## Abstract

An investigation is reported of the identification and measurement of region of interest (ROI) in quantitative phase-contrast maps of biological cells by digital holographic microscopy. In particular, two different methods have been developed for in vitro bull sperm head morphometry analysis. We show that semen analysis can be accomplished by means of the proposed techniques . Extraction and measurement of various parameters are performed. It is demonstrated that both proposed methods are efficient to skim the data set in a preselective analysis for discarding anomalous data.

©2011 Optical Society of America

## 1. Introduction

The assessment of male fertility potential is very important prior to performing artificial insemination (AI) or in vitro fertilization (IVF) to ensure good results. In this context, sperm morphology has been identified as a characteristic that can be useful in the prediction of fertilizing capacity. In fact, decreasing fertility due to poor semen morphology has been observed in men [1], stallions [2], and bulls [3]. Moreover, sperm head abnormalities have been associated with early embryonic loss, lowered fertility and embryo quality [4], and reduced capacity to bind to the ovum [5]. As sperm analysis is typically performed through human visual inspection, the process as a whole is characterized by a high degree of subjectivity and bias. Thus, the need for accurate objective assessment of sperm morphology has led to the development of computer-assisted sperm head morphometry analysis [6,7]. Sperm morphology is generally quantified in terms of the following morphological features of sperm head: head area, perimeter, width, length, width-length ratio, ellipticity, and so on. Several methods have been developed to increase the sensitivity of automated analysis, allowing the identification of minute differences between head of sperm. On the other hand, spermatozoa appear essentially transparent under a bright field microscope unless phase contrast microscopy is used. Conventional DIC microscopy [8] is inherently a qualitative technique due to the nonlinear relationship between the image intensity and the optical path length (OPL) sample. Phase contrast microscopy proposed by Zernike [9] was a major advance in intrinsic contrast imaging, as it revealed inner details of transparent structures without staining or tagging. Recently, the Digital Holographic (DH) microscope configuration has been successfully applied to obtain an accurate quantitative three-dimensional morphological analysis of sperm cells [10,11]. The possibility offered by DH to manage quantitative information in a user-friendly format without any harmful procedure that could alter the physiology of the cells such as, staining, labeling, or electrical or physical stresses, opens the possibility to use this approach as an in vivo technique for complete sperm analysis. In fact, one of the main advantage of the DH is the possibility to directly retrieve a digitalized quantitative phase-contrast of the morphology of the analyzed spermatozoon. Thus, this image can be used to perform an accurate computer-assisted sperm head morphometry analysis. The first step to perform accurate 3D morphometry analysis is to detect and extract in accurate way the 2D region of interest (ROI) in the phase-contrast maps. To this aim, in this paper two different techniques to identify and measure the region of spermatozoon head by DH approach are investigated and compared with each other.

For extracting the region-of-interest (ROI) in a particular phase-contrast map it is is necessary to develop and apply specific algorithms. The choice of the more appropriate algorithm could be related also with the adopted microscopy technique (Zernike, DIC, or DH).

One classical algorithm for detecting ROI was found a long time ago by Canny [12]. Many others algorithms have been proposed to automatically detect ROIs [13], where the effectiveness of approaches is related to the specifications of the analyzed images. For example, in [14] is proposed an approach to determining ROI within an object from truncated differential projection data on differential phase-contrast tomography. Instead in [15] an adaptive multigrids approach to improve the computational efficiency and the quantitative accuracy of Diffuse Optical Tomography image reconstruction is investigated. In [16] a fast and scalable alternating optimization technique to detect ROIs in cluttered Web images without labels is proposed. This method is an alternating optimization approach for scalable unsupervised ROI detection by analyzing the statistics of similarity links between ROI hypotheses.

Other results for remote sensing images, in which a new technique for ROI coding is applied, named partial multiply bitplane alternating shift, are obtained and described [17]. An algorithm for the ROI detection of biomedical images, which is based on alternating sequential filtering and watershed transformation, is proposed [18].

Due to the lack of benchmarks, the methods proposed for ROI detection were often tested on small data sets that are not available to others, making reasonable comparisons of these methods difficult. Examples from many fields have shown that repeatable experiments using published benchmarks are crucial to the fast advancement of the fields. To fill the gap, in [19] is presented a collaborative game approach, called Photoshoot, which collects human ROI annotations for constructing an ROI benchmark.

In our work reported here a novel aspect is the application of the two proposed ROI detection techniques of quantitative phase-contrast maps (QPMs) in DH. In fact, to the best of our knowledge, only few papers analyze of ROIs on QPMs coming from digital holograms or other interference microscopy techniques. An experimental method to visualize a 3D ROI by means of an astigmatic Gaussian beam is proposed [20]. This method allows one to reduce the amount of image planes to be reconstructed, thus saving computational time. ROI determination is performed without any computational step because particles that are located in the ROI can be distinguished from the others according to the hyperbolic shape of their diffraction pattern. Theoretical location of the ROI is determined by using the ABCD approach for in-line DH [21]. An efficient method for extracting 3D ROIs of object volume directly from a hologram is presented in [22]. This method consists of a rapid focus detection algorithm, a summation kernel for computing a sum image onto which of a sequence of object images are projected and an image segmentation algorithm. Sperm head detection and extraction from microscopic images, is usually performed using adaptive thresholding techniques directly on the input image [23,24]. In [23] the *n* different threshold values are given by the prior knowledge of the object's morphological structure. In [24] is proposed a modification of Ostu-based segmentation that remove impurities. However, the presence of severe speckle noise from phase reconstructed holograms makes this technique unpractical, unless denoising is performed. So it is important to design a procedure that is based on holographic properties of phase reconstruction.

In the following sections two different strategies, based on iterative methods, are investigated with the aim to estimate sperm heads. One of them is specifically developed to be adapted to sperm cell studied in the our experimental work. The methods are described, the results are reported and compared.

## 2. Proposed algorithms

We propose two different algorithms to extract a sperm head in phase-contrast map reconstructed by digital holograms. The first, which we will call Algorithm 1, is based on a nonlinear diffusion filter. This method, introduced by Perona and Malik in the late eighties [25] is a method for searching edges in general images without any privileged assumptions.

Their model has been thought to build a strong filter contribution in areas of the image where gray levels are locally coherent, while maintaining almost unchanged areas with high gradient, in order to preserve edges. Let $\Omega =\left(0,{a}_{1}\right)\times \cdot \times \left(0,{a}_{m}\right)$be our image domain in ${\Re}^{m}$, $f\left(x\right)\in {L}^{\infty}(\Omega )$ be a scalar image, and $u\left(x,t\right)$ its filtered version computed by solving a nonlinear diffusion equation with the original image as initial state and homogeneous Neumann boundary conditions

with *n* denoting normal to the image boundary $\partial \Omega $ and “*div*” the divergence operator. Time *t* is a scale parameter. Equation (2) refers to the initial condition and Eq. (3) to the behavior of the filter at the boundary conditions, while all other cases are governed by Eq. (1) and its diffusivity function *g*. To reduce smoothing along edges, diffusivity *g* is chosen as the inverse of ${\left|\nabla {u}_{\sigma}\right|}^{2}$, where

where “$\ast $” is the convolution operator and Eq. (4) is a Gaussian smoothed image *u* with

The diffusivity is then

Smoothing along both sides of an edge has strong effects with respect to smoothing across it. This selective process prefer intraregional smoothing to interregional one. Choosing *c* ≈3.315 ensures that the flux $\Phi \left(s\right)=s\text{\hspace{0.17em}}g\left({s}^{2}\right)$ is increasing for $\left|s\right|\le \lambda $ and decreasing for the opposite. The anisotropic diffusion filter used in this paper, implements the Additive Operator Splitting (AOS) scheme introduced in [26], which separates and discretize Eq. (1). For this scheme, the diffusivity need not be equal in all directions; we want to approximate the continuous process by a discrete algorithm. Again, the AOS scheme will separate the 2D diffusion into several one-dimensional diffusion processes along chosen directions. Threshold λ has been chosen as the following median absolute deviations:

with *f* the initial image. So far, the anisotropic diffusion filter results are thresholded with the automatic binarization mechanism introduced by Otzu [27]. Otzu proposed a criterion for maximizing the between-class variance of pixel intensity to perform picture thresholding.

The second method presented here, which we will call Algorithm 2, is based on a particular intrinsic structural feature in QPMs reconstructions of sperm cells. In the hypothesis in which in the field of view is present only one sperm, we can see that the QPM shows a maximum value near the sperm head, as is clearly visible in other interferometry studies, too [10,28–31]. In Fig. 1 are shown typical QPMs obtained by DH in liquid sperm. Clearly visible in both QPMs is a maximum into the head region.

It is important to notice that sperm cells in liquid have a very low phase-contrast due to the natural refractive index matching between the cell and the hosting medium, thus making it difficult to accomplish the task to identify and extract the ROI of the head. In fact the phase-difference into the head is comparable with the spatial fluctuation into the surrounding liquid medium. The extraction algorithm consists in denoising the QPM using a threshold filtering, with a fixed threshold value *S* equal to the half of maximum phase value, and then to create two binary rectangular masks on the phase reconstruction as indicated below.

The steps of Algorithm 2 are the following:

- • Find the maximum value on the phase map.
- • Choose the dimension of rectangular mask M
_{x}and M_{y}(pixels unit). - • Design two masks (labeled ${B}_{1}$ and ${B}_{2}$), one of them centered in the maximum value, the other that begin on the maximum value (see Fig. 2 ).
- • Denoising the image.
- • At first iteration: mask ${B}_{1}$ and ${B}_{2}$ are applied on the phase denoised map and mean values ${\mu}_{k}^{1}$,
*k*= 1,2 are estimated. - • At generic iteration (q): masks are rotated of an angle $\Delta \vartheta $, obtaining ${B}_{i}^{q}=\text{rotate}\left({B}_{i},{\vartheta}^{q}\right)$ where ${\vartheta}^{q}={\vartheta}^{1}+q\Delta \vartheta $,
*i*= 1,2. These rotated masks are applied on the phase map and mean values ${\mu}_{k}^{q}$,*k*= 1,2 are computed. The number of iteration is related to the angle step $\Delta \vartheta $ and is given by ${q}_{\mathrm{max}}=\frac{2\pi}{\Delta \vartheta}$. - • After last iteration, the quantities ${\tilde{\mu}}_{k}=\mathrm{max}\left\{{\left({\mu}_{k}^{q}\right)}_{q=1}^{{q}_{\mathrm{max}}}\right\}$,
*k*= 1,2 are evaluated and the best value of these two $\tilde{\mu}=\mathrm{max}\left\{{\tilde{\mu}}_{1},{\tilde{\mu}}_{2}\right\}$ is estimated. - • Extraction of head using the mask related to the maximum mean value $\tilde{\mu}$.

Then, we want to find the best ellipse that fits the region selected by each algorithm. Considering the areas *y* as a random variable, density estimation is performed with Parzen windows [32]. This will allow us to make no assumptions on the underlying ellipse area’s statistics and discover the form of the random variable’s density in an unsupervised manner starting only from ${\left\{{y}_{i}\right\}}_{i=1}^{N}$, which is the extracted data set with the two above described algorithms. The sample measurements is composed of *N* data and the density estimation is performed with Gaussian kernel *K*

where the estimate ${\widehat{p}}_{N}\left(y\right)\to p\left(y\right)$ for $N\to \infty $. The kernel function is defined over profile $K(\u2022)$ with bandwidth ${h}_{N}$, which represents a windows function used to interpolate data distribution, i.e. each sample contribute to the estimate of p based on the distance form ** x**. A critical choice is the bandwidth value (resolution), where large

*h*values results in too much smoothed estimated density, while lower values results in crisp densities. An optimal value is given by the median of absolute deviations ${h}_{N}=c\cdot MAD\left\{{y}_{1},\mathrm{...},{y}_{N}\right\}$, with

*c*a constant factor (in the experiments

*c*= 1/4) empirically found [33].

An elegant way to find the mode of the distribution without the need to compute the entire distribution is given by the mean-shift procedure introduced in [34] where two different kernel functions have been investigated: Epanechnikov and Gaussian. In our experiments we found the Normal kernel provides a smoothed version of the density estimate where its profile is

## 3. Experiments and results

In this section, we compare the results of the two developed procedures by computing the Probability Density Function (PDF) of head area and other characteristic morphological parameters. We give a first example of estimation of ROIs for several kind of cell's holographic phase reconstructions. Then we apply our proposed algorithms on the sperm's holographic phase reconstructions. We also consider two typical test cases to quantify the reliability of estimated probability distribution of area. It is important to note that the proposed algorithm is based on the assumption that the maximum phase value is present in the sperm head. This is a direct consequence of the holographic phase reconstruction, in which the intensity values of the phase map is related to the organic content of the cells. Same situation is verified for other type of cells that we will consider.

#### 3.1 Holographic setup and materials

We apply the two proposed algorithms on a data set composed by N = 14 holograms relative to bovine spermatozoa and its reference holograms [10,31,35,36]. The bovine sperm cells to be analyzed were prepared by the Institute “Lazzaro Spallanzani” after fixation in suspension of the seminal material with 0.2% glutaraldehyde solution in phosphate buffered saline (PBS) without calcium and magnesium (1:3 v/v) [37]. A drop with volume 6 *μL* has been deposed on a glass slide, and then, covered with a cover slip (20 *mm* × 20 *mm*). The cover slip has been linked to the glass slide by means of a strip of varnish. Holograms of such bovine sperm cells were created and acquired by means of setup sketched in Fig. 3
.

The laser source has wavelength λ = 633 *nm* and it has a nominal power of 10 *mW* but not all the power was used since it was reduced by a variable attenuator (not shown in Fig. 3) to avoid damage of sperm cell. The reference and object beams were plane wavefronts obtained by a beam expander. The first beam splitter was a cube-polarizing beam splitter and a λ/2 wave plate was in the reference beam to obtain equal polarization direction for the two beams, and thus, improve the fringe contrast. The used microscope objective was with a magnification of 50 × and numerical apertures of 0.70, respectively. The CCD detector was a 1392 × 1040 pixel array with pixel size Δx = Δy = 4.7 *μm*. From each recorded hologram both the intensity and the phase map of the observed object is numerically reconstructed [38–40].

#### 3.2 Reconstruction and analysis of quantitative phase contrast maps

We consider here four different holographic phase-contrast reconstruction maps of cells and we compare the two propose algorithms with a standard edge detection scheme developed by Canny [12] to analyze phase-contrast images obtained by our DH microscope. Firstly, the phase-contrast image is convolved with 5x5 Gaussian with the aim to suppress high-frequency noise. As next step, the resulting image is differentiated with respect to x and y axes to calculate an edge-strength function. This procedure does not allow automatic extraction of the ROI because the final result depends closely by the choice of mean and variance of Gaussian filter. By using classical ROI extraction algorithm such as the Canny method, we show that we were not able to find an automatic way to extract accurately a particular ROI by the phase reconstruction of spermatozoa cells under investigation in this work.

For all considered test cases we have to choose the appropriate mean and variance values for the Gaussian filter in the Canny edge detection, while for the two algorithms we propose here, the execution parameters remain constant. It can be noted also that, in each phase reconstructions in Fig. 4 , the highest levels of gray value in the images correspond to the presence of an object in its surroundings. As is clearly shown in Fig. 4 both proposed algorithm give a good results on the spermatozoa head extraction (see Figs. 4 a4–c4) while the Canny edge detection is not able to extract only the head (see Fig. 4 d1–d4).

Head area in pixels unit is estimated using both algorithms. The measurements of areas are computed by evaluating the best fit ellipse on the contour pixels of the connected components of the segmented region of interests from both algorithms. As final step we compute the probability distribution on the data set. Figures 5 and 6 show the results of both algorithms proposed here applied on the two phase map depicted in Fig. 1. The nonlinear diffusion filter uses the Gaussian diffusivity function which allows more accuracy of the edge position. The filter is overrun in the sense that strong effects is given to non-object region with the objective of providing a resulting flat region for the successive binarization phase. To this purpose a number of 80 iterations has been used. For the nonlinear diffusion Algorithm 1, Gaussian smoothing parameter we have σ = 0.1 and the time step size is 5. After automatic thresholding, connected components are extracted and the ROI undergoes area calculation. This has been performed in two ways, by computing the number of pixels of the segmented connected components and by computing the best fit ellipse.

The output of Algorithm 2 is the best mask to extract the sperm head. The mask dimensions are M_{x} = 40 pixels, M_{y} = 80 pixels in all map of data set. The total area of head is given by the area of best fit ellipse computed on the output of both algorithms (see Figs. 5c, 5f, and Figs. 6c and 6f for algorithm 1 and 2, respectively). Figure 7
and Fig. 8
show the estimated PDFs of the areas and other morphological parameters of the extracted ellipses of both algorithms using Gaussian kernels.

With reference to Fig. 8, ellipticity *E* and perimeter *P* are well-known geometric parameters of an ellipse, while the shape factor is defined by

where A is the area of the ellipse. For completeness, we also compute the average value of execution times, for both algorithms, which are *t*_{1} = 12.6262 *sec*, *t*_{2} = 7.6531 *sec*. Therefore, Algorithm 2 is computationally more efficient than Algorithm 1, but it loses in terms of accuracy of estimation of head area, as shown in Fig. 7.

Note that we have considered only the phase reconstructions that present one spermatozoa in its field of view. The more general case in which there are two or more spermatozoa in the image is not treated. In fact, although both algorithms are robust with respect to changes of the reconstruction distance, when two or more spermatozoa appear, they are almost never on the same reconstruction plane. Therefore the whole procedure should be modified by inserting an automatic search for the in-focus distance for each spermatozoa head that was extracted from the image.

#### 3.3 Skimming of anomalous data

Now we consider different recordings of holograms in which sperm heads are shown with structural distortions and rotations around axes. In both cases, it is not impossible take into consideration those data to derive the area of the head, and thus they should be discarded. We show that the proposed algorithm also works for these corrupt data, and we can identify anomalous distortions using statistics obtained in the previous section. In this way we can discard data that can affect the quantitative analysis. We consider here two typical real case studies, and we apply the two proposed algorithm to them. The pictures in Fig. 9 and Fig. 10 show the results obtained for a distorted sperm cell.

We define the confidence interval on our statistical distribution given in Fig. 3 as ${I}_{k}=\left[{\mu}_{h}-k{\sigma}_{h},{\mu}_{h}+k{\sigma}_{h}\right]$, where ${\mu}_{h}$ and ${\sigma}_{h}$, *h* = 1,2, related to the algorithm, are average value and deviation standard of PDF respectively.

We compare the two computed areas of distorted sperm cell for both algorithms with respect to the average values ${\mu}_{h}$. We find the value of *k* on the confidence interval to establish the percentage of reliability of the calculated area. In other words we determinate whether the two extracted heads are distorted or not.

This analysis using Algorithm 1 shows that the example in Fig. 9 has good data with probability P_{1} ≈0.05 (k≈2) because the computed area is 1056.7, while the example in Fig. 10 has good data with probability P_{1} ≈0 (k>3) because the computed area is 609.54. Then the results of Algorithm 2 show that the first example has good data with probability P_{2} ≈0.03 (k≈1) because the computed area is 1114.5 and the second example has good data with probability P_{2} ≈0 (k>3) because the computed area is 308.08. We have proved that both approaches allow efficient discarding of corrupted data. This is very useful when accurate analysis of morphological evaluation has to be performed on large amount of data. This preselective procedure for the quantitative phase contrast maps allows one to avoid taking corrupted data into consideration.

## 4. Conclusion

The paper has shown that the holographic reconstruction of quantitative phase contrast images coupled with image processing technique is a reliable technique to conduct semen analysis. Two different denoising and ROI extraction techniques have been proposed and applied. One is the popular iterative nonlinear diffusion which is intrinsically slow but scale invariant, while the second algorithm is very fast but with a slight limitation to scale variations of heads, and is based on an intrinsic parameter in the quantitative phase contrast map of the investigated cell. Both techniques have been reported to be the useful for accomplishing semen analysis by digital holography in microscopy. Therefore, both proposed method are very efficient to skim the data set in a preselective analysis.

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