# Computational Methods for Neuron Segmentation in Two-Photon Calcium Imaging Data: A Survey
## 5 Matrix Factorization-Based Methods
### Non-Negative Matrix Factorization-Based Approaches
proposed a matrix factorization-based approach for calcium signal extraction which tries to solve the problems of denoising, factorization and deconvolution simultaneously [31]. First, a parametric model is estimated for calcium transients under the assumption that sampling time is less than the decay time of the transients. This is achieved by approximating the calcium transients as the impulse response of an auto-regressive (AR) process of order p (p is either 1 or 2). Next, the spiking signal is estimated by solving a non-negative sparse constrained deconvolution problem. To achieve this, a T time steps field of view is considered. This field of view can be considered as \(d\times T\) matrix where \(d\) is the number of elements in one time step (all the pixels in one image/slice). The number of neurons in this field of view is initially assumed to be \(K\). The spatio-temporal activity of a neuron is assumed to be an AR process with two elements, one element is for the spatial footprint and the other element is for the temporal footprint in \(d\times T\) field. The spatial and temporal footprints are extracted by factorizing the field in two lower rank matrices in a constrained way. The first constraint is that the residuals between observed and reconstructed videos should not have higher noise power than a threshold. The second constraint is the spatial and temporal footprints cannot have negative values. The third constraint is that the spatial components need to be sparse. Keeping all these constraints in mind, the overlapping components are merged iteratively. To achieve this, those components whose temporal correlations are higher than a threshold are merged after each iteration. In order to address a situation where the neurons are densely packed and lazy initialization might result in loss of resolutions in terms of lost neurons, the neurons are treated as Gaussian kernels and the initialization seeks to minimize the number of Gaussian kernels which can explain the observed activity. The approach is tested on an in vitro dataset containing 207 spinal motor neurons. The calcium imaging stacks are acquired at 14.6 Hz. The authors report a strong correlation between the activity traces obtained by the proposed approach and raw traces. On a simulated dataset, the proposed approach achieves a correlation of 0.965 between the ground truth and the extracted traces. The authors have not reported the performance of their approach in terms of neuron detection, or on the Neurofinder challenge [61] or any other manually curated dataset. [PERSON] et al. proposed a wavelet transform powered image segmentation and sparse dictionary learning-based pipeline for identifying neuronal activity in calcium imaging [62]. They refer to this pipeline as \"ADINA\". The pipeline is composed of two components, cell sorting and calcium transients extraction. ADINA solves the cell sorting as a sparse dictionary learning problem where a dictionary element represents a group of co-activated cells (basis functions). The \(N\times T\) image stack (\(N\) is the number of pixels in an image and \(T\) is the number of images in the stack) is assumed to be made of two elements, an \(N\times K\) spatial element and a \(T\times K\) temporal element where \(K\) represents the number of dictionary elements. A representative set of sparse basis functions is learned by keeping either one of the components (spatial and temporal) fixed while iterating over the other. After extraction of the basis functions, those basis functions are identified that might represent more than one cell by non-maximal suppression [63]. Finally, the cells are refined by fitting a 2D Gaussian for each cell. The authors validated their approach on both synthetic and real data. On synthetic data, the authors report a sensitivity of 94.3%. On a real imaging stack acquired from hippocampal cultures of different mice, the authors have qualitatively validated their approach but since no ground truth was available, they did not report quantitative results.