Two original frames taken from a cropped well is depicted in Fig. 1(ai) and 1(aii). Well cropping is often approximate and the well boundaries may be partially or completely visible in the cropped image sequence, as can be seen in Fig. 1(bi). Modelling cells on a uniform, zeromean background requires that any existing background be estimated and subtracted.
A probabilistic cell model is proposed as the product of two probabilistic terms associated with the cell brightness and the surrounding background. We also propose a background estimation method which is driven by the proposed cell model to identify and reject the well structures. In this section the background estimation method is first presented followed by the proposed probabilistic cell model.
Pointwise Background Estimation for Cell Detection
To estimate the background, let image I
_{
k
}be defined on a fixed lattice L:
{I}_{k}=\{{I}_{ijk}(i,j)\in L\}
(1)
thus for each frame I
_{
k
}of an image sequence we can write
{I}_{k}={F}_{k}+B+{n}_{k}\cdot 1+{V}_{k}
(2)
where F
_{
k
}is the dynamic foreground, B is the fixed background, n
_{
k
}models the temporal variations in global lighting, and V
_{
k
}is spatiotemporal random additive noise. The temporal noise n
_{
k
}is estimated over all pixels in each frame k
{\widehat{n}}_{k}=\underset{\left\{ij\right\}}{{\displaystyle mean}}({I}_{ijk})\{(i,j)\in L\}
(3)
For temporal correction, the estimated temporal noise is subtracted from the original frame I
_{
k
}:
{g}_{k}={I}_{k}{\widehat{n}}_{k}\cdot 1
(4)
The true background B = [B
_{
ij
}] is composed of stationary distortions and illumination variations at each pixel location. The pointwise estimation {\widehat{B}}^{0}of the background can be estimated over K frames of temporal corrected sequence g = (g_{1}, g_{2},..., g_{K}):
{\widehat{B}}_{ij}^{0}=\underset{\{k\in [1,K]\}}{mode}({g}_{(i,j,k)})
(5)
and is subtracted out from g:
\widehat{F}=\text{g}{\widehat{B}}^{0}
(6)
An imperfectly cropped well, the corresponding estimated background, and the corrected image are depicted in Figs. 1(bi), (bii), and 1(biii) respectively.
Justification of mode as statistical measure
Empirically, the motion of blood stem cells is essentially random, especially when observed minutes apart. Since cell motion is rarely zero, the spatial variations in cell brightness, mean that the variability of an image pixel, located within a cell, is considerably higher than the variability of an image pixel lieing in the background, whose variability is due only to random noise. Therefore, excepting cases of unusually small cell motion, the distribution of brightness values at a pixel should be most sharply peaked at the background, which is therefore recovered by the mode of the sample histogram.
Cell Detection in a Uniform Background
Let I = (I
_{1}, I
_{2},..., I
_{
K
}) be a set of K images which we will assume to consist of cells on a plain, uniform background. A typical microscopic multiwell image sequence I in our experiments consists of 32 separated wells, in each of which two to four HSCs are injected. Singlewell image sequences are cropped from the original multiwell image sequence and are processed individually. Frames 1 and 50 of a typical cropped well are depicted in Fig. 1(a).
In our previously designed [8] cell detection method, stemcell center locations were inferred from an image I
_{
k
}by
\begin{array}{c}{P}_{old}({z}_{k}^{m}{I}_{k})=\\ {P}_{cb}({z}_{k}^{m}{I}_{k})\cdot {P}_{in}({z}_{k}^{m}{I}_{k})\cdot {P}_{cdf}({z}_{k}^{m}{I}_{k})\end{array}
(7)
where {z}_{k}^{m}=({x}_{k}^{m},{y}_{k}^{m},{r}_{k}^{m}) represents a cell with radius {r}_{k}^{m} located at coordinate ({x}_{k}^{m},{y}_{k}^{m}), and where P
_{
cb
}, P
_{
in
}and P
_{
cdf
}characterize the bright cell boundary, the dark cell interior, and the boundary uniformity, respectively.
As we can see in Fig. 2(a), all HSCs cannot, in fact, be characterized by model (8) as a well recognizable dark interior and a bright boundary and the previous method in [8] performs poorly to detect them.
Different HSC phenotypes can be characterized as approximately circular objects with high intensity variations against the background, as can be observed in Fig. 2(a). Assuming a uniform background, we designed a general cell detection model in our previous work [9]. This method performed well to detect different HSC phenotypes, however it suffered from discontinuity and spurious detection. Moreover the gray level image was converted to a binary image using Otsu's thresholding at the expense of losing gray level features that could be used to improve cell detection.
Therefore to model the HSCs we propose an improved model, characterized by the following criteria:

1.
The cell is round, with some radius.

2.
The intensities of cell pixels deviate (both brighter and darker) from those of background pixels.

3.
Most of the pixels around the cell are near the background mean.
We propose a revised probabilistic cell model as the product of two probabilistic terms P
_{
cell
}and P
_{
back
}which are associated with the cell brightness and the surrounding background respectively:
P({z}_{k}^{m}{I}_{k})={P}_{cell}({z}_{k}^{m}{I}_{k})\cdot {P}_{back}({z}_{k}^{m}{I}_{k})
(8)
This revised model (8) is similar in sprit to [8], but generalized to different HSC phenotypes investigated in this work. The description and computation of each probabilistic term follows.
Cell Probability as a Circular Anomaly
A cell is modelled as a circular anomaly, darker or brighter than the uniform background. Let G({z}_{k}^{m},{I}_{k}) return a circular set of pixels
\begin{array}{c}G(z,I)=\\ \{{I}_{ij}{\left(xi\right)}^{2}+{\left(yj\right)}^{2}\le {\left(r\right)}^{2}\}\end{array}
(9)
The cell model assumes the background to have been subtracted, therefore the background mean is zero, and deviations from zero suggest the presence of a cell. Thus we extract the meansquare intensities
\overline{G}=\frac{{\displaystyle {\sum}_{g\in G}{g}^{2}}}{G}
(10)
The cell probability P
_{
cell
}is proposed, based on observations, to be an exponential
{p}_{cell}({z}_{k}^{m}{I}_{k})=1exp\{\overline{G}({z}_{k}^{m},{I}_{k})\}
(11)
so that P
_{
cell
}has a strong response for cell pixels, and weak (close to zero) for background pixels.
Penalizing False Candidates
Except for cells which are very tightly packed, most cells will be surrounded by background. Therefore to distinguish a cell center from a point between two adjacent cells, we can test for the presence of background pixels in a ring around the cell, between radii of r and 2r:
\begin{array}{c}E(z,I)=\\ \{{I}_{ij}{r}^{2}\le {\left(xi\right)}^{2}+{\left(yj\right)}^{2}\le {\left(2r\right)}^{2}\}\end{array}
(12)
Not all of the pixels surrounding a cell are necessarily background, but we do expect at least half of them as background, and penalize the meansquare deviation from zero of this fraction. Let {E}^{{\scriptscriptstyle \frac{1}{2}}} be the half subset of E with intensities closest to the background mean of zero:
{E}^{\frac{1}{2}}=\{e\in Ee<Median(E)\}
(13)
We then calculate the mean square {\overline{E}}^{{\scriptscriptstyle \frac{1}{2}}}
{\overline{E}}^{\frac{1}{2}}=\frac{{\displaystyle {\sum}_{e\in {E}^{\frac{1}{2}}}{e}^{2}}}{{E}^{\frac{1}{2}}}
(14)
We assume that we have an image sequence with a zeromean background plus additive noise which, if Gaussian, leads {\overline{E}}^{{\scriptscriptstyle \frac{1}{2}}} to be χ^{2} when {E}^{{\scriptscriptstyle \frac{1}{2}}} contains mostly background, so the cell/background hypothesis test can be approximated by a simple exponential:
{P}_{back}({z}_{k}^{m}{I}_{k})=exp\{{\overline{E}}^{\frac{1}{2}}({z}_{k}^{m}{I}_{k})\}
(15)
Locating The Cell Centers
To locate the cell centers, we compute the probability map P({z}_{k}^{m}{I}_{k})by applying the cell model in (8) to image frame I
_{
k
}, and then find the cell centers as the thresholded local maxima in P({z}_{k}^{m}{I}_{k}).
The threshold is computed analogously to [8], in which the threshold is varied and selected to minimize the sum of missed detections and false alarms. For the proposed cell metric (8), a threshold of 0.25 was found to be very effective, giving a detection rate over 95%, and a false alarm rate of approximately 6%. Fig. 2(b) shows the application of the cell model to a cropped well interior with no boundaries (Fig. 2(bi)) and a coarsely cropped well before and after background correction is depicted in Figs. 3(ai) and 3(aiii) respectively. As it can be observed, the cell model performs very poorly (Fig. 3(aii)) where the cropped well contains visible well boundaries, however the cell model performs perfectly where it is applied to a cropped well interior with no boundaries (Fig. 2(bii)) or a background corrected coarsely cropped well (Fig. 3(aiv)).