 Methodology
 Open Access
Twodimensional multifractal detrended fluctuation analysis for plant identification
 Fang Wang^{1}Email author,
 Dengwen Liao^{2},
 Jinwei Li^{3} and
 Guiping Liao^{3}
https://doi.org/10.1186/s1300701500497
© Wang et al.; licensee BioMed Central. 2015
 Received: 15 October 2014
 Accepted: 21 January 2015
 Published: 26 February 2015
Abstract
Background
In this paper, a novel method is proposed to identify plant species by using the two dimensional multifractal detrended fluctuation analysis (2D MFDFA). Our method involves calculating a set of multifractal parameters that characterize the texture features of each plant leaf image. An index, I _{0}, that characterizes the relation of the intraspecies variances and interspecies variances is introduced. This index is used to select three multifractal parameters for the identification process. The procedure is applied to the Swedish leaf data set containing leaves from fifteen different tree species.
Results
The chosen three parameters form a threedimensional space in which the samples from the same species can be clustered together and be separated from other species. Support vector machines and kernel methods are employed to assess the identification accuracy. The resulting averaged discriminant accuracy reaches 98.4% for every two species by the 10 − fold cross validation, while the accuracy reaches 93.96% for all fifteen species.
Conclusions
Our method, based on the 2D MFDFA, provides a feasible and efficient procedure to identify plant species.
Keywords
 Plant identification
 Multifractal detrended fluctuation analysis
 Support vector machines and kernel methods
Introduction
The increasing interest in biodiversity and biocomplexity, together with the growing availability of digital images and image analysis algorithms, makes plant species identification and classification a topic that has attracted many researchers’ attention. In general, many parts of a plant such as flowers, seeds, roots, and leaves can be used to identify plant species [13]. In this paper, we focus on the usage of image of leaves as they are widely available. Leaf’s shape, color, vein properties, texture and contours are important features for plant identification. For example, leaf shapes were used in [46]; complex veins and contours of leaves were used in [7] and leaf texture was used in [811] for plant species identification. For plant species identification using digital morphometrics, we refer the reader to [1214] and the references therein.
Note that in [7], a monofractal method was used to extract plant leaf’s features from leaf images. This method was then used in [15,16]. It’s been recognized that the monofractal method cannot fully extract detailed information from the leaf image and therefore cannot be efficiently applied to process the images of the objects that are locally irregular [17]. To overcome this difficulty, several multifractal analysis (MFA) methods were proposed [1822]. For example, Backes et al. [18,19] used multiscale fractal dimensions to describe the texture property of leaf’s surface to identify plants, which turned out to be very efficient. Note that the classical MFA is based on capacity measurement or probability measurement and thus describes only stationary measurements [17]. For a leaf image, the surface itself is hardly stationary. Therefore, the multifractal detrended fluctuation analysis (MFDFA) method that can deal with nonstationary is a desirable method for leaf image analysis [23]. Though the MFDFA method has been successfully applied in many fields for nonstationary series and surfaces [2430], to the best of our knowledge, no work yet has applied the MFDFA on leaf images for plant identification and classification. In this paper, we attempt to identify plant species via leaf images by using the MFDFA. More precisely, we first adopt the MFDFA to extract important texture features from leaf images and obtain several key multifractal parameters, and then we apply the support vector machines and kernel methods (SVMKM) to distinguish leaves from different plant species. The widely used Swedish leaf data set [31] containing leaves from fifteen different Swedish tree species are used for our experiments. Our results show that the average accuracy is 98.4% for every two species by the 10 − fold cross validation; for the overall species, the average accuracy reaches 93.96% by the same validation criterion.
We organize the rest of this paper as follows: in Methods and materials we adopt the twodimensional (2D) MFDFA to calculate the multifractal parameters. In Results and discussion, we present and discuss our results. Our method is then further tested in Model test. A summary is provided in Conclusions.
Methods and materials
Multifractal detrended fluctuation analysis

Step 1: Regard a leaf image as a selfsimilar surface and represent it by an M × N matrix X = (X(i, j)), i = 1, 2,…, M and j = 1, 2,…, N. Partition the surface into M _{ s } × N _{ s } nonoverlapping square subsurface of equal length s, where M _{ s } ≡ [M / s] and N _{ s } ≡ [N / s] are positive integers (Here [u] stands for the largest integer that is less than or equal to u). Each subsurface is denoted by X _{ m,n } = X _{ m,n }(i, j) with X _{ m,n }(i, j) = X(r + i, t + j) for 1 ≤ i, j ≤ s, where r = (m1)s and t = (n1)s. Note that M and N are not necessarily multiples of the length s, therefore, the subsurfaces in the upperright and the bottom may not be taken into consideration. We can then repeat the partitioning procedure starting from the other three corners.

Step 2: For each subdomain X _{ m,n }, find its cumulative sum$$ {G}_{m,n}\left(i,j\right)={\displaystyle \sum_{k_1=1}^i{\displaystyle {\sum_{k_2=1}^j}_{m,n}Xm\left({k}_1,{k}_2\right)}}, $$(1)

where 1 ≤ i, j ≤ s, m = 1, 2, …, M _{ s } and n = 1, 2, …, N _{ s }. Then G _{ m,n } = G _{ m,n }(i, j) (i, j = 1, 2, · · ·, s) itself is a surface.

Step 3: For each surface G _{ m,n }, obtain a local trend G ^{~} _{ m,n } by fitting it with a prechosen bivariate polynomial function. In this paper, we choose the trending function as$$ {\tilde{G}}_{m,n}\left(i,j\right)= ai+bj+c, $$(2)

a.where 1 ≤ i, j ≤ s and a, b and c are free parameters to be determined by the leastsquares method. The residual matrix is then given by y _{ m,n } = y _{ m,n }(i, j) with$$ {y}_{m,n}\left(i,j\right)={G}_{m,n}\left(i,j\right){\tilde{G}}_{m,n}\left(i,j\right). $$(3)

Step 4: Define the detrended fluctuation function F(m, n, s) for the segment X _{ m,n } as follows:$$ {F}^2\left(m,n,s\right)=\frac{1}{s^2}{\displaystyle \sum_{i=1}^s{\displaystyle \sum_{j=1}^s{y}_{m,n}{\left(i,j\right)}^2}} $$(4)

and the qthorder fluctuation function$$ {F}_q(s)={\left[\frac{1}{M_s{N}_s}{\displaystyle \sum_{m=1}^{M_s}{\displaystyle \sum_{n=1}^{N_s}{\left[F\left(m,n,s\right)\right]}^q}}\right]}^{1/q},q\ne 0. $$(5)$$ {F}_q(s)= \exp \left\{\frac{1}{M_s{N}_s}{\displaystyle \sum_{m=1}^{M_s}{\displaystyle \sum_{n=1}^{N_s} \ln \left[F\left(m,n,s\right)\right]}}\right\},\ q=0. $$(6)

Step 5: Vary the value of s ranging from 6 to min(M, N)/4. If there is longrange powerlaw correlation for large values of s, then$$ {F}_q(s)\propto {s}^{h(q)}. $$
α _{max} = max{α(q), q∈[−15,15]} and α _{min} = min{α(q), q∈[−15,15]}. Note that the index ∆α is considered as an indicator to measure the absolute magnitude of the gray scale volatility. The larger value of ∆α, the smaller even distribution of probability measure and the more roughness image surface will be expected. The index ∆f is the Hausdorff dimension of the measure object, which measures the degree of confusion. Therefore both ∆α and ∆f are important multifractal parameters in describing the characteristics of an image in our study.
Experiment materials
We first transform the color image to gray scale so that each image can be viewed as a three dimensional surface with the first two coordinates (i, j) denoting the 2D position and the third coordinate z denoting the gray level of the corresponding pixel.
Multifractal nature of image surfaces
Each image is stored as a 2D matrix in 256 grey levels. This allows us to follow the procedure introduced in Multifractal detrended fluctuation analysis to calculate the associated h(q) and τ(q). If τ(q) is nonlinear in q, that is h(q) is not independent of q, then the image possesses the multifractal nature.
Results and discussion
As seen in Figure 4, comparing with h(2) and h(3), the estimations of h(−3), h(−2), h(−1) and h(1) vary in relatively wider dynamic ranges and thus demonstrate better abilities to distinguish textures among different species. Yet, one notes that there are relatively large variations in the standard deviations among the 75 samples for the h(q) exponents in Figure 5. This suggests that this indicator alone may not be adequate to identify the fifteen tree species. Also as seen in Figure 6 that the three parameters, α _{max}, ∆α, and ∆f admit wider dynamic ranges than the other three parameters do. The variations among the 75 samples in the same tree species are notably large as shown in Figure 7.
The calculated σ _{ bet.} , σ _{ in } and I _{0} for the 12 multifractal parameters
Parameters  h (−3)  h (−2)  h (−1)  h (1)  h (2)  h (3)  α _{max}  α _{min}  Δα  Δf  D _{1}  D _{2} 

σ _{ bet.}  0.0605  0.0403  0.0363  0.0255  0.0237  0.0233  0.0845  0.0183  0.0806  0.1327  0.0140  0.0259 
σ _{ in }  0.0368  0.0342  0.0381  0.0779  0.0496  0.0395  0.0722  0.0151  0.0646  0.2645  0.0264  0.0252 
I _{0}  1.6459  1.1810  0.9548  0.3280  0.4777  0.5891  1.1705  1.2132  1.2469  0.5015  0.5316  1.0284 
We choose the combination of three multifractal parameters with larger I _{0} values, namely, {h(−3), α _{min}, Δα}, as the feature descriptors for our classification purpose and apply the support vector machines and kernel methods (SVMKM) with the heavytailed radial basis function’htrfb’ as the kernel [36]. It is worth mentioning that the combination of 4 or more parameters does not lead to significant higher accuracies, but at a cost with much longer computational time and with no visual advantages. In this sense, the combination of the above three parameters is optimal. For the total sample set containing 75 × 15 = 1125 samples, we use the K − fold cross validation to evaluate the learning performance. This means that 100 (K − 1)/K% samples are randomly chosen as a training set and the remaining 100/K% samples are considered as a test set. The calculation process is then repeated 10 times to eliminate the impact of randomness.
The results of identification for the fifteen species of tree leaves by the method of SVMKM with K = 10
MI  MII  MIII  MIV  MV  MVI  MVII  MVIII  MIX  MX  MXI  MXII  MXIII  MXIV  MXV  

MI  69  0  2  0  1  0  0  0  3  0  0  0  0  0  0 
MII  0  70  0  0  1  0  1  0  1  1  0  1  0  0  0 
MIII  1  0  71  0  0  0  0  0  0  0  2  1  0  0  0 
MIV  1  0  1  68  1  0  0  3  1  0  0  0  0  0  0 
MV  0  2  0  0  69  0  0  2  1  0  0  1  0  0  0 
MVI  1  0  1  1  0  69  0  0  1  0  1  1  0  0  0 
MVII  0  2  0  1  2  0  70  0  0  0  0  0  0  0  0 
MVIII  0  0  0  1  2  0  1  70  0  0  1  0  0  0  0 
MIX  1  0  0  0  1  1  0  0  70  0  1  1  0  0  0 
MX  0  0  0  0  0  0  0  0  0  74  1  0  0  0  0 
MXI  1  0  1  0  0  0  0  0  3  0  70  0  0  0  0 
MXII  0  0  1  1  0  1  1  0  0  0  0  71  0  0  0 
MXIII  0  0  0  0  0  0  0  0  0  0  0  0  72  0  3 
MXIV  0  0  0  0  0  0  0  0  0  0  0  0  0  73  2 
MXV  0  0  0  0  0  0  0  0  0  0  0  0  2  2  71 
Model test
Conclusions
We should point out that most of the existing work on texture image recognition focuses mainly on the standard multifractal analysis. Our work has shown that the MFDFA is of particular practice for plant leaf identification as the MFDFA multifractal parameters can be combined to distinguish similar but different leaf textures.
Declarations
Acknowledgments
The authors wish to thank the two reviewers and the editorinChief Prof. Brian G. Forde for their comments and suggestions, which led to a great improvement to the presentation of this work.
FW was supported by the Young Scholar Project Funds of the Education Department of Hunan Province (14B087). JL and GL were supported by the Major project funds of Science and technology program of Hunan Province (2013FJ10064). DL was partially supported by a grant from the Forestry Department of Hunan Province, China.
Authors’ Affiliations
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