Differentiation of signal from noise is a key step in any image segmentation process (Pham et al., 2000) and is based on the assumption that pixels constituting the image can be divided into spatially non-overlapping regions which are characterized by statistically similar intrinsic features (Haralick and Shapiro, 1985; Pal and Pal, 1993).

During the segmentation process, the signal can be defined as a spatially congruent population of pixels with gray values, which are statistically different from the neighboring pixels, that is, from the background (Pham et al., 2000; Pal and Pal, 1993). A theoretically optimal segmentation process should be independent of the person performing the analysis and should be based on the intrinsic properties of the image rather than manually defined values. It should consider not only the intensities but also the spatial arrangement of the picture elements (i.e., pixels).

The current segmentation methods are either based on hard or soft computing (Sinha et al., 2020). Though there is huge surge of soft computation methods based on fuzzy logic, deep learning methods neuronal networks and artificial intelligence (Dey et al., 2018; Hemanth and Anitha, 2014; Devi et al., 2021), image analysis based on binary, hard computation methods have their own advantages (Jena et al., 2020) and best results can be expected from the combination of the two (Perfilieva et al., 2019). Many hard computing image segmentation methods depend on thresholding (Sinha et al., 2020; Sahoo et al., 1988). Although more advanced threshold-based methods (TBMs) utilize local not global thresholds or multilevel thresholding (Manikandan et al., 2014; Umaa Mageswari et al., 2013), the threshold is still defined using the intrinsic properties of the image (e.g., as deviation from the mean intensity in the picture), selection of the degree of deviation is subjective and can yield different results (Figure 1—figure supplement 1). Furthermore, even in local thresholding there is no guarantee that spatially inhomogeneous noise reaching or approaching the intensity of the signal will not be included as signal.

Here, we propose a novel hard computing method for unsupervised image segmentation based on Local Moran’s spatial correlation coefficient (Local Moran’s I) (Anselin et al., 2006). The method was originally invented for geoinformatics and was used to study large-scale spatial correlation of geological (Moran, 1948; Getis and Ord, 1992), social, or medical events for example suicide rates among counties (Helbich et al., 2015) or in geomorphological studies (Alvioli et al., 2016). The method, as used here, is based on the statistical evaluation of intensities among neighboring pixels, estimating the probability whether a given pixel intensity arrangement is due to a random process or not. The proposed method allows for selecting spatially congruent pixels with statistically similar intensity (i.e., signal and background) and pixels with intensity distribution not significantly different from random (i.e., noise).

In this paper, we systematically compare the performance of TBM and Local Moran’s method using images with different levels of noise. We show that, especially at high noise levels, Moran’s method convincingly outperforms TBM on both artificial and natural images. Finally, we also test Moran’s in a biological problem: clustering of voltage-gated ion channels in neuronal membranes. We show that the method can segment ion channel clusters, which are biologically relevant since their existence can be demonstrated using independent methods.

In order to demonstrate the way Local Moran’s method works for image segmentation, we used high magnification fluorescent (i.e., dark field) (Figure 1a–j) as well as bright field (Figure 1k–t) light microscopic images showing immunoreactivity for Kv4.2 potassium channel subunit in the ventral posteromedial (VPM) nucleus of the mouse thalamus. In our approach, the unit of the analysis is the individual pixel of the image.

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The method is based on testing the null hypothesis that pixel intensity distribution in the image is random, which is analyzed with the aid of convolution matrices. The results are tested by Monte Carlo simulation to identify whether the pixel intensity distribution is non-random in an image (Figure 1). In dark field images, contiguous clustered pixels with similarly high intensities represent object (red in Figure 1j), clustered pixels with low intensities indicate no object (e.g., lumen of a blood vessel) (blue in Figure 1j). Group of pixels with random intensity value distribution (which do not reach significance) represent noise (gray in Figure 1j). In bright field images, the picture is reversed, contiguous low- and high-intensity pixels represent the object and no object, respectively (blue and red in Figure 1t).

Local Moran’s I differentiates between object, no object, and noise in the following way: the method first normalizes each pixel intensity value (Figure 1c, m) (see Methods). Next, it defines a weight matrix in the local neighborhood (3 × 3 in this particular case) for each pixel (Figure 1d, n), where the pixel in question receives zero weight. Then it calculates a weighted intensity value for each pixel in the image using the summed values of the neighboring pixels (Figure 1e, o). Immediate neighbors of each pixel are assigned with the weight of 1. Next, it creates lagged gray values by dividing the matrix values with the number of neighboring pixels (Figure 1f, p). This was called ‘lagged’ in the original description (Anselin, 1995), since the distance (lag) was considered in the calculation of this particular value (which was 1 in our case). Finally, it generates the Local Moran’s I value for each pixel by multiplying the lagged gray values (Figure 1f, p) with the normalized values (Figure 1c, m).

The resulting Local Moran’s I value (Figure 1g, q) will represent how the intensity of the pixel and its neighbors differs from the distribution of intensities of all pixels in the image (clusteredness). We can test the salience of this value if we repeat the same procedure with randomly selected neighboring pixels. For each pixel, the method determines the probability (pseudosignificance) of obtaining this or higher Local Moran’s I value using Monte Carlo simulations of the intensity values of all pixels in the image (Figure 1h, r). Significant pixels (p <0 .05) with high intensities (in case of dark field images) or low intensities (in case of bright filed images) will be considered as object. These steps ensure that signal is not determined by an arbitrarily defined threshold, but by the combination of spatial arrangement and pixel intensities. As a consequence, in this method consideration of each pixel as signal will not depend purely on the absolute intensity value of the pixel but on the intensity values of the neighboring pixels and the statistical probability that this spatial arrangement can arise (or not) randomly.

Comparison of TBM and Moran’s method on artificial images

We compared the performance of Moran’s and TBM on artificial binary objects with various levels of Gaussian noise (Figure 2a). Since in this case we defined the objects (the ground truth), true positive rates (TPR = true positive pixels/number of object pixels) and false positive rates (FPR = false positive pixels/number of background pixels) could be directly determined.

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Since Moran’s method can be performed with convolution matrices of different sizes (i.e., Moran’s order, see Methods; Moran, 1948) we first determined the optimal matrix size for different object sizes at variable noise levels on circular artificial objects (Figure 2—figure supplement 1a). First-order matrix contains the immediate neighboring pixels of the central pixel (3 × 3), the second-order matrix contains all the immediate neighbors of the first-order pixels (5 × 5), and so on. We found that optimal Moran’s order was independent of the noise level, and it was 0.25 times (range: 0.238–0.253) of the object size (Figure 2—figure supplement 1b). Thus, for these measures, we selected Moran’s order 4. For the TBM, the threshold was set to 50% of the intensity level of the objects.

At zero noise, both methods had 1 TPR and 0 FPR values (Figure 2a–c). Increasing noise levels resulted in a parallel decrease in TRP values. However, in case of the TBM FPR increased dramatically (cyan dots in Figure 2b) due to the higher fluctuation of pixel values (Figure 2d), which indicates a highly unreliable exclusion of false positive pixels (Figure 2e, circles), while the FPR value in case of Moran’s method remained nearly constant (Figure 2e, triangles) with increasing noise. In case of TBM FPR increase was the consequence of that random noise frequently increased the pixel intensity values of the background above the threshold level (Figure 2), which were detected as positive pixels.

Next, we compared the two methods in a larger sample of images (n = 100) which contained 64 randomly distributed objects of different sizes. The diameter of the objects varied between 4 and 64 pixels in each image (Figure 2a). Here, we also examined the effect of Moran’s order on the quality of detection (Figure 2—figure supplement 2). Similar to same-sized objects in images containing variable-sized objects at 0 noise level, both methods selected the objects equally well (TRP = 1, SD = 0, FPR = 0, SD = 0, n = 100 images). However, increasing the noise level resulted again in a pronounced increase in FPR in the case of TBM (Figure 2b) whereas using Moran’s, this effect was evident only at the edges of detected objects and FPR just marginally increased even at very high noise levels (Figure 2c). Huge increase in false positive pixels in TBM with increasing noise resulted in significantly lower FPR values compared to that of Moran’s (Figure 2e, f, FPR values noise 80: threshold, 0.13333 ± 0.0009, Moran’s: 0.0085 ± 0.0003, p < 5.4*10⁻²¹⁵; noise 160: threshold, 0.3866 ± 0.0012, Moran, 0.0055 ± 0.0005, p < 2.1*10^–244). TPR of Moran’s was also significantly better at noise 80 (Figure 2e, f, TPR values noise 80: threshold, 0.9116 ± 0.0023; Moran’s, 0.9674 ± 0.0025, p < 2.7*10^–130). The TPR of TBM was marginally better at noise 160 (Figure 2e, f, noise 160: threshold, 0.6799 ± 0.0041; Moran’s: 0.6609 ± 0.0060, p < 5.04*10^–69), but the difference in FPR values was so large between the two methods that it far outweighed this effect. We defined the quality of detection as a vector in the FPR/TPR space starting from the FPR = 0, TPR = 1 point. Quality of detection was significantly better with Moran’s method at both noise levels in a population of 100 randomly generated images (Figure 2e, FPR/TPR vector noise 80: threshold, 0.1600 ± 2.2*10^–6; Moran’s: 0.0337 ± 5.8*10^–6, p < 5*10^–169; noise 160: threshold, 0.5019 ± 7.5*10^–6, Moran, 0.3391 ± 3.7*10^–5, p < 10^–156).

Finally, to gain insight into the performance of the two detection methods in a larger parameter space, we repeated the comparison using 13 Moran’s order, 13 threshold levels, and 13 noise levels (n = 100 images for each condition). We found that with the exception of the lowest noise levels (0–30) Moran’s method outperformed TBM at every noise levels irrespective of the threshold mostly independent of the exact Moran’s order used (range 4–13; Figure 2—figure supplement 3). These data demonstrate that, especially in high noise situations, Moran’s method successfully eliminates background noise and selects objects with extremely low FPR.

Comparison of TBM and Moran’s methods on natural images

Next, we tested the performance of the two methods on grayscale images of natural objects (Figure 3a) where, by definition, the ground truth is not known, and therefore the comparison of the two methods to a common reference is not possible. To circumvent this problem, we chose a high magnification confocal image of a nerve tissue immunostained with vesicular glutamate transporter (vGluT2), a ubiquitous marker of axon terminals with subcortical origin in the thalamus (Lavallée et al., 2005; Graziano et al., 2008; Rovó et al., 2012). We prepared images with three different added noise levels and asked 23 expert human observers, who were blind to the nature of the image and the nature of the investigation, to delineate objects (Figure 3a, b and Figure 3—figure supplement 1). Observers always started delineation in the images with the highest noise level. Next, we generated a heat map and labeled the number of experts selecting each individual pixel. For the purpose of this study, we defined ground truth as pixels which were selected by at least 12 experts (Figure 3—figure supplement 1) and compared the performance of the two methods to this common reference (Figure 3c–e). The size of the objects (n = 36) in the image varied between 1 and 303 pixels corresponding to 0.043 and 13.003 μm², respectively. For a systematic comparison of the two image segmentation methods, we analyzed the data using different threshold levels and matrix sizes (Figure 3—figure supplement 2) at three noise levels (0, 80, and 160).

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Similar to artificial objects, Moran’s method outperformed TBM at every noise level. The FPR values remained low with Moran’s method even at high noise levels at every Moran’s order tested (Figure 3f). Thus, even though at the best threshold levels, the TPR values were comparable between the two methods, lower FPR values resulted in a better quality of detection (shorter vector) in the TPR/FPR space (Figure 3f).

It is important to emphasize that in the case of TBM at increased noise levels, false positive pixels were homogeneously distributed in the image due to the increased background noise (Figure 3c). In contrast, with Moran’s, most false positive pixels surrounded the human-based selection areas (Figure 3d). This is largely the consequence of that Moran’s method detected the out of focus pixels as well, which were routinely not selected by the human experts.

These data demonstrate that when natural images were analyzed, the quality of detection was better with Moran’s method in every condition tested. The main strength of Moran’s was that it did not include pixels of random noise, like TBM.

Application of Moran’s method for a biological problem

Next, we tested Moran’s method to address a biologically relevant problem, the clustering of Kv4.2/4.3 subunits of the voltage-gated K⁺ channels in thalamocortical neurons. Voltage-gated ion channels are either distributed homogeneously, along a gradient (Lorincz and Nusser, 1979; Kerti et al., 2012) or form high-density clusters in neuronal membranes (Alonso and Widmer, 1997; Kollo et al., 2006; Kollo et al., 2008), endowing neurons with unique electrical properties (Chen et al., 2006; Hoffman et al., 1997; Johnston et al., 2003; Losonczy et al., 2008). However, the exact subcellular distribution of these subunits and the logic of their possible clustering in thalamic cells are unknown. Therefore, we tested whether Kv4.2/4.3 ion channel clusters, detected by Moran’s I method, form biologically relevant membrane protein aggregations.

Immunostainings for the Kv4.2 and Kv4.3 subunits displayed homogenous, diffuse labeling in the major somatosensory nucleus of the thalamus (VPM). At high magnifications, however, it became visually evident that Kv4 subunits also form strongly immunopositive clusters (2–10 µm; Figure 4a). To study the precise subcellular distribution of Kv4 subunits, we first examined the Kv4.2 subunit at the electron microscopic level using preembedding immunogold reactions. This approach confirmed the exclusive dendritic localization of this subunit (n = 123 profiles). As suggested by the light microscopic observations, large caliber dendrites of relay cells displayed a high density of immunogold particles for the Kv4.2 subunit (n = 9 animals, Figure 4b, c). Quantification of silver particles along the dendritic membranes (n = 3 animals) demonstrated more than three times higher density on thick (minor diameter over 1.2 µm) dendrites, preferentially innervated by large excitatory terminals containing round vesicles (RL terminals) compared to thin (minor diameter below 1.2 µm) dendrites which received very little of this type of inputs (Figure 4d).

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To exclude the possibility that the uneven (i.e., clustered) immunogold labeling is the consequence of uneven accessibility of the antibodies to the epitopes in thick tissue and to better visualize protein aggregates at high resolution, we performed another electron microscopic experiment using the diffusion-free freeze-fracture replica immunogold labeling technique (Masugi-Tokita and Shigemoto, 2007; Figure 4e–h). Examination of replicas of the VPM immunolabeled for the Kv4.2 or the Kv4.3 subunits revealed gold particles associated with the protoplasmic face of dendritic segments, consistent with the intracellular location of the epitopes. On large diameter dendrites, we observed uneven distribution of gold particles (Figure 4e) consistent with our observations using confocal microscopy and preembedding immunogold localization. Quantitative evaluation of the reactions in one rat demonstrated that the density of gold particles for the Kv4.3 subunit was twice as high in large, compared to small diameter dendrites (12.6 ± 8.0 gold/µm², n = 18 vs 6.0 ± 3.6 gold/µm², n = 18). A similar difference was found for the Kv4.2 subunit, for which the density of gold particles was 22.8 ± 8.8 gold/µm² (n = 9) in large diameter proximal dendrites, whereas it was 9.3 ± 3.2 gold/µm² (n = 6) in small, distal dendrites. Dendritic appendages were also decorated with gold particles (Figure 4h).

These data together demonstrate that Kv4.2 and Kv4.3 subunits display weak distal dendritic labeling, but the proximal, large diameter dendrites contain high densities of these ion channel subunits in VPM.

Next, we investigated whether the Moran’s method as described above can delineate Kv4.2 subunit clusters and quantitatively characterize them in the VPM (Figure 5). Moran’s method clearly segmented immunopositive and negative regions (Figure 5a–c). The immunopositive regions displayed large variability in size intensity and contained diffuse labeling and high-intensity clusters as suggested by electron microscopic investigation. To test to what extent Kv4.2 clusters detected by Moran’s method represent ion channel clusters identified on the proximal dendrites of VPM cells at the electron microscopic level (Figure 4), we visualized large excitatory axon terminals known to be associated with the proximal dendrites of VPM cells (Sherman and Guillery, 2006). These subcortical excitatory inputs were labeled with vesicular glutamate transporter type 2 (vGluT2, Figure 5d; Herzog et al., 2001; Fremeau et al., 2001). We selected the proximal dendrites by performing segmentation using Moran’s method in both channels of the Kv4.2/vGluT2 double immunofluorescent images. The data showed that large, high-density clusters of Kv4.2 ion channels were indeed associated with vGluT2-immunopositive terminals (Figure 5e, cyan arrows), whereas the smaller and less intense ones were not (Figure 5f, yellow arrows). Immunopositive clusters generally had highly variable sizes (5.59 ± 13.26 µm²) and intensities (745 ± 260 a.u.). The large majority of the non-colocalizing Kv clusters were below 1 µm² and faintly stained (Figure 5g) whereas more than the half of the colocalized Kv4.2 structures were above 5 µm² and more intensely labeled (Figure 5g). Kv4.2 clusters associated with vGluT2-positive terminals were significantly larger (12.21 ± 18.63 µm²) and more intense (894 ± 280 a.u.) than Kv4.2 regions outside the vGluT2 terminals (average size: 0.90 ± 1.31 µm², average intensity: 639 ± 182 a.u.) indicating that Moran’s I method could delineate functionally relevant ion channel clusters associated with a specific synaptic input (Figure 5h).

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To verify the association of afferents and ion channel clusters, we performed double immunostaining for Kv4.2 and vGluT2 at the electron microscopic level (Figure 5i, j). Indeed, membrane-associated silver intensified gold particles for the Kv4.2 subunit were associated with membrane surfaces covered by vGluT2-positive terminals, as demonstrated by the Moran’s method at the light microscopic level. Membrane segments contacted by vGluT2 boutons showed more than four times higher gold particle density (37.0 particles/100 μm membrane profile, total measured length: 601 μm) compared to surrounding membrane surfaces that were not contacted by vGluT2-positive axons (8.9 particles/100 μm, total measured length: 394 μm). These data reveal a novel association of voltage-gated ion channel clusters on the postsynaptic membrane surfaces with a specialized presynaptic terminal.

In this study, we demonstrate that Local Moran’s method is suitable for image segmentation and outperforms TBM in both artificial and natural images, especially when the noise is substantial. We used this method to successfully delineate and quantify biologically relevant membrane protein clusters. Since Moran’s selections are based on the relative intensity values of neighboring pixels, it can select contiguous points as objects, unlike TBM. TBM does not use spatial information of the pixels. Therefore, it selects random noise as well, provided it reaches the threshold. Unlike in TBM, selection of the signal will not depend on the absolute intensity of its pixels in Moran’s method since the image processing starts with a normalization step. This way, it uses inherent properties of the image to select clusters. This will allow the systematic, quantitative comparison of images where sample preparation and image acquisition cannot be standardized, such as in most histological samples.

Local Moran’s method has not been used for image segmentation so far. Its global version, however, has already been utilized in a wide range of biomedical image analyses. Global Moran’s method assigns a value to the whole image and describes whether pixel distribution is random or not, but does not segment pixels to object, no object and noise. Global Moran’s I were used to examine muscle and fat tissue on three-dimensional (3D) MRI (Magnetic Resonant Images) images (Santago et al., 2016) or to measure anisotropy in ultrasound imaging (Ahn and Kaptchuk, 2011). In addition, Global Moran’s method was also used to analyze the homogeneity of the temporal information in Ca²⁺ signals in periodontal fibroblasts (Ei Hsu Hlaing et al., 2019) or to detect correlated neuronal activity (Schmal et al., 2017). Based on these, the utilization of Local Moran’s method, as described here, will not be restricted to segmenting objects on two-dimensional images but can be extended for 3D image segmentation and analyzing temporal signals.

The strongest argument to use Local Moran’s I instead of threshold-based segmentation is the effective elimination of false positive pixels from the background. Although the intensity of noisy pixels can reach or exceed that of the objects, white noise does not cause a pattern. Detection of the lack of spatial correlation is a powerful tool to ignore false positive salient pixels in the background.

As indicated above, Moran’s method works without user-dependent intensity values. However, the size of the convolution matrix needs to be set by the user. In this study, we examined the effect of matrix size in a wide range of applications (Figure 3f, Figure 3—figure supplements 1, 2). We determined that selection of the signal is relatively insensitive of the exact matrix size within a certain range (Moran’s order 3–8), especially in noisy conditions. A priori knowledge of the expected object size helps to define the best matrix size since we found a linear correlation between object size and Moran’s order for optimal image segmentation (Figure 2—figure supplement 1).

We found that Local Moran’s method can delineate and quantitatively characterize Kv4.2 ion channel clusters in the somatosensory thalamus formed in the proximity of large vGluT2-positive afferent nerve terminals. The selective association of a given presynaptic terminal type and the postsynaptic accumulation of ion channel clusters were verified by independent electron microscopic methods. The functional relevance of this unique association remains to be determined: it may participate in timing the precision of sensory integration, but recently the non-synaptic role of voltage-gated ion channel accumulation has also been proposed (Cserép et al., 2020).

The code for the analysis is available online as a user-friendly MATLAB script at: https://github.com/dcsabaCD225/Moran_Matlab/blob/main/moran_local.m (copy archived at Csaba, 2024). All source data relevant to figures and charts are included in source data files.

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Author details

1. Csaba Dávid

   1. Lendület Laboratory of Thalamus Research, HUN-REN Institute of Experimental Medicine, Budapest, Hungary
   2. Department of Anatomy, Histology and Embryology, Semmelweis University, Budapest, Hungary

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4221-9468

2. Kristóf Giber

   Lendület Laboratory of Thalamus Research, HUN-REN Institute of Experimental Medicine, Budapest, Hungary

   Contribution

   Investigation

   Competing interests

   No competing interests declared

3. Katalin Kerti-Szigeti

   1. Laboratory of Cellular Neurophysiology, HUN-REN Institute of Experimental Medicine, Budapest, Hungary
   2. Novarino Group, Institute of Science and Technology, Klosterneuburg, Austria

   Contribution

   Investigation

   Competing interests

   No competing interests declared

4. Mihály Köllő

   1. Laboratory of Cellular Neurophysiology, HUN-REN Institute of Experimental Medicine, Budapest, Hungary
   2. Sensory Circuits and Neurotechnology Laboratory, Francis Crick Institute, London, United Kingdom

   Contribution

   Investigation

   Competing interests

   No competing interests declared

5. Zoltan Nusser

   Laboratory of Cellular Neurophysiology, HUN-REN Institute of Experimental Medicine, Budapest, Hungary

   Contribution

   Supervision, Funding acquisition, Methodology, Writing - review and editing

   Competing interests

   No competing interests declared

6. Laszlo Acsady

   Lendület Laboratory of Thalamus Research, HUN-REN Institute of Experimental Medicine, Budapest, Hungary

   Contribution

   Conceptualization, Resources, Supervision, Funding acquisition, Methodology, Writing - original draft, Writing - review and editing

   For correspondence

   acsady@koki.hu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0679-2980

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