Combining Mathematical Morphology and the Hilbert Transform for Fully Automatic Nuclei Detection in Fluorescence Microscopy

Quantitative biology relies heavily on the analysis of fluorescence microscopy images. Key to a large number of experiments is the ability to detect and count the number of cells or nuclei in an image or region of interest. Frequently, such as for high-throughput and large-scale imaging experiments, this is the first step in an automated analysis pipeline. As such, the scientific community requires novel, robust and automatic approaches to this essential step.

Manual cell counting and annotation, by visual inspection, is difficult, labour intensive, time consuming and subjective to the annotator involved. As such, a large number of groups have proposed a variety of approaches for automatic nuclei detection in fluorescent microscopy images. In recent years proposed methods have included:

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  variations upon thresholding-based segmentation, which are limited in performance due to intensity inhomogeneity and nuclei/cell clustering [5];

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  H-minima transforms [8];

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  voting-based techniques [18], which both show good results but are sensitive to parameters;

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  gradient vector flow tracking and thresholding [9, 10], such PDE-based methods require strong stopping and reinitialisation criteria, set in advance, to achieve smooth curves for tracking;

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  using Laplacian of Gaussian filters [17], which is low computational complexity but struggles with variation in size, shape and rotation of objects within an image; and

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  graph-cut optimisation approaches [12], which requires the finding of initial seed points for each nucleus.

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  convolutional neural networks, which generally require copious amounts of manually labelled training data, and struggle to separate overlapping objects [7]

In our experience, nuclei size (scale) is an important contributor to 1. whether or not an algorithm is successful 2. how robust an algorithm is to image variation and 3. to the running time of an algorithm. We have also found that many excellent algorithms exist for detecting small blobs, on the scale of a few pixels diameter, that fail or are less reliable for medium or large blobs, i.e. over ten pixels diameter, when such objects tend to display significant eccentricity. At this and larger scales the number of algorithms available decrease or make some assumptions on the data.

In this paper we apply a new ellipse detection algorithm based on mathematical morphology and the Hilbert transform to the task of counting medium blobs, i.e. nuclei, on the scale of $1050$ pixels in fluorescence microscopy images. This new algorithm requires no preprocessing and no training. Simplistically, the algorithm performs three steps: 1. Mathematical morphology is used to create an object sparse search space over different orientations and scales; 2. The Hilbert transform is used to search this space and range object edges and for determining the ellipse/nucleus geometry at all pixels of the image; 3. Prominent ellipses/nuclei are extracted and post-detection pruning is used to eradicate over detection. We call this method Hilbert Edge Detection and Ranging (HEDAR). All code is available on GitHub at https://github.com/ChasNelson1990/Ellipse-Detection-by-Hilbert-Edge-Detection-and-Ranging.

In the rest of this paper we introduce HEDAR (Section 2) and compare this method to several traditional and new algorithms for blob/nuclei counting in fluorescence microscopy (Section 3). We have used the Broad Bioimage Benchmark Collection [11] throughout this paper to ensure comparison with previously published results, which we have reported where appropriate.

Our new algorithm, Hilbert Edge Detection and Ranging, is a three step process. First, we erode the original 2D, greyscale image $I$ with a bank of line structuring elements $SE_{d,\theta}$ of different integer pixel lengths $d\in[1,d_{\max}]$ and rotations $\theta\in[0,180)$. We then stack the resulting eroded images into a four dimensional array $M$, such that $M(x,y,d,\theta)=(I\ominus SE_{d,\theta})(x,y)$. We note that in 2D pixel-space we can define the limit on the number of rotations as $N_{\theta}=2d$, i.e. the perimeter of the bounding box of the half-circle covered by all (symmetric) line structuring elements of length $d$. Thus, for a given $x,y,\theta$ where the point $(x,y)$ lies inside a bright ellipse, $M(x,y,:,\theta)$ yields a morphological intensity signal that drops off at the distance $s$ from $(x,y)$ to the boundary of the ellipse along the direction $\theta$.

Next, we calculate the 1D Hilbert transform over all morphological signals. The transform will be maximal at a step in the morphological signal from high to low (and minimal at a step from low to high). By identifying the maxima in each signal we can range the distance $s$ from the current pixel to the nearest edge along that angle. Mathematically, we can define this as,

The major axis $a$ of any ellipse, i.e. nucleus, centred at $(x,y)$ in the image will be,

We can concurrently identify the orientation $\phi$ of any nucleus centred at $(x,y)$ in the image as,

We then extract the distance to any edge at $90\mathrm{°}$ to the major axis $a$ to determine the minor axis $b$ of any nucleus centred at $(x,y)$ in the image. Using the Hilbert transform makes this process robust to noise and blurring.

Like other mathematical morphological image processing approaches, such as neuriteness in vessel enhancement, so far the algorithm has assumed the presence of an ellipse at all points (c.f. assuming a vessel in neuriteness), the next step is to identify likely ellipses/nuclei. To accomplish this we element-wise multiply the minor and major axes at every pixel. This produces a heat-map with high prominence peaks at the centres of any possible nuclei.

The prominence map is masked with a minimum threshold size $t$ for Hilbert steps in morphological signals. This acts similarly to local thresholding of the image: as each step must be at least as large as the threshold, we only consider pixels where the possible nucleus is brighter than its local background by at least $t$. This suppresses false positives caused by background noise. After smoothing the masked prominence map with a $3\times 3$ pixels median filter we then identify the regional maxima. Each maximum represents a nucleus and we determine the major axis, minor axis and orientation from previous steps.

In post-processing, we cycle through all detected ellipses, removing the more overlapped ellipse of any pair with intersection-over-union greater than $50\text{\,}\mathrm{\char 37\relax}$.

We report results on image sets BBBC001v1 [3], BBBC002v1 [3], BBBC004v1 [15] and BBC039v1 [2] of the Broad Bioimage Benchmark Collection [11].