# Applying Multifractal Analysis to Remotely Sensed Data for Assessing PYVV Infection in Potato (Solanum tuberosum L.) Crops

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Plant Material and Treatments

#### 2.2. Data Collection

#### 2.2.1. Spectroradiometric Data

^{®}panel was used for converting the reflected radiation into relative reflectance values. At the same time, a visual assessment of disease symptoms in PYVVi plants was continuously conducted and compared to Ctrl plants.

**Figure 1.**Aperture of the spectroradiometer sensor for each experiment. (

**a**) Experiment with a 25° solid angle aperture, (

**b**) Experiment with a 1° solid angle aperture.

#### 2.2.2. Multispectral Imagery

_{NIR}− R

_{red})/(R

_{NIR}+ R

_{red}), R=reflectance, ranging from −1 to +1) that track changes in chlorophyll concentration [22]; the soil adjusted vegetation index (SAVI = {(R

_{NIR}− R

_{red})/(R

_{NIR}+ R

_{red}+ L)*(1 + L)} , where L ranges from 0 to 1), proposed as a soil-line vegetation index to reduce the background effect [23]; and the infrared percentage vegetation index (IPVI = R

_{NIR}/(R

_{NIR}+ R

_{red})), a ratio-based index that holds a limited range with no negative values (0 < IPVI < 1), whose main disadvantage is its sensitivity to atmospheric noise [24]. These three SVIs had proved to be good indicators of infection by PYVV in potato [8].

#### 2.2.3. Spectroradiometric Data Pre-Processing

- S
_{j}(λ_{i}) and G_{j}(λ_{i}) are the corrected and raw signals for the j^{th}plant at the i^{th}wavelength, respectively - G
_{max j}and G_{min j}are the maximum and minimum raw measures of the j^{th}plant, and - G
_{Total max}and G_{Total min}are the maximum and minimum raw measures of all the plants within a treatment, measured in a sampling date. - A
_{j}is the ratio: response range of the j^{th}plant to total population. - B
_{j}is the regression intercept.

_{j}is the resultant reflectance, for the j

^{th}plant, after application of moving average; S

_{j}is the reflectance value obtained from Equation 1, through the λ

_{k}wavelength values in the analyzed moving window; and k is the counter that allows the analyzed window to move from −20 to +20. The λ

_{i}wavelength value ranged from 390 to 1,020 nm.

#### 2.3. Wavelet and Multifractal Data Analysis

^{th}order statistical moment [30]. A summarized explanation about wavelet-multifractal theory is presented in the Appendix for interested readers. For more details on the subject, see the papers by [12] and [31]. Thus, following the works of [12] and [14], wavelet and multifractal formalisms were applied to the data, aiming at detecting such singularities and obtaining the multifractal spectrum for each group, infected and control.

**Figure 2.**Heterogeneous signal of a reflectance spectrum, obtained from a potato plant, showing two types of singularities: cusp-like and step-like features, the so-called singularities in the signal.

#### 2.4. Statistical Analysis

## 3. Results and Discussion

#### 3.1. Reflectance Measurements

^{th}order of statistical moment. When ΔDh(q) = 0 and Δh(q) = 0, for a given q, it is taken as an indication of a normal evolution of reflectance (healthy plants). Conversely, any value other than zero (ΔDh ≠ 0 or Δh ≠ 0) is an indicator of stress (infected plants).

**Figure 3.**Scale invariance assessment for the reflectance data of healthy and infected plants. Dh(q, scale) is the numerator of Equation 6 in the Appendix for each order moment q > 0 along the wavelength scale. The slopes correspond to the Hölder exponents.

**Figure 4.**Passive reflectance of plants of both the first (left) and second experiments (right). Notice that the observable differences in the raw reflectance spectra registered by a sensor aperture of 25° (left) do not allow a clear treatments discrimination. In contrast, the spectra obtained by a sensor aperture of 1° (right) shows more clearly the differences in the spectra, at earlier dates.

**Figure 5.**Multifractal singularity spectra of plants. The first experiment was with a sensor aperture of 25° (left) and the second experiment was with a sensor aperture of 1° (right) .

#### 3.2. Visible and Near-Infrared Reflectance

#### 3.3. Recovery Period

**Figure 6.**Analysis of discrete bands (blue, green, red and NIR) of reflectance spectra from plants. In the first experiment, (

**a**) and (

**b**), the difference between treatments was evident at the 25th day after infection. For the second experiment, (

**c**) and (

**d**), the difference between treatments was noticed on the 23rd day after infection.

#### 3.4. Spectral Vegetation Indices

**Figure 7.**Spectral vegetation indices calculated from the images registered by the three-band (red, green, NIR) multispectral Tetracam

^{®}agricultural camera. Viral infection was evidenced 15 days after inoculation, 22 days before symptoms were visible.

## 4. Conclusions

## Acknowledgments

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## Appendix

#### The Continuous Wavelet Transform and the Wavelet Transform Modulus Maxima Multifractal

_{th}derivative of the Gaussian function for which we used η = 2, the Mexican hat wavelet.

_{0}, and this scales as

_{0}) describes the local degree of singularity or regularity around the point t

_{0}and is called the Hölder exponent (or singularity strength). The singularities can be measured by calculating their h(t

_{0}) exponent, which permits the characterization of singularities in time regardless of whether the derivative exists at that point. The Hölder exponent is a useful tool for mathematically encapsulating the notion of ‘‘sharp changes’’ in a time series. Locally, the Hölder exponent h(t

_{0}) is then governed by the singularities that accumulate at t

_{0}. This results in unavoidable oscillations around the expected power-law behavior of the wavelet transform amplitude. The exact determination of h from log-log plots on a finite range of scales is therefore somewhat uncertain [32]. WTMM methods circumvent these difficulties. As proven by [31], the WTMM (i.e., the local maxima of |TW(t,a

_{0})| at a given scale a

_{0}) detect all the singularities of a large class of signals. Thus, they are likely to contain all the information on the hierarchical distribution of singularities in the signal.

_{mmi}, and t and mm are the location parameter and location index, respectively. Then, the scaling exponent, h, and the singularity spectrum, D(h), are calculated from [29,32].

_{mm}denotes all positions of local maxima of |TW(t,a

_{0})| at some scale a

_{0}, being mm the index referring to the modulus maxima and q is the statistical moment. The moment q provides a microscope for exploring different regions of the singular measure. For q > 1, u(q) amplifies the more singular regions of the measure, whereas for q < 1, it accentuates the less singular regions, and for q = 1, the measure μ(1) replicates the original measurement [40].

^{®}. The reflectance data obtained from the above-mentioned experiments were processed with the WTMM method using the second derivative of the Gaussian function (Mexican hat) as mother wavelet analyzer. Derivatives are a reliable method for removing the noise or disturbance from remote sensing data, such as the additive baseline shift and the linear baseline increase [2,44].

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Chávez, P.; Yarlequé, C.; Piro, O.; Posadas, A.; Mares, V.; Loayza, H.; Chuquillanqui, C.; Zorogastúa, P.; Flexas, J.; Quiroz, R.
Applying Multifractal Analysis to Remotely Sensed Data for Assessing PYVV Infection in Potato (*Solanum tuberosum L.*) Crops. *Remote Sens.* **2010**, *2*, 1197-1216.
https://doi.org/10.3390/rs2051197

**AMA Style**

Chávez P, Yarlequé C, Piro O, Posadas A, Mares V, Loayza H, Chuquillanqui C, Zorogastúa P, Flexas J, Quiroz R.
Applying Multifractal Analysis to Remotely Sensed Data for Assessing PYVV Infection in Potato (*Solanum tuberosum L.*) Crops. *Remote Sensing*. 2010; 2(5):1197-1216.
https://doi.org/10.3390/rs2051197

**Chicago/Turabian Style**

Chávez, Perla, Christian Yarlequé, Oreste Piro, Adolfo Posadas, Víctor Mares, Hildo Loayza, Carlos Chuquillanqui, Percy Zorogastúa, Jaume Flexas, and Roberto Quiroz.
2010. "Applying Multifractal Analysis to Remotely Sensed Data for Assessing PYVV Infection in Potato (*Solanum tuberosum L.*) Crops" *Remote Sensing* 2, no. 5: 1197-1216.
https://doi.org/10.3390/rs2051197