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Automatic Leaf Epidermis Assessment Using Fourier Descriptors in Texture Images

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Scientific Reports
May 2016



The identification of plant species is not a trivial task, since it is carried out by experts and depends on the presence of fruits, flowers and leaves. However, fruits and flowers are not available throughout the year, while leaves are accessible most of the year. In order to assist the specialized work of species identification, methods of texture image analysis are used to extract characteristics from samples of imaged leaves and thus predict the species. Texture image analysis is a versatile and powerful technique able to extract measurements from patterns in the images. Using this technique, recent research has found a close relationship between texture and plant species (da Silva et al., 2015 and 2016). Here, we describe the procedure to extract texture features from microscopic images of leaves using Fourier (Cosgriff, 1960; Azencott, 1997; Gonzalez and Woods, 2006). It is important to highlight that other methods for texture extraction can be used as well.

This protocol is split into two parts: (A) leaf epidermal dissociation and (B) automatic method for leaf epidermal image analysis.

Keywords: Epidermis dissociation (表皮解离), Stomata visualization (气孔可视化), Plant identification (植物鉴定), Leaf epidermis (叶表皮), Texture image analysis (纹理图像分析)


The protocol for plant material dissociation was described by Franklin (1945) for wood; botanical anatomists have adapted the method for leaf epidermal dissociation. Dissociation methods allow the visualization of the epidermal cells as well as their attachments in a clearer way than diaphanization and epidermal imprints methods. In addition to the dissociation method described herein, there are others using Jeffrey’s solution (1:1 solution of chromic acid and nitric acid, each at 10%) or only nitric acid (Johansen, 1940), but these methods use strong acids and more hazardous to human health and the environment. Moreover, in our experience, epidermal dissociation with Franklin solution is more effective for a greater number of species.

As leaf epidermal traits have a great taxonomic value, texture analysis from epidermal images can be a suitable method for plant identification, once texture analysis is a powerful technique that allows the extraction of measurements from patterns in the images. In the study realized by (da Silva et al., 2016), three different methods were used in the process of texture feature extraction from leaf epidermal images: Fourier descriptors, corrosion-inspired texture analysis and local binary patterns; among them, Fourier descriptors were the best option, achieving higher success rate in plant species identification. A brief description of the methods for epidermal dissociation and texture feature extraction from microscopic epidermal images using Fourier descriptors has been reported in (da Silva et al., 2016).

Materials and Reagents

  1. Razor blade
  2. Glass vial
  3. Petri dish
  4. Needles and brush
  5. Glass slide (Perfecta, catalog number: 7105 )
  6. Cover glass (Perfecta, catalog number: 7004 )
  7. Gloves
  8. Leaf samples of different plant species (see species studied in Supplementary Table 1)
  9. Glacial acetic acid (Dinâmica Química, catalog number: 1242 )
  10. Hydrogen peroxide (30 volumes) (Dinâmica Química, catalog number: 2138 )
  11. Glycerin (Chem-Impex International, catalog number: 00599 )
  12. Safranin (Vetec, catalog number: 379 )
  13. 1% aqueous Safranin solution (see Recipes)


  1. Analytical balance (Shimadzu, model: AUY220 )
  2. Oven (Fanem, model: 315 SE )
  3. Light microscope (ZEISS, model: Axio Scope.A1 ) with equipment for photomicrography (ZEISS, model: Axiocam ICc3 )


  1. MATLAB R2014a (Version 8.3) (MATLAB and Statistics Toolbox Release, 2014a)
  2. Weka 3 (Hall et al., 2009), a data mining software in Java


  1. Leaf epidermal dissociation
    1. With the aid of a razor blade, remove from each leaf two or three samples of approximately 1 cm2 but less from its middle region, including the midrib (Figure 1A). Fresh or dry leaves can be used. Put the leaf fragments in a glass vial containing 1:1 glacial acetic acid and hydrogen peroxide (Franklin solution) in a drying oven at 60 °C (add enough volume so that the vial is left with about 1 cm of solution). Generally, this procedure is completed when the epidermis is beginning to peel off from the leaf fragments or when the leaf fragments become transparent and with air bubbles inside (Figures 1B-1E). The procedure can last from 12 h to a few days, depending on the morphological characteristics of each leaf. Some plants with thicker and sclerified leaves will require higher temperatures (up to 90 °C). Thus, if a leaf fragment incubated at 60 °C fails to produce an isolated epidermis, it must be heated up to 80-90 °C (it is not necessary to change the dissociation solution, however, it may be necessary to add more solution as it evaporates more at higher temperatures).
      1. For example, the species Clusia criuva, Duguetia furfuracea and Roupala montana required a temperature of 90 °C for leaf epidermal dissociation.
      2. We recommend that the fragments be removed from the middle region of the leaves by standardization; this region is usually used by botanists in leaf anatomy studies. It is possible to have differences in the size and density of the stomata according to the sampled region of the leaf.
    2. After this procedure, pour a little of the solution along with the leaf fragments in a Petri dish containing distilled water. Separate the adaxial and abaxial faces of the epidermis from the leaf fragment with a brush or needles (Figure 1F). In Petri dish, never remove the epidermis from the aqueous medium to maintain it stretched. Put the abaxial epidermis on a slide with a needle; the outer side of the epidermis must be placed upwards on the slide (Figure 1G). Whenever you deal with the epidermis on the slide, the epidermis must be stretched.
      Note: The inclusion of the midrib in the leaf fragment helps to distinguish between the adaxial and abaxial surfaces of the epidermis.
    3. To remove residues of glacial acetic acid and hydrogen peroxide solution, wash the fragment with distilled water holding the epidermis with a needle. Remove water excess from the slide with soft paper. Put one drop of 1% aqueous Safranin (see Recipes) on the epidermis (Figure 1H), stain for 10 min. Then, wash the epidermis with distilled water to remove the stain excess. Dry the slide with soft paper and, for mounting, put one to two drops of glycerin on the epidermis and carefully put the cover glass (Figure 1I).
      Note: Use gloves and protection glasses whenever you deal with Franklin solution or its components. Use gloves whenever you deal with Safranin.
    4. The epidermis is observed and photographed under an optical microscope. Since the plant epidermis has distinct characteristics, mainly of thickness, each image must be taken with adequate light intensity so that it does not become dark or too clear. All the images were taken at 200x magnification.
    5. The synthesis of the procedure is presented in Figure 1.

      Figure 1. Procedure for epidermal decoupling. A. Fresh leaf fragment; B-D. Stages of mesophyll digestion by the solution of glacial acetic acid and hydrogen peroxide; E. Leaf fragment ready for epidermal separation; F. Faces of the epidermis being dissociated from leaf fragment; G. Epidermis being placed on the slide; H. Staining; I. Mounting; J. Ready slide. Scale bars: A-E = 1 cm; F-J = 2 cm.

  2. Leaf epidermis image analysis
    1. The computational and statistical methods have been implemented using the software MATLAB R2014a (Version 8.3) (MATLAB and Statistics Toolbox Release, 2014a) and the classification procedures have been performed using Weka 3 (Hall et al., 2009), a data mining software in Java.
    2. Pre-processing method
      1. The first step is to convert the color image into a gray scale image. The pre-processing procedure was conducted by performing a histogram stretching to increase the image contrast. Let an image f (x, y), g(x, y) be the enhanced image of f (x, y) given by

        where, bpp is the number of bits per pixel of the image f (x, y). In one image with 256 gray levels, bpp = 8. In the enhanced image g(x, y), 1% of data is saturated at low and high intensities of the original image.
      2. Next frame shows the code in MATLAB to obtain the stretching of the input image.

      3. Figure 2 shows two examples, using Baccharis linearifolia and Esenbeckia pulmila, of the pre-processing stage.

        Figure 2. Pre-processing. Images of Baccharis linearifolia (A-C) and Esenbeckia pulmila (D-F). A and D: Original images; B and E: Gray images of the first column; C and F: Stretching of the images in the second column. Images are at magnification of 200x (scale bars = 50 µm).

    3. Image texture characterization
      As an image is represented as numerical matrix, a variety of methods can be used to extract patterns of texture in images. In this protocol, Fourier descriptors will be described.
      Fourier Descriptors
      1. Proposed by Cosgriff (1960), Fourier descriptors represent periodic functions given by the sum of sines and cosines of a periodic signal obtained from the discrete Fourier Transform (DFT). Discrete Fourier transform of a function 1D f (x) is given by:

        M is the length of the signal, u, x ∈ [0, 1..., M - 1] and j is a complex number. Inverse transform, from F (u) to f (x) is given by:

        For two-dimensional functions, as images, the Fourier transform is given by:

        M and N are the signal dimensions. Inverse transform of a 2D signal is:

      2. When the Fourier Transform is applied to the image, lower frequency coefficients remain at the extremities of the spectrum. The Figure 3A shows the original image and the Figure 3B, the spectrum of the Fourier transformation. Then, a shift operation is performed on the resultant image moving the origin of the Fourier Transform to the central coordinates ([M/2] for 1D signals and [M/2, N/2] for 2D signals) at frequency domain, as shown in Figure 3C.

        Figure 3. Fourier Transform. A. Original image; B. Fourier transform of (A); C. Shift of (B). Images are at magnification of 200x (scale bars = 50 µm).

      3. Low frequency components, found at the center of the shifted Fourier Transform, contain the most relevant information of the behavior of a signal. High frequency components are found at the extremities and represent abrupt changes and noise. These complex values can be used as characteristics of texture (Azencott et al., 1997; Gonzalez, and Woods, 2006). Here, the central values are prioritized as can be seen in the following. Two approaches are used to obtain the characteristics, also called descriptors, of the image texture.
      4. Fourier Circular: After shift operation, G circular rings are used to obtain the sum of all spectrum absolute values from the origin to each circular ring. G is given by min ([M/2J], [N/2J]), where the image has M x N pixels size. So, radius equal to 1, 2..., G providing G descriptors. The circular rings can be seen in Figure 4A.
        Fourier Circular-Angular Complementing the circular rings, F (u, v) is partitioned into eight angles equally spaced over the image, as in Figure 4B, resulting in Figure 4C. Eight circular rings have been used with radius equal to 3, 6, 9, 12, 15, 18, 21 and 24 pixels of distance. Thus 64 sectors are obtained, totaling 64 descriptors by summing the spectrum absolute values of each sector. The order in which the descriptors vector is composed does not interfere with the result. In this implementation, check the code list, the radius loop is inside the quadrant loop.

        Figure 4. Circular and angular rings to obtain the Fourier descriptors. A. Circular; B. Angular; C. Circular-Angular.

      5. The next three frames show the complete code in MATLAB to obtain the Fourier Circular-Angular descriptors.

Data analysis

  1. For each image of the leaf, a set of characteristics is extracted to represent it. Then, these characteristics are compared to identify to which species of plant this leaf belongs. To proceed with the experiments, the species of the samples in the training set are known and the goal is to predict the species of the test set that are unknown. The descriptors of one image belonging to the test set are compared to all samples of the training set using the Nearest Neighbor algorithm (Aha et al., 1991) to predict the species. The species is assigned to a given sample according to the closest sample in the training set.
  2. The strategy to distribute the samples in a training and test set is the k-fold cross-validation scheme (Hastie et al., 2001), represented in Figure 5. In this approach, the set of samples is equally divided into k folds. Then, k-1 folds are joined forming the training set while the remaining fold is assigned to the test set. This procedure is performed k times varying the test fold. At the end, all the samples have been classified generating the success rate by calculating the ratio between the number of samples correctly classified by the number of samples.

    Figure 5. k-fold cross-validation scheme

  3. The data analysis procedure can be summarized by creating a file with all the characteristics of each sample to be read by Weka and perform the classification. Next frame shows the template to construct the file to be read by Weka to perform the classification of the samples.

  4. Synthesis of the computer procedure
    1. Pre-process all the images to enhance the contrast of the image.
    2. Extract characteristics using Fourier descriptors from each image, as shown in Figure 6.
    3. Classify all the samples using Nearest Neighbor in a k-fold cross validation scheme and check the success rate.
    4. Training the system and apply it to the leaf identification or analysis.

      Figure 6. Extraction of characteristics using Fourier descriptors


The success in dissociation of the epidermis in different species depends on the level of individual training. Each species behaves differently during the procedure. As necessary, oven temperature can be increased to speed up and to facilitate the process. Thus, the temperature can influence the result; thicker leaves usually require higher temperatures (up to 90 °C) than thinner ones (60 °C).


  1. 1% aqueous Safranin solution
    Add 1 g Safranin in 100 ml distilled water in a bottle
    Mix well and store in the dark in a refrigerator (can be kept for several months under these conditions)


This protocol was adapted from da Silva et al. (2016). The authors gratefully thank the financial support of São Paulo Research Foundation (FAPESP), with grant Nos.: 2011/01523-1, 2011/23112-3 and 2011/21467-9, National Council for Scientific and Technological Development (CNPq) with grant Nos.: 307797/2014-7 and 484312/2013-8 and PROPE/UNESP (14/2012/Renove), and Coordination for the Improvement of Higher Education Personnel (CAPES). The authors declare that there are no conflicts of interest or competing interest.


  1. Aha, D. W., Kibler, D. and Albert, M. K. (1991). Instance-based learning algorithms. Mach Learn 6(1): 37-66.
  2. Azencott, R. Wang, J. P. and Younes, L. (1997). Texture classification using windowed Fourier filters. IEEE T Pattern Anal 19(2):148-153.
  3. Cosgriff, R. L. (1960). Identification of shape. Ohio State University Research Foundation, Columbus. ASTIA AD 254: 792.
  4. da Silva, N. R., da Silva Oliveira M. W., Almeida Filho, H. A., Pinheiro, L. F. S., Rossatto, D. R. Kolb, R. M. and Bruno, O. M. (2016). Leaf epidermal images for robust identification of plants. Sci Rep 6:25994.
  5. da Silva, N. R., Florindo, J. B., Gómez, M. C., Rossatto, D. R., Kolb, R. M. and Bruno, O. M. (2015). Plant identification based on leaf midrib cross-section images using fractal descriptors. PLoS One 10(6): e0130014.
  6. Franklin, G. L. (1945). Preparation of thin sections of synthetic resins and wood-resin com- posites, and a new macerating method for wood. Nature 155(3924):51.
  7. Gonzalez, R. C. and Woods, R. E. (2006). Digital Image Processing (3rd Edition). Prentice-Hall.
  8. Hall, M., Frank, E. Holmes, G. Pfahringer, B. Reutemann, P. and Witten, I. H. (2009). The weka data mining software: An update. ACM SIGKDD Explorations Newsletter 11(1): 10-18.
  9. Hastie, T. Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning. Springer.
  10. Johansen, D. A. (1940). Plant microtechnique. McGraw-Hill.
  11. MATLAB and Statistics Toolbox Release. (2014a). The MathWorks, Inc., Natick, Mas-sachusetts, United States.


植物种类的鉴定不是一项微不足道的任务,因为它是由专家进行的,取决于水果,花和叶子的存在。然而,水果和鲜花全年都不可用,而叶子一年中大部分时间都可以使用。为了辅助物种鉴定的专业工作,纹理图像分析的方法被用来从成像叶片的样本中提取特征,从而预测物种。纹理图像分析是一种多功能且功能强大的技术,能够从图像中的图案中提取测量结果。使用这种技术,最近的研究发现了纹理和植物物种之间的密切关系(da Silva et al。,2015和2016)。在这里,我们描述了使用傅里叶(Cosgriff,1960; Azencott,1997;冈萨雷斯和伍兹,2006)从叶片的显微图像中提取纹理特征的程序。强调其他纹理提取方法也是很重要的。



由于叶表皮性状具有很高的分类学价值,因此纹理分析是一种功能强大的技术,可以从图像的图案中提取测量结果,因此表皮图像的纹理分析可以成为植物鉴定的合适方法。在(da Silva et al。2016)实现的研究中,从叶表皮图像提取纹理特征的过程中,使用了三种不同的方法:傅里叶描述符,腐蚀启发纹理分析和局部二进制模式;其中傅里叶描述符是最好的选择,在植物种类鉴定中取得较高的成功率。已经在(da Silva等人,2016)中报道了使用傅立叶描述子从显微表皮图像中提取表皮解离和纹理特征的方法的简要描述。

关键字:表皮解离, 气孔可视化, 植物鉴定, 叶表皮, 纹理图像分析


  1. 剃刀刀片
  2. 玻璃小瓶
  3. 培养皿
  4. 针和刷
  5. 玻璃幻灯片(Perfecta,目录号:7105)
  6. 盖玻璃(Perfecta,目录号:7004)
  7. 手套
  8. 不同植物物种的叶样品(见补充表1
  9. 冰醋酸(DinâmicaQuímica,目录号:1242)
  10. 过氧化氢(30卷)(DinâmicaQuímica,目录编号:2138)
  11. 甘油(Chem-Impex International,目录号:00599)
  12. Safranin(Vetec,目录号:379)
  13. 1%的番红素水溶液(见食谱)


  1. 分析天平(岛津,型号:AUY220)
  2. 烤箱(法恩姆,型号:315 SE)
  3. 光学显微镜(ZEISS,型号:Axio Scope.A1)和显微摄影设备(ZEISS,型号:Axiocam ICc3)


  1. MATLAB R2014a(版本8.3)(MATLAB和统计工具箱发布,2014a)
  2. Weka 3(Hall等人,2009),Java中的数据挖掘软件


  1. 叶表皮解离
    1. 在一个剃刀刀片的帮助下,从每片叶子上取下两个或三个大约1cm2的样品,但是从中部区域,包括中脉(图1A),更少。新鲜或干燥的叶子可以使用。将叶片置于含有1:1冰醋酸和过氧化氢(富兰克林溶液)的玻璃小瓶中,置于60℃的干燥箱中(加入足够的体积以使小瓶留有约1cm的溶液)。通常,当表皮开始从叶片上剥离或叶片变得透明并且内部有气泡时,该过程就完成了(图1B-1E)。程序可以持续12小时到几天,这取决于每片叶子的形态特征。一些植物的叶片较厚和变硬需要较高的温度(高达90°C)。因此,如果在60℃温育的叶片不能产生分离的表皮,则必须将其加热至80-90℃(不需要改变解离溶液,但是可能需要加入更多的溶液因为它在更高的温度下蒸发更多)。
      1. 例如,Clusia criuva,Duguetia furfuracea和Roupala montana等物种需要90°C的温度才能使叶表皮解离。
      2. 我们建议通过标准化将碎片从叶子的中间区域去除;这个地区通常被植物学家用于叶解剖学研究。根据叶子的采样区域,可能在气孔的大小和密度方面有所不同。
    2. 在此过程之后,将一点溶液和叶片一起倒入含有蒸馏水的培养皿中。用刷子或针将叶表皮的正面和背面从叶片上分开(图1F)。在培养皿中,切勿从水性介质中除去表皮以保持其延伸。用针将下表皮置于载玻片上;表皮的外侧必须向上放在载玻片上(图1G)。每当你处理幻灯片上的表皮,表皮必须被拉伸。
    3. 为了去除冰醋酸和过氧化氢溶液的残留物,用针用蒸馏水洗涤表皮。用软纸清除载玻片上多余的水分。将一滴1%的番红素水溶液(见食谱)放在表皮上(图1H),染色10分钟。然后,用蒸馏水清洗表皮,除去色斑。用软纸擦干玻片,为了安装,在表皮上放一到两滴甘油,小心地放上盖玻片(图1I)。
    4. 观察表皮并在光学显微镜下拍照。由于植物表皮具有明显的特征,主要是厚度,所以每个图像必须在足够的光照强度下拍摄,以使其不变暗或不太清晰。所有的图像都是200倍放大。
    5. 程序的合成如图1所示。

      图1.表皮解耦过程A.新鲜叶片; B-d。用冰醋酸和过氧化氢溶液叶肉消化的阶段; E.叶片准备好表皮分离; F.表皮的表面与叶片片断分离; G.表皮被放置在幻灯片上; H.染色; I.安装; J.准备幻灯片。比例尺:A-E = 1厘米; F-J = 2厘米。

  2. 叶表皮图像分析
    1. 已经使用软件MATLAB R2014a(版本8.3)(MATLAB and Statistics Toolbox Release,2014a)实施了计算和统计方法,并且使用Weka 3(Hall等人,2009年)进行了分类程序),Java中的数据挖掘软件。
    2. 预处理方法
      1. 第一步是将彩色图像转换成灰度图像。预处理程序通过执行直方图拉伸来增加图像对比度。设图像f(x,y),g(x,y)为f(x,y)的增强图像,由

        其中,bpp是图像f(x,y)的每像素的比特数。在256灰度级的图像中,bpp = 8。在增强图像g(x,y)中,1%的数据在原始图像的低强度和高强度下饱和。
      2. 下一帧显示在MATLAB中的代码,以获得输入图像的拉伸。

      3. 图2显示了两个使用前处理阶段的 Baccharis linearifolia 和 Esenbeckia pulmila 的例子。

        图2.预处理 Baccharis linearifolia (A-C)和 Esenbeckia pulmila (D-F)的图片。 A和D:原始图像; B和E:第一列的灰色图像; C和F:在第二列中拉伸图像。图像放大200倍(比例尺= 50微米)。

    3. 图像纹理表征
      1. 由Cosgriff(1960)提出,傅立叶描述符表示由离散傅里叶变换(DFT)获得的周期性信号的正弦和余弦之和给出的周期性函数。函数1D f(x)的离散傅立叶变换由下式给出:

        M是信号的长度,u,x∈[0,1 ...,M-1],j是一个复数。从F( u )到f( x )的逆变换由下式给出:



      2. 当傅立叶变换应用于图像时,低频系数保留在频谱的末端。图3A显示了原始图像,图3B显示了傅里叶变换的频谱。然后,对于将傅立叶变换的原点移动到中心坐标(对于1D信号为[M / 2],对于2D信号为[M / 2,N / 2])的合成图像进行移位操作,如如图3C所示。

        图3.傅立叶变换:一种。原始图像; B.(A)的傅里叶变换; (B)的转换。图像放大200倍(比例尺= 50微米)。

      3. 位于移位傅里叶变换中心的低频分量包含信号行为的最相关信息。高频分量在四肢发现,代表突变和噪音。这些复杂的值可以用作纹理的特征(Azencott等,1997; Gonzalez和Woods,2006)。在这里,中心值被优先考虑,如下所示。两种方法被用来获得图像纹理的特征,也称为描述符。
      4. 傅里叶循环:移位运算后,使用G个圆环来获得从原点到每个圆环的所有谱绝对值之和。 G由min([M / 2J],[N / 2J])给出,其中图像具有M×N像素大小。因此,半径等于1,2,...,G提供G描述符。圆环可以在图4A中看到。

        图4.获取傅立叶描述符的圆环和角环。 :一种。圆; B.角度; C.圆角。

      5. 接下来的三个框架显示在MATLAB中完整的代码来获得傅立叶圆角描述符。


  1. 对于叶的每个图像,提取一组特征来表示它。然后,比较这些特征,以确定这片叶子属于哪一种植物。为了进行实验,训练集中样本的种类是已知的,目标是预测未知的测试集的种类。属于测试集合的一个图像的描述符与使用最近邻算法(Aha等人,1991)的训练集合的所有样本进行比较以预测物种。
  2. 在训练和测试集合中分发样本的策略是图5中所示的k-fold交叉验证方案(Hastie等人,2001)。在这种方法中,样本集合平分为k倍。然后,加入k-1个折叠形成训练集,而剩下的折叠分配给测试集。这个程序执行k次,改变测试倍数。最后,通过计算正确分类的样本数量与样本数量之间的比例,所有样本都被分类为生成成功率。

    图5. k-fold交叉验证方案

  3. 数据分析过程可以概括为创建一个文件,每个样本的所有特征都要由Weka读取并执行分类。下一帧显示的模板来构建Weka读取的文件来执行样本的分类。

  4. 计算机程序的综合
    1. 预处理所有图像以增强图像的对比度。
    2. 使用来自每个图像的傅里叶描述符来提取特征,如图6所示。

    3. 使用最近邻法在k-fold交叉验证方案中对所有样本进行分类,并检查成功率。
    4. 培训系统并将其应用于叶识别或分析。





  1. 1%番红素水溶液


该协议摘自da Silva et al。(2016)。作者非常感谢圣保罗研究基金会(FAPESP)的财政支持,批准号为:2011 / 01523-1,2011 / 23112-3和2011 / 21467-9,国家科学和技术发展委员会(CNPq)批准号:307797 / 2014-7和484312 / 2013-8以及PROPE / UNESP(14/2012 / Renove),以及改进高等教育人员的协调(CAPES)。作者声明不存在利益冲突或利益冲突。


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引用:da Silva, N. R., Oliveira, M. W., Filho, H. A., Pinheiro, L. F., Kolb, R. M. and Bruno, O. (2017). Automatic Leaf Epidermis Assessment Using Fourier Descriptors in Texture Images. Bio-protocol 7(23): e2630. DOI: 10.21769/BioProtoc.2630.