Automatic Stomatal Segmentation Based on Delaunay-Rayleigh Frequency Distance

Article Automatic Stomatal Segmentation Based on Delaunay-Rayleigh Frequency Distance

Miguel Carrasco 1,* , Patricio A. Toledo 1 , Ramiro Velázquez 2 and Odemir M. Bruno 3

1 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibañez, Av. Diagonal Las Torres, 2700 Santiago, Chile; [email protected] 2 Facultad de Ingeniería Josemaría Escrivá de Balaguer 101, Campus Aguascalientes, Universidad Panamericana, Aguascalientes 20290, Mexico; [email protected] 3 Scientiﬁc Computing Group, São Carlos Institute of Physics, University of São Paulo, P.O. Box 369, São Carlos, SP 13560-970, Brazil; [email protected] * Correspondence: [email protected]; Tel.: +562-2331-1269

Received: 3 November 2020; Accepted: 17 November 2020; Published: 20 November 2020

Abstract: The CO2 and water vapor exchange between leaf and atmosphere are relevant for plant physiology. This process is done through the stomata. These structures are fundamental in the study of plants since their properties are linked to the evolutionary process of the plant, as well as its environmental and phytohormonal conditions. Stomatal detection is a complex task due to the noise and morphology of the microscopic images. Although in recent years segmentation algorithms have been developed that automate this process, they all use techniques that explore chromatic characteristics. This research explores a unique feature in plants, which corresponds to the stomatal spatial distribution within the leaf structure. Unlike segmentation techniques based on deep learning tools, we emphasize the search for an optimal threshold level, so that a high percentage of stomata can be detected, independent of the size and shape of the stomata. This last feature has not been reported in the literature, except for those results of geometric structure formation in the salt formation and other biological formations.

Keywords: stomatal segmentation; image segmentation; Delaunay-Rayleigh frequency

1. Introduction Stomata are the gates through which gas exchange through the leaf takes place, involving carbon dioxide intake and water vapor loss (reviewed in Reference [1–3]). The carbon dioxide intake drives plant growth and productivity [4], while the water vapor loss through a process known as transpiration regulates leaf temperature, nutrient uptake, and root-to-shoot signaling [5,6]. The remaining of the epidermis is covered by an impervious cuticle, with restricted gas exchange properties [7,8]. Gas-exchange capacity depends on stomatal density (i.e., stomata number per unit area), size, and patterning (spacing) [9–11]. Methods for automatic segmentation of stomatal density and size have been earlier presented [12,13], whereas patterning (spacing) is currently performed manually [14]. From a computational point of view, stomatal analysis has been explored mainly on pixel or object analysis. Although localization and processing tools exists, these are limited to search for stomatal position through techniques based on chromatic and morphological combinatorial operations [15–17], texture analysis [16], fractal analysis [18], segmentation using object-oriented method in multiple resolutions [19], and, more recently, by Deep Convolutional Neural Networks [12,13,20–22]. Today, no technique allows full spectrum analysis of diﬀerent stomatal types since they have a great variation according to species, shape, position, and, in general, noise and technique present in the microscopy

Plants 2020, 9, 1613; doi:10.3390/plants9111613 www.mdpi.com/journal/plants Plants 2020, 9, 1613 2 of 17 image [23–26]. At large, both the location and the analysis of the stomata require demanding work on the part of the biologists. This task is not simple, since there are great differences in the same species by the microscopy technique used [27]. This research proposes a new segmentation algorithm based on the spatial stomatal statistical distribution. We define a geometric model, revisited in recent years, implying a mechanism present in nature and allowing an ideal distribution for gas transfer [5,14,28]. The study and analysis of stomata is the first step towards phenotypic plasticity and the relationship with climate change, soil use, resource, drought, CO2 concentration, and light level [4,8,29–33]. Consequently, segmentation process automation and stomatal morphology analysis is a relevant task, especially in high volume analysis of images needed as input for other studies [20]. The latter might not be viable by hand because of analysis complexity. Although we use a specific microscopy database, both for the design and for the analysis of the computational algorithm, our findings have potential use for any kind of stomatal images since it uses as key information stomata’s geometric configuration between analyzed species. We generate a manual segmentation in different species of monocot and dicot families to test our hypothesis. The results section presents the main findings of our algorithm through a fully automatic system for the Jatoba Hymenaea Courbari species. Subsequently, we manually analyze the algorithm proposed in other species in order to understand the existence of an underlying pattern between them. The Materials and Methods’ section presents a description of the steps of the algorithm and, finally, the general conclusions of the method.

2. Results Delaunay-Rayleigh Threshold Binarization (DRTB) algorithm has been evaluated over a set of 31 optic microscopy images from Jatoba Hymenaea Courbaril tree localized in Caribbean, Central and South American zones. The images were taken with an Olympus E-330 camera with optic microscope with 3136 2352 resolution pixels, at the Biosciences Institute at University of São Paulo, Brazil (USP). × The images’ set has 3087 regions hand-classified as stomata (see Supplementary Materials section) with its coordinates (x,y). The abaxial surfaces of the epidermis were obtained from three or four leaves (one per individual) of each species. From each collected leaf, a sample of approximately 1 cm2 was removed from the leaf’s middle region, between the margin and midrib. Dissociation of the leaf epidermis was performed using a 1:1 solution of glacial acetic acid and hydrogen peroxide at 60 ◦C for 12 h, or the time required to completely decouple the epidermis [34]. The algorithm’s performance was evaluated by two statistical indicators: precision (also known as positive predictive value, PPV) and recall (or true positive rate, TPR). The procedure consists of manually identifying stoma’s center (x,y) coordinates. Then, the segmentation-algorithm identifies this position and proceeds with a comparison. If the distance between the manual coordinate is less than a threshold with respect to the coordinate of the segmented region, we consider that it has been classified correctly. This case is considered as True Positive (TP). In case the algorithm does not find the stoma, even when it is present in the image, we consider this case as False Negative (FN). Finally, when the algorithm classifies a region where there is no stoma, we consider this coordinate as False Positive (FP) (see Figure1). Therefore, the evaluation metrics are:

TP PPV = , (1) FP + TP

TP TPR = . (2) TP + FN PlantsPlants 20202020,, 99,, 1613x FOR PEER REVIEW 33 of 1717 Plants 2020, 9, x FOR PEER REVIEW 3 of 17

Figure 1. Illustration of precision and recall concepts. True positive (TP) occurs when the distance FigurebetweenFigure 1.1. manual IllustrationIllustration-coordinate of precisionprecision (yellow andand cross) recallrecall is concepts. concepts.less than True a threshold positivepositive with ((TP respect)) occursoccurs to whenwhen the coordinate thethe distancedistance of betweenthebetween segmented manual-coordinatemanual-region-coordinate (ellipses). (yellow (yellow False cross) cross) negative is is less less than(FN than) aoccurs thresholda threshold when with with the respect algorithm respect to theto doesthe coordinate coordinate not find of the of segmented-regionstoma,the segmented even when-region (ellipses).it is (ellipses).present False in False negativethe image. negative (FN Finally,) ( occursFN) occurswhen when the when the algorithm algorithm the algorithm classifies does not does a find region not the find stoma,where the eventherestoma, whenis evenno stoma, it when is present we it isconsider inpresent the image. this in thecoordinate Finally, image. when Finally,as False the whenPositive algorithm the (FP algorithm classifies). The ratio a classifies region TP/(FP where a+ regionTP there) is knownwhere is no stoma,asthere precision is we no considerstoma, (positive we this predictiveconsider coordinate this value ascoordinate False (PPV Positive)), andas False the (FP ratio ).Positive The (TP ratio /(TPFPTP ).+ FN/The(FP) isratio+ calledTP )TP is /( knownrecFPall + (trueTP as) precisionis positive known (positiverateas precision (TPR predictive)). (positive value predictive (PPV)), value and the(PPV ratio)), and (TP /theTP ratio+ FN ()TP is called/TP + FN recall) is called (true positive recall (true rate positive (TPR)). rate (TPR)). From this set, the proposed algorithm is capable of 2752 detections (recall) of 89.14 8%. On the From this set, the proposed algorithm is capable of 2752 detections (recall) of 89.14± ± 8%. On the otherFrom hand, this the set, number the proposed of false algorithm positive regions is capable (precision) of 2752 detections is 72.8 10%. (recall As) of shown 89.14 ± on 8% Figure. On the2, other hand, the number of false positive regions (precision) is 72.8 ± 10%.± As shown on Figure 2, the theperformanceother performance hand, the is variablenumber is variable withof withfalse each eachpositive sample, sample, regions given given (structuralprecision structural )conditions is conditions; 72.8 ± 10%.; however however, As shown, a a high high on classification classificationFigure 2, the raterateperformance isis achievedachieved is variable as a result with of multileveleach sample, threshold given structural traveling conditions design (see ; howeverexamples , a in high Figure Figures classifications 33 andand4 4).). InInrate contrast,contrast, is achieved lowerlower as resultsresultsa result occuroccur of multilevel atat lowerlower threshold imageimage qualityquality traveling wherewhere design thethe stomata(seestomata examples areare locatedlocated in Figure becausebecauses 3 and theythey 4). areareIn contrast, outout ofof focus.focus. lower The results combined occur at performance lower image of qualityboth measures where the reflects reflects stomata this are fact,fact located, and it because is shownshown they inin FigureFigureare out3 3 (specimen).of (specimen) focus. The. combined performance of both measures reflects this fact, and it is shown in Figure 3 (specimen).

FigureFigure 2. 2. Performance of the proposed algorithm applied to 3131 specimens specimens of of HymenaeaHymenaea Courbaril Courbaril (Jatoba(JatobaFigure Database). 2.Database). Performance Overall of therecall proposed and precision algorithm are 89.14%89.14% applied andand to 72.8%,72.8%31 specimens, respectively.respectively. of Hymenaea MaximumMaximum Courbaril recallrecall performanceperformance(Jatoba Database). isis achievedachieved Overall atat specimensspecimensrecall and #8,precision#8, #9,#9, and are #10 89.14% with 100% and and 72.8% maximum , respectively. precision Maximum performance recall isisperformance reached reached at at 98% is 98%achieved by specimen by specimenat specimens #3. Worst #3. #8, Worst recall #9, and and recall #10 precision with and 100% precision performance and maximum perfor is 70%mance precision and is 58%, 70% performance respectively, and 58%, bothrespectivelyis reached at specimen at, both 98% #30. at byspecimen specimen #30. #3. Worst recall and precision performance is 70% and 58%, respectively, both at specimen #30.

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Figure 3. Analysis of selected specimens. (Left) Best performance is achieved at the upper-right area FigureFigure 3. 3.Analysis Analysis of selected specimens. specimens. (Left) (Left) Best Best performance performance is achieved is achieved at the atupper the- upper-rightright area of the diagram with high recall and precision. Worst performance is achieved at the lower-left area. areaof ofthe the diagram diagram with with high high recall recall and andprecision. precision. Worst Worst performance performance is achieved is achieved at the lower at the-left lower-left area. Orange cross indicates mean (recall/performance 83%/72%) and standard deviation (8%/10%). (Right) area.Orange Orange cross cross indicates indicates mean mean(recall/performance (recall/performance 83%/72%) 83% and/72%) standard and standard deviation deviation (8%/10%). (8%(Right)/10%). Specimen #1 represents an average performance (85%/72%) with clear regions but high diffusion. (Right)Specimen Specimen #1 represents #1 represents an average an average performance performance (85%/72%) (85%/72%) with with clear clear regions regions but buthigh high diffusion. diffusion. Specimen #3 has 98%/89% detection performance, and stomata shows sharp borders with clear SpecimenSpecimen #3 #3has has 98% 98%/89%/89% detection detection performance, performance, and and stomata stoma showsta shows sharp sharp borders borders with clearwith regions clear regions leading to good detection metrics. Specimen #12 shows 94%/58% with high recall and lower leadingregions to leading good detection to good metrics.detection Specimen metrics. Specimen #12 shows #12 94% shows/58% 94%/58% with high with recall high and recall lower and precision, lower precision, explained by diffuse borders and poor region definition. Specimen #30 has the poorest explainedprecision, by explained diffuse borders by diffuse and borders poor region and poordefinition. region Specimen definition. #30 Specimen has the #30 poorest has the performance poorest performance with 70%/58% with very diffuse borders and poor regions with high false positive rate. withperformance 70%/58% withwith very70%/58% diffuse wit bordersh very diffuse and poor borders regions and withpoor highregions false with positive high false rate. positive rate.

Figure 4. Example of segmentation output. Our algorithm is able to detect stomatal centroids (blue Figure 4. Example of segmentation output. Our algorithm is able to detect stomatal centroids (blue Figuredots) 4. andExample segmented of segmentation areas (red ellipses). output. With Our algorithmthe geometric is able information to detect stomatalfrom centroid centroids coordinates, (blue dots) dots) and segmented areas (red ellipses). With the geometric information from centroid coordinates, andstatistics segmented and tessellations areas (red ellipses). are built. With the geometric information from centroid coordinates, statistics andstatistics tessellations and tessellations are built. are built.

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3. Discussion Previous results reveal a high precision rate in the detection of Hymenaea Courbaril stomata. However, is this distribution similar on other species? To answer this question, we analyzed seven species with different microscopy techniques. Experimental results show that not only Hymenaea Courbaril has a stable distribution, we applied a manual segmentation to 2842 stomata with 12 different configurations shown in Figure5 and described in Table1. Despite morphological and chromatic differences, the spatial distributions are similar, as it is shown in Figure4. These results are consistent with the findings found by Croxdale [9], where the relationship between area and number is analyzed in the first instance. This result is relevant since it provides statistical evidence that would allow us to design an optimal algorithm for the relative distance between stomata, which is consistent with Peat & Fitter [35] developments. The only requirement is a preprocess with efficient acceptance/rejection range for stomatal segmentation. This last task could be performed by any segmentation algorithm, such as those mentioned above. It is known that environmental conditions during growth may also affect stomatal dispersion pattern among species [36,37]. Based on this background, stomatal spacing may be employed as a marker for studying plant adaptation to growth environment [38]. Next, we present a brief discussion of manual segmentation in different types of plants of the Monocot family, Plantscompared 2020, 9 to, x FOR the automaticPEER REVIEW evaluation of the Dicot family. 6 of 17

Figure 5. HandHand segmentation to to 2842 stomata within 12 species as described in Table Table1 . 1. Despite Despite morphological and and chromatically chromatically di ﬀ differenceserences, the, the spatial spatial distributions distributions are similar are similar to the to example the example shown shownin Figure in4 Figure( Hymenaea 4 (Hymenaea Courbaril Courbaril)). .

3.1. Manual Segmentation The automatic algorithm evaluation gives 푇푃푅 = 89.14 ± 8% and 푃푃푉 = 72.8 ± 10%. Those results are relevant given image type. Our results are statistically valid, considering the total number of stomata studied. To assess if this relation is relevant for other species, we evaluate 12 different configurations on 7 different plant species. Despite interspecies variations, a relevant result is the Root-Mean-Square Deviation (RMSD) falling with segmented region number. In terms of distribution parameters, they are stable in the range 3 < 휎 < 8 as long as region number is lower than 100. As future work, it remains to design an optimal search algorithm based on a distribution parameter in the appropriate range, depending on the type of species, the analysis of second (autocovariance) and higher order moments of geometric characteristics, and the cross covariances with other relevant properties captured by microscopes, like color. According to the analyzed data, the segmented region number affects the algorithm performance. In general, the higher the segmented region number the lower RMSD between Rayleigh distribution and frequency histogram; which confirms the true nature of the spatial distribution of stomata (see Figure 6). Moreover, Rayleigh distribution ideal parameters (that minimize RMSD) have a narrow range between 3 < 휎 < 8; see Figure 7. For each species, this parameter is different; however, results tend to cluster as the segmented region number reaches 100.

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Table 1. Stability of Rayleigh parameter for various species. From 2842 regions analyzed, a stability range 3 < σ < 8 might be proposed. Maximum Rayleigh parameter is 11.1 (Tradescantia Pallida) minimum value is 3.3 (Tradescantia Zebrina). Family name: C: Commelinaceae, M: Marantaceae, F: Fabaceae.

Stomata’s Rayleigh # Species Analyzed RMSD Group Family Number Parameter 1 Tradescantia Zebrina 753 0.00832 3.3 Monocot C 2 Tradescantia Pallida—under 24 h of light 394 0.00983 3.6 Monocot C 3 Tradescantia Pallida—in natural condition 245 0.03327 7.1 Monocot C 4 Callisia reppens 65 0.05358 8.8 Monocot C 5 Callisia reppens 29 0.12969 9.7 Monocot C 6 Callisia reppens 104 0.02037 4.3 Monocot C 7 Tradescantia Zebrina 69 0.05905 8.3 Monocot C 8 Tradescantia Pallid 25 0.13832 11.1 Monocot C 9 Ctenanthe Oppenheimiana 138 0.02531 7.1 Monocot M 10 Calisia reppens—using stereoscope 15 586 0.01578 3.3 Monocot C × 11 Tradescantia Pallida using stereoscope 15 295 0.01902 5.4 Monocot C × 12 Hymenaea Courbaril 139 0.02420 6.1 Dicot F Total Regions 2842 µ = 0.04473

3.1. Manual Segmentation The automatic algorithm evaluation gives TPR = 89.14 8% and PPV = 72.8 10%. Those results ± ± are relevant given image type. Our results are statistically valid, considering the total number of stomata studied. To assess if this relation is relevant for other species, we evaluate 12 different configurations on 7 different plant species. Despite interspecies variations, a relevant result is the Root-Mean-Square Deviation (RMSD) falling with segmented region number. In terms of distribution parameters, they are stable in the range 3 < σ < 8 as long as region number is lower than 100. As future work, it remains to design an optimal search algorithm based on a distribution parameter in the appropriate range, depending on the type of species, the analysis of second (autocovariance) and higher order moments of geometric characteristics, and the cross covariances with other relevant properties captured by microscopes, like color. According to the analyzed data, the segmented region number affects the algorithm performance. In general, the higher the segmented region number the lower RMSD between Rayleigh distribution and frequency histogram; which confirms the true nature of the spatial distribution of stomata (see Figure6) . Moreover, Rayleigh distribution ideal parameters (that minimize RMSD) have a narrow range between 3 < σ < 8; see Figure7. For each species, this parameter is di fferent; however, results tend to cluster as the segmented region number reaches 100. Plants 2020, 9, 1613 7 of 17 PlantsPlants 20202020,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 77 ofof 1717

FigureFigure 6. 6. RootRoot-Mean-SquareRoot--MeanMean--SquareSquare Deviation Deviation ( ((RMSD)RMSRMSDD)) versus versus segmented segmented stomata stomata number number and and Rayleigh Rayleigh parameter.parameter. ( (aa)),),, the thethe RMS RMSDRMSDD decreases decreases as as a a power power law law withwith numbernumber ofof segmentedsegmented stomata.stomata. ( ((bb)),),, RMS RMSDRMSDD increasesincreasesincreases exponentially exponentiallyexponentially with withwith Rayleigh RayleighRayleigh parameter. parameter.parameter. High HighHigh RMS RMSDRMSDD species speciesspecies (red (red(red labels) labels)labels) tend tendtend to to have have lower lower numbernumber of of segmented segmented regions regions andand higherhigher RayleighRayleigh parameter.parameter.

FigureFigure 7. 7. RelationshipRelationship between between segmented segmented region region and and Rayleigh Rayleigh distribution distribution histogram.histogram. First First column column (stomata)(stomata) shows shows different didifferentﬀerent speciesspecies analyzed.analyzed. Second Second column column (Delaunay (Delaunay center center-of-mass)center--ofof--mass)mass) shows shows mass mass centerscenterscenters andand correspondingcorresponding tessellations. tessellationstessellations.. Third ThirdThird column columncolumn (Zoom (Zoom(Zoom Region-of-Interest, RegionRegion--ofof--IntInteresterest ROI),, ROI)ROI) shows showsshows region regionregion of ofinterest.of interest. interest. Fourth Fourth Fourth column column column (Distance (Distance (Distance distribution) distribution) distribution) shows sensibility shows shows sensibility sensibility of histogram of of histogram tohistogram spatial distribution to to spatial spatial distribution(moredistribution at https: (more(more//github.com atat https://github.com/mlacarrasco/drtb/tree/main/stomatasDB/output)https://github.com/mlacarrasco/drtb/tree/main/stomatasDB/output)/mlacarrasco/drtb/tree/main/stomatasDB/output). ..

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3.2.3.2. Defocus Stomata in Automatic Mode InIn the the automaticautomatic segmentation segmentation mode, we observe relevant didifferencesﬀerences in in the the maximum maximum and and minimumminimum performance of of the the proposed proposed algorithm. algorithm. When When analyzing analyzing these these differences diﬀerences in detail, in detail, we weobserve observe that that they they are are due due to tothe the lack lack of offocus focus of ofthe the stom stomataata in in some some samples samples of of the the study study ( (FigureFigure 8).).

FigureFigure 8.8. ComparisonComparison between high andand lowlow focusedfocused stomata.stomata. In specimen #10,#10, stomata are clearlyclearly visiblevisible withwith sharpsharp edgesedges withwith veryvery lowlow RMSDRMSD (see(see FigureFigure7 ).7). Specimen Specimen #29 #29 has has di diffusedﬀused edges.edges.

3.3.3.3. Main Advantages and Disadvantages AsAs a firstfirst advantage, we highlight that our method is based on the the stomata stomatall geometric geometric properties, properties, easilyeasily verifiable,verifiable, suchsuch as as centroids centroids spatial spatial distribution; distribution which; which does does not not depend depend on on the the shape shape and and size size of theof the stomata. stomata. Second, Second, the the image-preprocessing image-preprocessing method method is ais well-established a well-established procedure procedure in thein the field field of imaging,of imaging and, and it is inspired it is inspired by the by property the property of diffusion, of diffusion, which is one which of the is fundamental one of the fundamental mechanisms ofmechanisms stomatal physiology. of stomatal Third, physiology stomatal. geometry Third, stomata is describedl geometry by a Delaunay is described tessellation, by a which Delaunay is a technictessellation, known which to captures is a technic diff usive-processes,known to captures such diffusive as those-processes occurring, such at the as cellularthose occurring level. Fourth, at the thecellular use oflevel. the Fourth, median the as theuse mainof the statistic median decreases as the main outliers-weight, statistic decreases which outliers increases-weight, robustness. which Regardingincreases robustness. the disadvantages, Regarding our the proposal disadvantages, might be classifiedour proposal as standard might be as classified it was not as implemented standard as throughit was not a deep-learning implemented scheme through (although a deep-learning it would scheme be possible (although to do so). it would Despite be our possible efforts toto capturedo so). aDespite large number our efforts of samples, to capture we recognizea large number that the of number samples, of w speciese recognize is limited, that althoughthe number high of variability species is betweenlimited, samplesalthough can high be observed.variability Regarding between samples the distribution’s can be observed discrimination. Regarding method, the it distribution’s is known that RMDSdiscrimination has some method, bias, which it is known could bethat avoided RMDS has with some an Akaike bias, which scheme could or similar.be avoided The with preprocessing an Akaike phasescheme of or our similar. proposal The is preprocessing designed for Hymenaea phase of our Courbaril proposal, as futureis designed work, for and Hymenaea this process Courbaril might, beas automatedfuture work, in suchand athis way process that it is might possible be to automated accept segmentation in such a wayin other that types it is of possible stomata. to accept segmentation in other types of stomata. 4. Materials and Methods 4. MaterialsThe natural and wayMethods to deal with structural complexity found in stomata images is noise analysis. The latter is associated, by large, to simplification and information-reduction processes, like anisotropic The natural way to deal with structural complexity found in stomata images is noise analysis. diffusion, wavelet transform techniques, and nonlinear, statistical, or adaptive filters [39–41]. Even when The latter is associated, by large, to simplification and information-reduction processes, like image segmentation is possible, for example, by morphological processing, the use of geometric anisotropic diffusion, wavelet transform techniques, and nonlinear, statistical, or adaptive filters [39– properties is desirable. Therefore, we propose a novel segmentation method that seeks an optimal 41]. Even when image segmentation is possible, for example, by morphological processing, the use binarization threshold through Rayleigh-distribution distance-minimization. This idea was inspired of geometric properties is desirable. Therefore, we propose a novel segmentation method that seeks by Staff et al. [42], where they analyzed stomatal pore center coordinates, but, in this case, only areas of an optimal binarization threshold through Rayleigh-distribution distance-minimization. This idea the inner triangles and their angles were reported. In Reference [43], a statistical mechanics explanation was inspired by Staff et al. [42], where they analyzed stomatal pore center coordinates, but, in this of this phenomenon is given through particle interaction, in this case, the distance between stomata, case, only areas of the inner triangles and their angles were reported. In Reference [43], a statistical as a means of regulation and state of the tissue in general. An algorithm summary is shown in Figure9. mechanics explanation of this phenomenon is given through particle interaction, in this case, the

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Plantsdistance2020 ,between9, 1613 stomata, as a means of regulation and state of the tissue in general. An algorithm9 of 17 summary is shown in Figure 9.

Figure 9. 9. GeneralGeneral process implemented in detection detection algorithm. algorithm. (Left) (Left) Input Input RGB image from microscope. (Center) (Center) Sequence Sequence of processing: First, Preprocessing stage: stage: Perona-MalikPerona-Malik filteringfiltering followed byby MeanshiftMeanshift clustering.clustering. Second, s segmentationegmentation algorithm proposed (DRTB) (DRTB):: binarization, labeling, tessellation,tessellation, distance analysis,analysis, and segmentationsegmentation stage plusplus optimaloptimal leveling.leveling. (Right) Final output image with segmented regionregion andand centroidcentroid basedbased tessellation.tessellation. 4.1. Preprocessing 4.1. Preprocessing Standard noise-reduction and boundary-preservation techniques are used given the common Standard noise-reduction and boundary-preservation techniques are used given the common noise properties present in most stomata images. We propose a two-step methodology based on noise properties present in most stomata images. We propose a two-step methodology based on Perona-Malik (PM) diffusion and Meanshift-Hadamard intensity-reduction. Perona-Malik (PM) diffusion and Meanshift-Hadamard intensity-reduction. Step I. Perona-Malik: The first step is PM iterative filter application [44]. This algorithm reduces Step I. Perona-Malik: The first step is PM iterative filter application [44]. This algorithm reduces noise level preserving and, at the same time, boundary structure through diffusion adaptation [45], noise level preserving and, at the same time, boundary structure through diffusion adaptation [45], meaning a lower diffusion constant near boundaries, and higher otherwise. Being an interactive meaning a lower diffusion constant near boundaries, and higher otherwise. Being an interactive filter, filter, level simplification of image structures is possible, which is a leading characteristic in stomata level simplification of image structures is possible, which is a leading characteristic in stomata search. search. Filter design criteria defined by Perona and Malik [46] are causality, immediate localization, Filter design criteria defined by Perona and Malik [46] are causality, immediate localization, and and piecewise smoothing. The last three properties are relevant in our problem given that stomata piecewise smoothing. The last three properties are relevant in our problem given that stomata have have well defined structures hard to segment, mainly by their surrounding structures. As for causality, well defined structures hard to segment, mainly by their surrounding structures. As for causality, PM filters all regions classified as noise. Immediate localization allows sharp boundaries, despite scale PM filters all regions classified as noise. Immediate localization allows sharp boundaries, despite changes, and the most useful property here is piecewise smoothing since it allows smaller structures to scale changes, and the most useful property here is piecewise smoothing since it allows smaller collapse into larger ones sharing a visual similarity criterion. structures to collapse into larger ones sharing a visual similarity criterion. 3 Let P be an image defined over the space Mn,m3(N) of arrays with n rows and m columns with pixels Let 푃 be an image defined over the space 푀푛,푚(ℕ) of arrays with 푛 rows and 푚 columns with in the natural numbers. Depending on the color-space, {Red, Green, Blue} (RGB) or {Hue, Saturation, pixels in the natural numbers. Depending on the color-space, {Red, Green, Blue} (RGB) or Value} (HSV)superscripts are used to refer the components. The PM filter ∂t defined from Mn,m(N) {Hue,Saturation,Value} (HSV)superscripts are used to refer the components. The PM filter 휕푡 into itself is a solution of the heat-equation parameterized by t representing the number of times the defined from 푀푛,푚(ℕ) into itself is a solution of the heat-equation parameterized by 푡 representing filter is applied (time in the original partial differential equation context) and a diffusion parameter the number of times the filter is applied (time in the original partial differential equation context) and controlling the noise level (see Reference [46] for more details) (see Figure10). As a result of PM filter a diffusion parameter controlling the noise level (see Reference [46] for more details) (see Figure 10). application over RGB space, the red channel increases separation levels with respect to background, As a result of PM filter application over RGB space, the red channel increases separation levels with andrespect this to result background, is relevant and as this the applicationresult is relevant of PM as filter the improvesapplication stomata of PM profilefilter improves detection, stomata as it is exemplifiedprofile detection in Figure, as it 11is .exemplified in Figure 11.

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Figure 10. 10. ImageImageImage preprocessingpreprocessing preprocessing withwith with PeronaPerona Perona-Malik--Malik (PM) (PM) filtering. filtering. PM PM has has three three main main parameters: parameters: Δ,, which∆, which represents represents the the diffusion diffusion level level;;; κ,,κ which, which represents represents an an advance advance-step--stepstep (time(time (time inin in thethe the original original PDEPDE framework)framework)framework);;; and and the the iteration iteration number number (advance (advance-step--stepstep times times iterationiteration iteration numbernumber number isis is thethe totaltotal timetime inin the the PDE framework). Four Four examples examples are are shown shown to to exemplify exemplify diffusion diffusion action action over over an an image image;;; higher ∆Δ parameter means more diffuse diffuse image.image.

Figure 11. ImageImageImage preprocessingpreprocessing preprocessing withwith PeronaPerona Perona-Malik--Malik (PM) (PM) filtering. ﬁltering. After After the the PM PM usage usage (see (see Figure Figure 1010))) thethe image image is is decomposed decomposed into into red, red, green green,green,, and blue channels (RGB decomposition decomposition))) with with a a standard standard routineroutine (see (see shared shared code code at at GitHub GitHub).).). TheThe red red channel channel P푃R푅 isisis used usedused later laterlater in inin the thethe next nextnext process. process.process.

Step II. MMeanshift-Hadamard:eanshift--Hadamard: The The second second step step is is the the application application of of Meanshift Meanshift 푀M operator over thethe redred channelchannel toto obtainobtain anan imageimage withwith with aa a reducedreduced reduced intensityintensity intensity levellevel level butbut but withwith with sharpsharp sharp defineddefined deﬁned boundariesboundaries boundaries,,, which eases eases the the segmentation segmentation process. process. This This step step uses uses an anunsupervised unsupervised clustering clustering algorithm algorithm over over the intensityintensitythe intensity-space--space through through a minimization a minimization process process between between each each energy energy cluster cluster [47][47] [.47. WithWith]. With thethe the aimaim aim atat improvingimprovingat improving signalsignal signal-background--background splitting, splitting, a Hadamard a Hadamard division division [48][48] operator [48] operator between between saturation saturation (HSV space)(HSV space)and red and (RGB red space) (RGB channel space) channelss is applied. is applied. This operation This operation enhances enhances signal splitting, signal splitting, as it can be as seen in Figure 12.. TheThe finalfinal operatoroperator isis thethe consecutiveconsecutive applicationapplication ofof PM,PM, Meanshift and Hadamard division (3):):

푠푠 푅 퐹 = 푃 ⊘ 푀((휕푡푡푃 )),, ((3)) where 퐹 isis thethe resultantresultant preprocessedpreprocessed image.image.

Plants 2020, 9, 1613 11 of 17 it can be seen in Figure 12. The ﬁnal operator is the consecutive application of PM, Meanshift and Hadamard division (3): s R F = P M ∂tP , (3) wherePlants 2020F is, 9 the, x FOR resultant PEER REVIEW preprocessed image. 11 of 17

FigureFigure 12. 12.Image Image preprocessing preprocessing with with Meanshift-Hadamard Meanshift-Hadamard division. division. After After the the PM PM usage usage (see (see Figure Figure 10), R 푅 R 푅 the10) ﬁltered, the filtered red-channel red-channel∂tP is 휕 used푡푃 is as used input as for input Meanshift; for Meanshift the output; theM output∂tP 푀is( combined휕푡푃 ) is combined with the 푆 saturationwith the saturation channel of channel the original of the image originalPS imagethrough 푃 the through Hadamard the Hadamard division, anddivision, this later and imagethis later is subjectimage tois subject clustering. to clustering.

ThisThis two-step two-step algorithm algorithm might might be be modiﬁed modifiedbased based upon upon stomata-image stomata-image type. type. However, However, the the proposedproposed technique technique is isuseful useful as segmentation as segmentation method method as it ashelps it helps with the with post-labeling the post-labeling process, process, which iswhich a main is focusa main of focus the research. of the research.

4.2.4.2. Delaunay-RayleighDelaunay-Rayleigh ThresholdThreshold BinarizationBinarization (DRTB(DRTB Algorithm) Algorithm) OurOur main main algorithm algorithm uses uses stomata stomata spatial spatial localization localization as as a a methodmethod to to findfind anan optimaloptimal binarizationbinarization threshold.threshold. As As input, input, it it uses uses the the grey-scale grey-scale image imagFeand 퐹 and outputs outputs an optimal an optimal umbralization umbralization level level such such that errorthat in error Delaunay-distance in Delaunay-distance frequencies frequencies be minimum be with minimum respect with to an ideal respect Rayleigh to an distribution ideal Rayleigh [28] (Figuredistribution 13). Next, [28] ( weFigure present 13). detailedNext, we description present detailed of DRTB description algorithm of phases.DRTB algorithm phases. Step III. Threshold level binarization: First, phase is binary-image building given a threshold. Literature gives various binarization techniques; however, most of them use as input information the relationship between intensity levels of the image. Instead, our algorithm explores a geometric-analysis of image embedded structures as optimal binarization-level search method. Initially, all binarization level-space values are traversed by normalization of intensity-levels in [0, 1] domain and then we explore such space. Binarization operator (F) at level l is defined as: Hl ( 0, F l, (F) = ≤ (4) Hl 1, F > l which has the same size of F but with values in [0, 1]. All pixels with values higher that l are given the value 1 and the rest 0. From the image (F), we can obtain a series of nR regions with areas given by Hl r = [r1, r2, r3, ... , rnR ]. Step IV. Binary labeling and filtering: Valid regions are those whose area be not an outlier. We propose [49] methodology for outlier detection. First, median absolute deviation (MAD) is determined as: MAD = b MEDIAN(r MEDIAN(r)), (5) −

Figure 13. Binary segmentation and Delaunay tessellation. (Left) Grey-scale image (see Figure 12) is subject to the process of binarization at the 푙-level. (Right) Different binarization threshold results in

Plants 2020, 9, x FOR PEER REVIEW 11 of 17

Figure 12. Image preprocessing with Meanshift-Hadamard division. After the PM usage (see Figure 푅 푅 10), the filtered red-channel 휕푡푃 is used as input for Meanshift; the output 푀(휕푡푃 ) is combined with the saturation channel of the original image 푃푆 through the Hadamard division, and this later Plants 2020image, 9, 1613is subject to clustering. 12 of 17

This two-step algorithm might be modified based upon stomata-image type. However, the measuredproposed in technique pixels with is value useful1, andas segmentationb = 1.4826 is a method normality as constant. it helps with Area the outlier post detection-labeling criterion process, is defined as: which is a main focus of the research. ri MEDIAN(r) − < 3, i = 1, ... nR. (6) 4.2. Delaunay-Rayleigh Threshold BinarizationMAD (DRTB Algorithm) ThisOur means main algorithm that all regions uses stomata not fulfilling spatial the localization criterion will as a bemethod discarded to find in an the optimal following binarization analysis. threshold.Step V. AsDelaunay input, it tessellation: uses the grey This-scale phase imag consistse 퐹 and in theoutputs generation an optimal of a triangular umbralization tessellation level such for allthat regions error defined in Delaunay as segmented-distance inliers. frequencies Mass centers be minimum for each withi-th region respect are to determined an ideal through Rayleigh centraldistribution moment [28 estimation] (Figure 13 [)50. Next,]. we present detailed description of DRTB algorithm phases.

FigureFigure 13. 13.Binary Binary segmentation segmentation and and Delaunay Delaunay tessellation. tessellation. (Left) (Left) Grey-scale Grey-scale image image (see (see Figure Figure 12 1)2) is is subjectsubject to to the the process process of of binarization binarization at at the thel -level.푙-level. (Right) (Right) Di Differentﬀerent binarization binarization threshold threshold results results in in diﬀerent tessellations. The best tessellation is found through an optimization procedure applied over the RMSD, with respect to an ideal Rayleigh distribution and the empirical histogram.

Let mi the mass center corresponding to i-th region (see Figure 14). A Delaunay algorithm uses an iterative process to generate a triangular tessellation considering a given region. The output is h 1 2 3i a set of 3 vertices for each triangle in terms of its coordinates mi , mi , mi ; given a two-dimensional set of points, it is easy to build a tessellation joining the points (vertices) with lines (edges) in a triangulation. For each triangle, a circumcircle might be drawn. A Delaunay tessellation is such a triangulation that no vertex from the tessellation is found inside the circles. Figure 14 shows a triangulation composed by 17 vertices. For each triangle, its circumcircle is drawn. A circle meets three vertices, and no vertex is inside. Delaunay tessellations tend to maximize the smallest inner angle possible. Diffusion operators complies with a maximum principle, which are relevant in existence and uniqueness of solutions under this operation; it is possible to extend these properties to numerical solution calculations obtained over Delaunay tessellations, mainly because of the geometric regularity Plants 2020, 9, 1613 13 of 17 imposed over the angle distribution [51]. This kind of tessellation are used in partial differential equation analysis and morphogenesis studies, and we expect it might play a role given the importance ofPlants gas-di 2020ffusion, 9, x FOR in PEER stomatal REVIEW physiology. 13 of 17

FigureFigure 14. 14.Delaunay Delaunay tessellation tessellation over over ROI ROI (red (red square). square). (Left) (Left) Meanshift Meanshift image image and and binarized binarized image. image. (Center)(Center) Zoom Zoom over over the the ROI ROI shows shows the the segmented segmented regions regions and and its its centroids centroids (yellow (yellow dots). dots). A A Delaunay Delaunay tessellationtessellation is is built built from from centroids; centroids note; note that that centroid centroid positions positions are are dependent dependent on on the the binarization binarization level levell.

(Right)푙. (Right) After After the the centroids centroids are are ﬁxed fixed and and the the tessellation tessellation is is built, built, the the set setd i 푑of푖 of all all found found distances distances are are usedused to to calculate calculate a histogram,a histogram, which which is is used used for for RMSD RMSD analysis. analysis.

StepStep VI. VI.Distance Distance analysis: analysis: As As result result of of the the last last process, process, a a set set of of triangles triangles in in obtained obtained (Figure (Figure 14 1),4), eacheach vertex vertex an an inlier-region. inlier-region. We We analyze analyze the the set set of of all all distances: distances: h i 푑 = [훿(푚1, 푚2 ), 훿(푚2, 푚3 ), 훿(푚11, 푚33) ], 푖 = 1, 2, 3, . . . 푛, (7) d푖i = δ m푖i , mi푖 , δ mi푖 , mi푖 , δ mi푖, mi 푖 , i = 1, 2, 3, ... n, (7) where 훿(∙,∙) is the Euclidian distance between vertices, and 푛 is the number of tessellated triangles. where δ( , ) is the Euclidian distance between vertices, and n is the number of tessellated triangles. Let 푑 =· [·푑1, 푑2, ⋯ , 푑푖, ⋯ , 푑3푛], a vector composed of all edges, and let 푛푖 the number of occurrences Let d = [d1, d2, , di, , d3n], a vector composed of all edges, and let ni the number of occurrences of of distance 푑푖···; then: ··· distance di; then: 푛n푖i 푝p((푑di|푙l) = (8) 푖 | P33푛n (8) ∑푖=i=11푛n푖 i isis the the occurrence occurrence probability probability of of di푑푖in in the the vector vectord 푑for for a a given given threshold thresholdl 푙obtained obtained from from the the binarizationbinarization process. process. StepStep VII. VII.Error Error estimation: estimation: Consider Consider the the Rayleigh Rayleigh distribution distribution of of parameter parameterσ : 휎:

푑2 푑 − d푖2 푖 2휎2i (9) ℛ(푑푖|휎) = diℯ 2; (di σ) =휎2 − 2σ ; (9) R | σ2 e our experimental results point at its presence in all analyzed images (manual or automatic process); oursee experimental Reference [28] results for other point applications at its presence of this in all probability analyzed imagesin natural (manual formations. or automatic Initial calibration process); seepoints Reference at 휎 = [282 ]and for other푡 > 0 applications. By root mean of- thissquare probability (RMS) deviation in natural (RMSD) formations. minimization Initial calibration, we assess points at σ = 2 and t > 0. By root mean-square (RMS) deviation (RMSD) minimization, we assess the the proximity of 푝(푑푖|푙) to ℛ(푑푖|휎) as a function of threshold 푙 and 휎-parameter: proximity of p(di l) to (di σ) as a function of threshold l and σ-parameter: 2 | R | ∑3푛 (ℛ(푑 |휎)−푝(푑 |푙)) 푅푀푆퐷(휎, 푙) s= √ 푖=1 푖 푖 . (10) P3n 3푛 2 i=1( (di σ) p(di l)) See the Discussion for commentsRMSD on(σ the, l) =significanceR of the| −parameter| . 휎. (10) 3n Step VIII. Optimal level selection: The ideal stomata distribution is associated with the minimum RMSD. The optimal parameter ̂푙 happens when such distance is minimum: 푎푟푔푚푖푛 ̂푙 = (푅푀푆퐷 ). (11) 0 < 푙 < 100 휎,푙 Once the best level is found, the image is finally binarized. An example is shown in Figure 15.

Plants 2020, 9, 1613 14 of 17

See the Discussion for comments on the signiﬁcance of the parameter σ. Step VIII. Optimal level selection: The ideal stomata distribution is associated with the minimum RMSD. The optimal parameter ˆl happens when such distance is minimum:

argmin ˆl = RMSD . (11) 0 < l < 100 σ,l

Once the best level is found, the image is ﬁnally binarized. An example is shown in Figure 15. Plants 2020, 9, x FOR PEER REVIEW 14 of 17

FigureFigure 15. 15.Sensibility Sensibility analysis analysis of binarization of binarization level and level RMSD and optimization RMSD optimization procedure. procedure. (Left) Diff erent (Left) histogramsDifferent histograms corresponding corresponding to different binarizationto different levels.binarization (Upper levels. right) (Upper RMSD sensibilityright) RMSD with sensibility respect towith binarization respect to level.binarization The level level. of The binarization level of binarization minimizing minimizing the RMSD the error RMSD is usederror foris used the for final the tessellation.final tessellation. (Lower (Lower right) Final right) output Final ofoutput the proposed of the proposed algorithm algorithm with the with positions the positions of stomata of fixed stomata at tessellationfixed at tessellation nodes. nodes.

5.5 Conclusions. Conclusions WeWe showed showed an an optimal optimal segmentation segmentation algorithm algorithm based based on on stomata-position stomata-position frequency-analysis frequency-analysis throughthrough Delaunay Delaunay tessellation tessellation and and Rayleigh Rayleigh distribution. distribution. WeWe tested tested the the algorithm algorithm with withHymenaea Hymenaea CourbarilCourbarilbefore before a a preprocess preprocess technique technique aimed aimed atat noisenoise reduction.reduction. Our proposal is centered around around a aDelaunay Delaunay-Rayleigh-Rayleigh Threshold Threshold Binarization (DRTB(DRTB algorithm)algorithm) that that allows allows an an optimal optimal binarization binarization thresholdthreshold with with a a posterior posterior segmentation. segmentation. The The automatic automatic segmentation segmentation shows shows the the presence presence of of a a stable stable RayleighRayleigh distribution, distribution despite, despite image image di differences.ﬀerences. This This result result is is relevant relevant given given that that DRTB DRTB algorithm algorithm mightmight be be applied applied to to other other species, species, as as was was shown shown in in the the manual manual experimental experimental phase phase with with seven seven didifferentﬀerent species. species.

Author Contributions: M.C. and P.A.T. performed manuscript conceptualization; R.V. performed formal analysis; P.A.T. analyzed the data; M.C., P.A.T., and R.V. wrote, edited the manuscript; O.M.B. performed project administration and funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding: This work has been partially financed by STICAmSud project number 14STIC-12.

Conflicts of Interest: The authors declare no conflict of interest.

Plants 2020, 9, 1613 15 of 17

Supplementary Materials: Our solution can be accessed online at https://github.com/mlacarrasco/drtb, and images database are available online at https://github.com/mlacarrasco/drtb/tree/main/database. Author Contributions: M.C. and P.A.T. performed manuscript conceptualization; R.V. performed formal analysis; P.A.T. analyzed the data; M.C., P.A.T., and R.V. wrote, edited the manuscript; O.M.B. performed project administration and funding acquisition. All authors have read and agreed to the published version of the manuscript. Funding: This work has been partially financed by STICAmSud project number 14STIC-12. Conflicts of Interest: The authors declare no conflict of interest.

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