Maximizing Kolmogorov Complexity for accurate and robust bright field cell segmentation
 Hamid Mohamadlou^{1},
 Joseph C Shope^{4} and
 Nicholas S Flann^{1, 2, 3}Email author
DOI: 10.1186/147121051532
© Mohamadlou et al.; licensee BioMed Central Ltd. 2014
Received: 19 March 2013
Accepted: 18 December 2013
Published: 30 January 2014
Abstract
Background
Analysis of cellular processes with microscopic bright field defocused imaging has the advantage of low phototoxicity and minimal sample preparation. However bright field images lack the contrast and nuclei reporting available with florescent approaches and therefore present a challenge to methods that segment and track the live cells. Moreover, such methods must be robust to systemic and random noise, variability in experimental configuration, and the multiple unknowns in the biological system under study.
Results
A new method called maximalinformation is introduced that applies a nonparametric information theoretic approach to segment bright field defocused images. The method utilizes a combinatorial optimization strategy to select specific defocused images from each image stack such that set complexity, a Kolmogorov complexity measure, is maximized. Differences among these selected images are then applied to initialize and guide a level set based segmentation algorithm. The performance of the method is compared with a recent approach that uses a fixed defocused image selection strategy over an image data set of embryonic kidney cells (HEK 293T) from multiple experiments. Results demonstrate that the adaptive maximalinformation approach significantly improves precision and recall of segmentation over the diversity of data sets.
Conclusions
Integrating combinatorial optimization with nonparametric Kolmogorov complexity has been shown to be effective in extracting information from microscopic bright field defocused images. The approach is application independent and has the potential to be effective in processing a diversity of noisy and redundant high throughput biological data.
Background
Cell segmentation is the identification of cell objects and their observable properties from biological images. Current cell segmentation methods perform most accurately when applied to high contrast and minimal noise images obtained from samples where the cells have fluorescentlylabeled cell nuclei and stained membranes, and are distinct with minimal adherent membranes. However, these ideal conditions rarely exist.
Fluorescently tagging cells using green fluorescent protein (GFP) leads to robust identification of each cell during segmentation. While GFP tagging is widespread, there are disadvantages when applying the method repeatedly to the same sample since under repeated application of highenergy light the cells can suffer phototoxicity. Such light can disrupt the cell behavior through stress, shorten life and potentially confound the experimental results [1–3]. Significantly, a requirement for GFP labeling adds a step before a new cell line can be studied, thus making it difficult to apply this method in a clinical setting.
The alternative is to use bright field microscopy, the original and the simplest microscopy technique, wherein cells are illuminated with white light from below. However, using only bright field imaging of unstained cells presents a challenging cell detection problem because of lack of contrast and difficulty in locating both cell centers and borders, particularly when cells are tightly packed. Bright field imaging, while eliminating phototoxicity, leads to an excess of segmentation errors that significantly reduce biological and medical utility.
We seek to remedy the disadvantages and harness the experimental advantages of bright field microscopy of living cells by applying informationtheoretic measures over defocused images to improve segmentation accuracy. The approach applies Kolmogorov complexity to identify the most informative subset of images within the focal stack that maximize information content while minimizing the effect of noise.
The paper first briefly reviews existing methods for segmentation of living cells, with a focus on recent approaches to defocused bright field images. Next, measures of Kolmogorov complexity are introduced and applied to image data. The new maximalinformation method is then defined and evaluated by comparing its performance with a recent method sephaCe[3] over image sequence data sets from three separate experiments. An analysis and a discussion of the results follows.
Cell segmentation methods
Several cell segmentation approaches have been developed over time for detection of live cells in microscopy images [4–7]. Most of the approaches binarize an image with certain thresholding techniques, and then use a watershed or levelset based method on either intensity, gradient, shape, differences in individual defocused images (referred to as frames) [3, 8], or other measures. The algorithms then remove small artifacts with size filters, and apply merge and split operations to refine the segmentation [4–6].
Florescent microscopy cell segmentation
Most studies can primarily be categorized into a few key approaches. Wavelets are used for decomposing an image in both the frequency and spatial domain, and can be an effective tool since wavelets are robust to local noise and can discard low frequency objects in the background. Genovesio et al. [9] developed an algorithm to segment cells by combining coefficients at different decomposition levels. Wavelet approaches work well with whole cell segmentation, but have difficulty to segment internal cell structures. In Xiaobo et al. [10] a watershed algorithm was introduced for cell nuclei segmentation and phase identification. Using adaptive thresholding and feature extraction, Harder et al. [11] classified cells into four cell classes comprising of interphase cells, mitotic cells, apoptotic cells, and cells with clustered nuclei. In Solorzano et al. [12] the level set method determines cell boundaries by expanding an active contour around each detected cell nuclei.
While these cell segmentation algorithms have been developed for fluorescence microscopy images, defocused bright field cell segmentation demands more complex and advanced level of image processing. Broken boundaries, poor contrast, partial halos, and overlapping cells are some of the shortcomings of available algorithms [3, 8] when applied to images lacking fluorescent reporters.
Defocused bright field microscopy approaches
Selinummi et al. [13] introduced zprojection based method to replace whole cell florescent microscopy with bright field microscopy. This method computes an intensity variation over a stack of defocused images (referred to as the zstack) to obtain a contrastenhanced image called a zprojection. Since variability of pixel intensity inside a cell is high compared to the background, the resulting zprojection image has high contrast and can substitute for an image obtained through whole cell florescent microscopy. The zprojection approach is straightforward and free from parameters setting. However, in order to distinguish between adherent cells, a second channel of nuclei florescent microscopy is required. As a final step CellProfiler[14] software is applied to both the zprojection and nuclei florescent channel to produce cell segmentation. While the zprojection approach avoids whole cell florescence, it still requires an additional nuclei channel of florescent microscopy and so does not eliminate potential problems with cell toxicity.
Implementation
A recent method that needs only brightfield defocused images has been introduced in sephaCe[3]. This system is capable of both the detection and segmentation of adherent cells and can be downloaded from (http://www.stanford.edu/~rsali/sephace/seg.htm) as a free and open source image analysis package. In contrast to Selinummi et al. where all the frames of the zstack are utilized, sephaCe selects only a subset of five frames as input to the image processing system. sephaCe selects this subset using a hardcoded strategy independent of each data set and each individual zstack contained within that data set. Therefore sephaCe does not adapt to the inevitable equipment and biological sample variation. While parameters of the image processing method can be tuned for specific data sets somewhat ameliorating the problem, a more general purpose nonparametric frame selection method is needed for highthroughput processing of diverse data sets. This work introduces a new adaptable frame selection method that applies an information theoretic measure to select frame subsets specific to the idiosyncracies of each zstack. This method is referred to as maximalinformation.
Where p(I(x,y)) is the probability of pixel intensity values. Entropy value is expected to be maximized for strongly out of focused images and minimized for the infocus image. Let the infocus image frame be I^{0}.
After detecting the infocus image, four additional images from the zstack are selected, two above the infocus frame and two below. To initialize the level set algorithm, a difference image is generated from two strongly defocused images selected at a fixed distance of ±25 μ m from the infocus frame, referred to as I^{++} and I^{}. This image is binarized using the Otsu [17] thresholding method and then small artifacts are removed by labeling connected components and applying size filter.
To guide the level set algorithm in expanding the initial cell boundaries, another difference image is generated between two slightly defocused images ±10 μ m from the infocus frame, referred to as I^{+} and I^{}. Details on how this difference image is applied to compute local phase and local orientation images that direct the border expansion is given in [8] and [3].
Motivation for the maximal information approach
However, in experiments over a diversity of images (given in Section Results) this fixed selection of outoffocus frames is demonstrated to produce poor segmentation. A fixed strategy cannot take into account random and systemic noise, variability in experimental configurations including microscope configurations, and multiple unknowns in the biological system under study. Some of these conditions are illustrated in selected frame images in Figure 1(c). Two possible reasons to account for the irregular entropyfocus plane relationship in Figure 1(b) are:

Biological variability where cells do not adhere to the flat surface of the culture medium but vary in the zdimension as they change morphology and form cellcell adhesive bonds. That is, a focused frame for one cell could be a defocused frame for other cells. In Figure 1(c), the bright upper cell is positioned higher than the rest. Therefore a semirandom level of sharpness resides in the all defocused images.

Systemic noise introduced by microscopy and imaging. For instance in Figure 1(c), frame 6 has strip noises introduced by the camera. Strip noise residing in the image increases the entropy value from the 5’th frame to 6’th frame while a decrease is expected.
Applying this fixed distance strategy to select strongly defocused frames can add unwanted initial active contours resulting in oversegmentation and also can miss initial active contours resulting in undersegmentation. Likewise, fixed selection of weakly defocused frames can add anomalies into the local phase and orientation images and thus misdirect the contour expansion to include or exclude cells, particularly when cells are tightly packed.
Overall, the fixed approach in selecting initial images in the sephaCe package is brittle and errorprone. The unavoidable variation requires an adaptable method rather than a fixed approach. The maximalinformation method uses an optimization based approach that searches the combinations of zstack frames to select the four frames that contain the highest information, evaluated using Kolmogorov informationtheoretic measure [18]. This process is repeated for each individual zstack and so adapts to the distinctiveness of each sample. Since the maximalinformation method is adaptive, it can be applied to a diversity of data sets utilizing different microscopes, lighting conditions and biological samples.
Kolmogorov information set complexity
Set complexity [19], denoted Ψ, is applied to quantify the amount of information contained within each possible set of four image frames. The measure is general purpose and nonparametric in that it computes the information content of set of objects so long as they can be encoded as strings. Set complexity has been applied to understand the organization and information content of biological data sets including developmental pattern formation [20], genetic regulatory network dynamics [21], and gene interaction network structure [22]. The Kolmogorov complexity [18] of a string is the length of shortest algorithm that can be used to generate the string. Exact computation is undecidable, but it can be approximated by the compression size of a string. Bzip2 and zip compressor with block size of 900 Kbytes have been tested and shown robust for this purpose.
 1.
$C({s}_{i}^{s}+{s}_{j}^{s})\simeq C\left({s}_{i}^{s}\right)\simeq C\left({s}_{j}^{s}\right)$ then $\mathit{\text{NCD}}({s}_{i}^{s},{s}_{j}^{s})\simeq 0.0$
 2.
$C({s}_{i}^{r}+{s}_{j}^{r})\simeq C\left({s}_{i}^{r}\right)+C\left({s}_{j}^{r}\right)$ then $\mathit{\text{NCD}}({s}_{i}^{r},{s}_{j}^{r})\simeq 1.0$
 3.
$C({s}_{i}^{r}+{s}_{j}^{s})\simeq C\left({s}_{i}^{r}\right)$ and $C\left({s}_{j}^{s}\right)\simeq 0.0$ then $\mathit{\text{NCD}}({s}_{i}^{r},{s}_{j}^{s})\simeq 1.0$
Where s^{ r } is from the set of random strings and s^{ s } are simple strings containing regular patterns.
Set complexity captures the relationships among strings in the set, discounting when strings are very similar (N C D close to 0.0) and so contain the same information, or highly dissimilar so that they have nothing in common and appear random (N C D closer to 1.0). The value is maximized when each string is intrinsically complex (high C(S_{ i })) and the similarity between the strings lies between maximally dissimilar and maximally similar N C D(s_{ i },s_{ j })≃0.5, which occurs when C(s_{ i }+s_{ j })≃C(s_{ i })/2C(s_{ j }), assuming C(s_{ i })>C(s_{ j }).
The NCD values for the four image frames given in Figure 2
NCD  I ^{++}  I ^{+}  I ^{}  I ^{} 

I ^{++}  0.0  0.1429  0.2154  0.1071 
I ^{+}  0.0  0.0  0.2615  0.1296 
I ^{}  0.0  0.0  0.0  0.2000 
I ^{}  0.0  0.0  0.0  0.0 
The maximalinformation segmentation method
To select the four most informative frames from a zstack with n frames, the method searches the space of all possible combinations of two frames from above the infocus frame (I^{++} and I^{+}) and two frames from below the infocus frame (I^{} and I^{}), evaluates each set for Ψ, then picks the maximizing combination. The method is given in Algorithm 1.
Algorithm 1 The maximalinformation algorithm to select the four zstack frames needed to initialize the level set method for segmentation. Let the input zstack be I containing n frames. The algorithm returns the infocus frame and four defocused frames. Note that all compression calculations are calculated once and cached.
First each image in the zstack is binarized using the Otsu [17] thresholding method and then converted to a string (linearization) by concatenating each column of the image to the next column [27]. Many methods of linearization were explored in [27] and column concatenation was found to be effective because spatially located regularities are picked up by compression. Bzip2 is applied to compute the compression size of each individual string and also each pairwise concatenated string (for N C D, Equation 2). From these cached compression values, pairwise N C D values are determined.
The O(n^{2}) compression step dominates the computation time since strings must be written to file before processing; the final Ψ calculation involves only matrix operations and is very fast, even though more combinations must be computed. For the three data sets studied in this work, the preprocessing and level set algorithms of sephaCe take approximately 10 seconds per zstack. The maximalinformation frame selection method adds approximately 20 seconds per zstack to the run time. Timings were on an Intel Pentium G640 Processor 2.8 GHz (3 MB cache).
Results
Set complexity analysis of image data
To understand how Kolmogorov Complexity measures could reveal information in zstacks, an initial study was performed by computing the N C D between each pair of 21 frames for three data sets each containing 192 zstacks. The data sets used for in this work are human embryonic kidney cells (HEK 293T) sampled at 5 minute intervals for 16 hours. Each zstack sequence is from a distinct experiment. Data was obtained using a Leica DM6000 microscope with each zstack containing 21 image frames each separated by 10 μ m, with resolution 1024 × 1024 12bit greyscale pixels. Since the zstack was sampled at a 10 μ m resolution, the strongly defocused frames for sephaCe were set at ±30 μ m.
Set complexity values for two different approaches
Fixed defocused distance (sephaCe)  Selected by maximalinformation  

Mean  278.5049  345.1289 
Variance  10620.73  12336.47 
Observations  192  192 
Pearson correlation  0.9603  
P(T<=t) onetail  1.19825E67  
t Critical onetail  1.6536  
P(T<=t) twotail  2.3965E67  
t Critical twotail  1.9736 
Precision and recall analysis
Segmentation results for three data sets for human embryonic kidney cells (HEK 293T)
Data set one  Maximalinformation  SephaCe  Correlation  t stat  P(T≤ t)onetail 

Correct segmentation t p  9.12  5.76  0.3970  9.4557  0.0 
Unexpected areas f p  0.68  0.80  0.2355  0.5492  0.2939 
Missing cells f n  1.60  4.72  0.0909  9.0929  0.0 
Precision Pr  93.20%  89.36%  0.3295  1.4461  0.0805 
Recall Re  85.37%  54.34%  0.2903  8.2830  0.0 
Data set Two  Maximalinformation  SephaCe  Correlation  t stat  P ( T ≤ t )onetail 
Correct segmentation t p  13.35  12.60  0.4344  3.4701  0.0012 
Unexpected areas f p  1.15  2.20  0.1633  4.0977  0.0003 
Missing cells f n  0.50  1.25  0.2939  3.4701  0.0012 
Precision Pr  92.30%  85.45%  0.1690  4.3714  0.0001 
Recall Re  96.40 %  91.08%  0.2822  3.4407  0.0013 
Data set three  Maximalinformation  SephaCe  Correlation  t stat  P ( T ≤ t )onetail 
Correct segmentation t p  15.56  11.86  0.4549  10.18  0.0 
Unexpected areas f p  1.72  2.00  0.3642  0.9434  0.1759 
Missing cells f n  2.81  6.36  0.4926  9.9501  0.0 
Precision Pr  91.66%  86.23%  0.3887  2.6898  0.0 
Recall Re  85.94%  65.21%  0.4256  10.12  0.0 
In Table 3 the average correctly segmented cells for maximalInformation is higher than sephaCe method and demonstrates the advantage of extracting more informative frames in the zstack. The average of both missing and unexpected cell segmentation for maximalinformation are lower than sephaCe method. All three of these measures of quality are shown to be significantly better for maximalinformation than for the sephaCe using a paired onetail Ttest (values that are less than 10^{8} are reported as 0.0 in the table).
In addition, Table 3 includes the intermethod correlation of t p, f p, f n over the zstack data sets. High correlation implies that the performance of both methods is consistent in that they perform poorly on the same set of “difficult” images, and well on the same set of “easy” images. Results in Table 3 show that true positives are highly correlated implying that the cells correctly identified by maximalinformation include some of the set of cells recognized by sephaCe.
Conclusions
This work has presented a method for identifying live cells in bright field defocused images. The method applies Kolmogorov complexity measures to identify specific outoffocus frames that encode the maximum information. These frames are then used to initialize active contours and guide contour expansion for levelset segmentation algorithms as applied in the sephaCe method.
The new maximalinformation approach is compared with a selection strategy employed in the original sephaCe that picks outoffocus frames using fixed offsets from the estimated infocus frame. An empirical study using a large data set of embryonic kidney cells (HEK 293T) zstacks taken from different experimental runs has demonstrated that the adaptive method significantly improves the recall and precision of the segmentation.
Kolmogorov set complexity identifies the most informative frames by exploiting similarity measures between all pairs of frames contained within the N C D matrix. Each selected frame is sufficiently dissimilar (high N C D) to other frames in the set so as to provide unique and synergistic information about each cell in the zstack. Recall that the dissimilarity is due to changes in cell appearance as the focal plane is moved through the cell profile. By selecting the best degree of dissimilarity, the differences between frames (used to initialize and guide the active contour of the levelset method) maximize sensitivity to the presence and shape of cells. Kolmogorov set complexity also tempers the effects of noise by discounting frames that have too higher dissimilarity since this is most likely due to noise.
The method introduced here is generally applicable because it relies on fundamental nonparametric informationtheoretic properties and treats data as simple strings, ignoring the actual semantics. Robustness is achieved by viewing frame selection as combinatorial optimization problem with set complexity as the scoring function. The full potential of the method in dealing with noise, variability in experimental configurations, and multiple unknowns across a diversity of biological data will be explored in further studies.
Availability and requirements
Project name: maximalinformation
Project home page:https://sites.google.com/site/maximalinformation,
Operating system(s): Platform independent
Programming language: Matlab
Other requirements: requires sephaCe[3] downloadedfrom (http://www.stanford.edu/~rsali/sephace/seg.htm
License: GNU GPL
Any restrictions to use by nonacademics: Contactcorresponding author
Declarations
Acknowledgements
This work was supported by the Luxembourg Centre for Systems Biomedicine, the University of Luxembourg and the Institute for Systems Biology, Seattle, USA. Research reported in this publication was partially supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number P50GM076547. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Thanks to Ilya Shmulevich and Pekka Ruusuvuori for helpful discussions, and to Adrian Ozinsky for image data.
Authors’ Affiliations
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