaces, and there is no ultrasensitive method to capture cell images, there always exist localization errors when analyzing cellular movement. Therefore, a highly sensitive method must be developed to remove unnecessary signals such as noise through advanced filtering or an advanced device to capture a spatiotemporally high-resolution image [6, 7, 8]. The spatiotemporal resolution of a dynamic target object is determined by its photo-physical properties, the signal to noise ratio, and the speeds of the moving objects [9]. Advancement of imaging techniques enables imaging of moving cells and macromolecules. Furthermore, with a live cell imaging system, which provides a controlled environment for cells under a microscope, it is possible to run a long-term experiment with living cells. The long-term recording of cell migration or macromolecules requires processing of a large number of time-lapse images. Tracking algorithms for analyzing such images have two types of errors, determinate (accuracy or bias error) and indeterminate (precision or standard deviation) [9; 10; 11]. Determinate errors are responsible for inaccuracies inherently caused by the algorithm, whereas indeterminate errors, which are caused by individual fluctuating random measurements, result from the noise of the image [9; 12]. Single-particle tracking algorithms have been developed to localize particles with subpixel displacement [10; 11; 13]. A microscopic image of a point object appears as a diffraction pattern in that intensity distribution following the point spread function (PSF), which is radially symmetric and shows Gaussian distribution [12; 14]. Direct Gaussian fitting of the intensity distribution is a superior algorithm in terms of both accuracy and precision [9]. First, the most common Gaussian fitting method for the measured PSF intensity profile is the nonlinear least squares algorithm [15; 16]. The main principle of this algorithm is to search for parameters that minimize weighted squared errors between the fit and the data. Least squares fitting is accurate but is a very costly algorithm [17; 18]. Second, while the maximum likelihood method is more accurate [19], Gaussian fitting requires much computational cost due to the large number of iterations and superfluous parameters like amplitude and width of the function necessary to locate the particle center [18; 20]. Third, the radial symmetry method localizes the center of the PSF through only one linear matrix calculation. The radial symmetry method has been shown to be as accurate as Gaussian fitting and more rapidly calculable [18]. Alternatively, segmentation of the target image is also applicable to localize the cellular image. Edges of objects that have different intensities from the background can be detected using this method. Many different thresholding algorithms have been developed and applied to analyze the shapes of target images from grey scale images [21; 22]. The simplest segmentation is intensity thresholding [23; 24]. Despite its simplicity, intensity thresholding is difficult to apply to cell images due to poor contrast or the halo effect.