## Abstract

The fluctuations in the elastic light scattering spectra of normal and dysplastic human cervical tissues analyzed through wavelet transform based techniques reveal clear signatures of self-similar behavior in the spectral fluctuations. The values of the scaling exponent observed for these tissues indicate the differences in the self-similarity for dysplastic tissues and their normal counterparts. The strong dependence of the elastic light scattering on the size distribution of the scatterers manifests in the angular variation of the scaling exponent. Interestingly, the spectral fluctuations in both these tissues showed multi-fractality (non-stationarity in fluctuations), the degree of multi-fractality being marginally higher in the case of dysplastic tissues. These findings using the multi-resolution analysis capability of the discrete wavelet transform can contribute to the recent surge in the exploration for non-invasive optical tools for pre-cancer detection.

© 2011 OSA

## 1. Introduction

The use of optical techniques for the study of biomedical systems is a rapidly developing field that has seen a dramatic expansion in the recent years, partly due to tremendous progress in the field of lasers, fiber optics and associated technologies. Both medicine and biotechnology require appropriate instrumentation to analyze and monitor biological systems for deviations from normality. Optical methods, due to their non-invasive nature, are providing novel approaches for medical imaging, diagnosis and therapy. Considerable efforts have been made in the recent past to use optical techniques such as fluorescence spectroscopy [1–6], Raman spectroscopy [7] and elastic scattering spectroscopy [8] for quantitative and early diagnosis of various diseases. Several optical imaging techniques like coherence gated imaging, polarization gated imaging and diffuse optical tomography are also being actively pursued for obtaining high resolution (micron scale) images of biological objects and their underlying structure [9–13].

For optical diagnosis, elastic and inelastic light scattering spectra from tissues are exploited. The in-elastically scattered light (via processes like fluorescence and Raman) contain useful biochemical information about the sample that can be employed for probing subtle biochemical changes as signatures of disease progression. On the other hand, elastically scattered light from biological tissues contain rich morphological and functional information of potential biomedical importance [8, 14–27]. Both the angular and wavelength dependence of the elastically scattered light from tissue can be analyzed to extract and quantify subtle morphological changes taking place during progression of a disease [16–20, 23, 24], and thus may be explored as a sensitive tool for early diagnosis. This would however involve appropriate modeling of light scattering in complex random media like tissues, and the development of suitable approaches to extract/interpret the morphological information contained in the elastic light scattering signal.

The spatial fluctuation of the refractive index in biological tissues arising from scatterers ranging in sizes from a few nanometers to a few micrometers give rise to elastic scattering [8, 22]. The lack of sufficient knowledge about the complex dielectric fluctuations in the tissues pose a formidable problem in the exact modeling of light scattering. Nevertheless, several efforts have been made in the recent years using electromagnetic (EM) theory based approaches like Mie theory and Born approximation to model and understand the scattering process from biological tissues [28–32]. It has also been shown that the refractive index fluctuations in biological tissues are fractal in nature which can be used to understand the structural changes in tissues induced by diseases [18, 29, 33–35].

Since the tissue morphology dependent refractive index fluctuations are recorded in the elastic scattering spectra [36], analysis of elastically scattered spectral fluctuations using sophisticated fluctuation analysis tools might facilitate extraction and quantification of subtle morphological changes associated with early stages of cancer. The scaling behavior which is generally assumed to be global (mono-fractal), has been shown to manifest in the local fluctuations in various physical processes [37, 38] and has been characterized using Multi-Fractal De-trended Fluctuation Analysis(for example see [39]). Wavelet Based Multi-Fractal Detrended Fluctuation Analysis (WB-MFDFA) is one other state-of-the-art technique that can be used for extracting and quantifying the self similarity at varying length scales associated with the structural changes associated with cancer progression due to the inherent use of fractal like transformation kernels.

Wavelet transform due to its multi-resolution analysis capability using the Daubechies’ basis which extract the polynomial trends (for example, Db-4 and Db-6 extract the linear and quadratic trends respectively) has been shown to characterize the scaling behavior and self-similarity of empirical data sets quite faithfully [40,41]. Indeed, it has been initially explored to analyze tissue fluorescence spectra in an attempt to distinguish between normal and dysplastic tissue [42–46]. In this work, we employ this multi-resolution property of wavelets to ascertain the changes in the self-similarity of dysplastic human cervical tissues as opposed to healthy human cervical tissues by analyzing the esoteric nature of the fluctuations in tissue light scattering spectra.

This article is organized as follows: Fourier and power spectrum analysis is reviewed in 2.1, discrete wavelet transform in 2.2, wavelet based power law analysis in 2.3, wavelet based multi-fractal de-trended fluctuation analysis in 2.4 and correlation based analysis in 2.5. Sec. 3 describes the experimental methods for light scattering measurements from tissues. Sec. 4 deals with our findings from the analysis and contains a discussion of the same in the context of the differences between the normal and dysplastic samples. In Sec. 5 we conclude with the prospect of pre-cancer detection using light scattering techniques combined with novel fluctuation analysis methods.

## 2. Theory

#### 2.1. Fourier analysis and power law spectrum

Fourier Analysis has traditionally been a preferred tool for analysis of experimental data sets. Here, we just briefly review the Discrete Fourier Transform (DFT) and its power spectrum. For a data set *x*(*n*),*n* = {1,2,...}, the DFT is a linear transformation over an orthogonal basis given by:

*α*is related to the Hurst scaling exponent

*H*∈ (0, 1), and the fractal dimension

*D*which identify the nature of fractal behavior by

_{f}*α*= 2

*H*+ 1 = 3 –

*D*[37]. Though this analysis assumes a global scaling behavior, due to the presence of inhomogeneities in the intra-cellular structure, the scaling behavior turns out to be localized in the spatial frequency domain. In multi-fractal analysis, a local Hurst exponent is calculated which quantifies the local singular behavior and provides useful and minute information which is hidden between different scales. We therefore applied a more general wavelet based approach for the multi-fractal analysis.

_{f}#### 2.2. A brief review of wavelet transforms

Wavelet Transforms [47–50], in both the forms of Discrete and Continuous transforms in the recent years have emerged as an invaluable tool in the field of data analysis and interpretation. Here, we will briefly review the Discrete Wavelet Transform (DWT).

In DWT, two functions, namely, *ϕ*(*n*) and *ψ*(*n*), called the father and the mother wavelets respectively, form the kernels. They satisfy the admissibility conditions: ∫ *ϕ*(*n*)*dn* < ∞, ∫ *ψ*(*n*)*dn* = 0, ∫ *ϕ*
^{*}(*n*)*ψ*(*n*)*dn* = 0, ∫ |*ϕ*(*n*)|^{2}
*dn* = ∫ |*ψ*(*n*)|^{2}
*dn* = 1. The scaled and translated versions of the mother wavelet *ψ*(*n*) are called the daughter wavelets

*s*scale, the height and width of the daughter wavelet are 2

^{th}*and 2*

^{s}^{s/2}of that of the mother wavelet respectively.

*s*and

*m*are the scaling and translation parameters.

The DWT for a function *f*(*t*) is given by,

*a*(called the approximation coefficient) extracts the trend and

_{m}*d*

_{s,m}(called the detail coefficient) extracts the details or fluctuations from the signal. Here,

*l*= ⌊log(

*N*)/log(2)⌋ is the upper bound for taking the maximum scale for analysis above which the edge effects corrupt the wavelet coefficients giving rise to spurious results. This mathematical artifact is explained by the cone of influence [49].

#### 2.3. Wavelet Fluctuation Power Law Analysis

The Daubechies’ family of wavelets are made to satisfy vanishing moments conditions, and hence are able to isolate polynomial trends from a time series (see [47]). The different wavelets in this family isolate trends of different polynomial orders, for example, *Db* – 4 isolates a monomial trend while *Db* – 6 isolates a quadratic trend. We first obtain the profile from the fluctuations by taking their cumulative sum:
$Y(n)={\Sigma}_{m=1}^{n}X(m),n\in \{1,2,\dots ,N-1\}$, where, *Y* and *X* are profile and fluctuation signal respectively and *N* is the data length. In this method, we first obtain the fluctuations at every scale by a wavelet reconstruction taking the approximation coefficients (using Db-4 wavelet) and subsequent subtraction from the signal. A flowchart for this fluctuation extraction is shown in Fig. 1. We then fit the Fourier power spectrum of these fluctuations to obtain the power law exponent as a function of scale, i.e. *α* ≡ *α*(*s*) for the identification of the short and long term correlations. This technique sheds light on the scaling behavior of the fluctuations in different spatial frequency regimes.

#### 2.4. Wavelet Based Multi Fractal De-trended Fluctuation Analysis

The Multi Fractal De-trended Fluctuation Analysis (WB-MFDFA) algorithm proposed by Manimaran *et al.* [40,41] has been employed gainfully to extract the multi-fractal nature of a variety of physical systems. In order to apply the WB-MFDFA algorithm, we then obtain the fluctuations using the extraction algorithm shown in Fig. 1. The asymmetric nature of the wavelet function and the edge-effects (due to the cone of influence) encountered during the convolution can affect the precision of the fluctuations. Hence, we repeat this procedure on a reversed profile and then take the average to get the fluctuations at every scale which are denoted by *F _{s̃}*. We used

*s̃*to represent the wavelet scale so as not to confuse with the segment length

*s*which is related to the wavelet scale

*s̃*by the number of filter coefficients for a given wavelet. These fluctuations are then segmented into

*M*non-overlapping sections such that

_{s}*M*= ⌊

_{s}*N*/

*s*⌋, where,

*s*and

*N*are the window size and the length of the fluctuations respectively. Subsequently, we obtain the

*q*order fluctuation function

^{th}*F*(

_{q}*s*) by

*q*can and both positive and negative integers. The fluctuation function for self similar processes follows a scaling law, the scaling function given by

*F*(

_{q}*s*) ≈

*s*

^{h(q)}. We should note here that the smaller fluctuations in the light scattering spectrum will be influenced by the negative

*q*values, whereas the positive values of

*q*will impact the larger fluctuations. The scaling function

*h*(

*q*) calculated at

*q*= 2 corresponds to the Hurst exponent [37].

*H*= 0.5 represents uncorrelated (white noise,

*f*

^{0}) or brown noise

*f*

^{−2}, while

*H*> 0.5 represents long range correlations or persistent behavior.

*H*< 0.5 reveals short range correlations or anti-persistent behavior. Mono-fractals are scale independent and hence their

*h*(

*q*) is independent of

*q*. They can be characterized by a single parameter like the fractal dimension. However, for multi-fractals, the

*h*(

*q*) is not independent of

*q*and they require a more complex function like the singularity spectrum for its characterization [38].

The classical multi-fractal scaling exponent *τ*(*q*) defined by standard partition function based formalism [52,53] is related to the Hurst exponent h(q) by *τ*(*q*) = *qh*(*q*) – 1. The multi-fractality can also be characterized by using the singularity spectrum *f*(*β*), which is related to *τ*(*q*) through a Legendre transform: *β* = *d*/*dq*[*τ*(*q*)] and *f*(*β*) = *qβ* – *τ*(*q*); where *β* is the singularity strength or Hölder Exponent [39]. *f*(*β*) denotes the dimension of the subset of the series to be characterized. *β*, *f*(*β*) and *h*(*q*) are related by

*β*will have a constant value for mono-fractal series, whereas for multi-fractal series

*β*values will have a distribution. Just like in the case of

*h*(

*q*), a constant

*τ*(

*q*) will signify a mono-fractal, while a multi-fractal will depend on the order of the moments. The singularity spectrum is broader when correlations are present in the series and if the correlations are absent or lost, then the singularity spectrum becomes narrower.

#### 2.5. Correlation based analysis

The correlation matrix analysis method has been extensively used to study various physical and biological systems (for example see [54]). For two variables *x _{i}* ∈

*X*and

*y*∈

_{j}*Y*, the correlation matrix is given by:

## 3. Experimental materials and methods

Pathologically graded (CIN or dysplastic) biopsy samples of human cervical tissues cut into slices were used for light scattering measurements and analysis. The tissue sections (thickness ∼ 5*μm*, lateral dimension ∼ 4*mm* ×6*mm*) were prepared on glass slides. The standard method employed for the preparation of the sections is tissue dehydration, embedding in wax and subsequent sectioning under a rotary microtome [55]. The prepared slides were then examined under the microscope by the pathologists for histopathological characterization. The normal counterparts were obtained from the adjacent normal area of the resected tissues. The tissue samples were obtained from G.S.V.M. Medical College and Hospital, Kanpur, India. A total number of fifteen tissue samples were collected from patients in the age group 35 – 60 years. Out of these, eight samples were included in the spectroscopic study. Among these, four were histopathologically characterized as dysplastic and the other four samples were pathologically normal. All the cervical tissues included in this study were squamous type.

A schematic representation of the experimental set-up is shown in Fig. 2. The angle resolved elastic scattering spectra were recorded using a goniometric arrangement for angular range *θ* : 10° – 150° at 10° intervals. White light output from a Xe-lamp (Newport USA, 50 – 5000 W) was collimated using a combination of lenses and were made incident on the sample kept at the center of the goniometer. The spot-size incident on the sample was controlled by a variable aperture (∼ 1*mm*). The scattered light from the sample was collimated by a pair of lenses and was then focused into a collecting fiber probe, the distal end of which was coupled to a spectrometer (Ocean Optics HR2000).

The resolution of the spectrometer was 1.8 nm (∼ 4 pixel resolution, spectral coverage 200 – 1100 nm, number of pixels −2048). Note that the analyzed spectral fluctuation window (in nm) was always larger than the resolution of the spectrometer. This follows because in our analysis, we have used *Db* – 4 wavelets and the minimum window size (corresponding to level 1) was 8, which in term of wavelength was ∼ 3.5*nm*. The spectroscopic data for the wavelength range 400 −800 nm was used in fluctuation analysis.

The selection of the angular interval (Δ*θ* = 10°) in our angle-resolved elastic scattering spectroscopic studies, was based on the range of scattering vector (
$q=2\times \frac{2\pi}{\lambda}\mathit{sin}\left(\frac{\theta}{2}\right)$) spanned by the varying wavelength and scattering angles (*λ _{min}* = 400 nm,

*λ*= 800 nm, Δ

_{max}*λ*= 400 nm;

*θ*= 10°,

_{min}*θ*= 150°; resulting in Δ

_{max}*q*∼ 29

*μm*

^{−1}). The idea was to cover the entire range of variation of Δ

*q*by varying scattering angle

*θ*. For the smallest scattering angle used in our study (

*θ*= 10°), the required angular interval Δ

_{min}*θ*was found to be ∼ 10°. Note that at larger scattering angles, one may not need to acquire spectra at angular interval of 10°, to capture the range of variation Δ

*q*. However, in order to be consistent, the angular interval of Δ

*θ*= 10° was used for the range of scattering angles studied. The scattering spectra recorded from the tissue samples at each scattering angles were normalized by the lamp spectra recorded from the glass slides alone. Since for fluctuation analysis spectral shape rather than the absolute intensity information is important, the angle-resolved scattering spectra were not normalized for absolute scattered intensities. Thus normalized angle-resolved scattering spectra were then subjected to the fluctuation analysis.

## 4. Results and discussions

Typical peak normalized elastic light scattering spectra recorded from normal and dysplastic tissues are shown in Fig. 3. Representative spectra for forward and back-scattering are shown for *θ* = 10° and *θ* = 150° in Figs. 3(a) and 3(b) respectively. The broader structures observed in the scattering spectra possibly originates due to contributions of the regular and large scale scattering inhomogeneities (such as epithelial cell nuclei) [15]. In contrast, the signatures of the small scale inhomogeneities (index variations in the sub-cellular micro-structures or intracellular organelles) manifest as much more complex and finer fluctuations in the scattering spectra and are thus difficult to identify. Never-the-less the apparent changes in the spectral shape between the forward and the backward scattering angles underscores the fact that the larger and the smaller sized scattering structures contribute differently to the scattering spectra recorded at the forward and the backward scattering angles. Indeed, it has been shown that the contribution of larger scatterers (like cells, nuclei) are more dominant in the spectra recorded at forward angles, whereas the spectra recorded at back-scattering angles are typically more influenced by the smaller sized scatterers [26].

Although it is difficult to make a one-to-one correspondence between the intensity fluctuations and local refractive index fluctuations in the morphological structures, some information about the morphology can still be inferred from the light scattering spectral fluctuations. It has been shown that with Born approximation, the light scattering spectrum can be represented as a Fourier transform power spectrum of the local refractive index fluctuations. Thus, the observed spectral fluctuations contain information about the corresponding refractive index fluctuations in the Fourier domain as discussed below.

The results of the Fourier analysis (following Sec. 2.1) on the light scattering spectra for different scattering angles are shown in Fig. 4. The variation of the determined scaling exponent *α*(*θ*) as a function of scattering angle *θ* for normal and dysplastic tissues are shown in this figure. The values are the average values over the sample size and the error bars represent standard deviations. Interesting differences can be noted in the values for *α* between the normal and the dysplastic tissues. While the value for *α* for the dysplastic tissues are higher at forward scattering angles, the reverse is the trend in backscattering angles. For example, in the forward scattering angles (10° – 70°), the dysplastic tissues show ∼ 1.186 ≤ *α _{c}* ≤ 1.322, in the same region however, the normal tissues show ∼ 1.106 ≤

*α*≤ 1.242. Similarly, in the backscattering angles (130° – 150°), the values for

_{n}*α*lie in the range ∼ 1.338 ≤

*α*≤ 1.397 for normal, and ∼ 1.119 ≤

_{n}*α*≤ 1.251 for dysplastic tissues. The higher values of the power-law coefficient

_{c}*α*corresponds to higher values of the Hurst scaling exponent

*H*, indicative of “coarseness” of the fluctuations tending towards sub-fractal behavior; whereas at higher angles (130° – 150°),

*α*values are observed to be smaller in dysplastic samples than that for normal samples in the same region indicating more “roughness” in the fluctuations, signifying a trend towards extreme fractality.

The higher *α* values in the angular range (10° ≤ *θ* ≤ 70°) for the dysplastic tissues are due to the more dominant contribution of the large scale scattering structures. This possibly can be related to the proliferative nuclear morphology in dysplasia [15]. This follows because such morphological changes are expected to lead to coarser fluctuations in light scattering spectra as is evident from the higher value of *α* in the dysplastic tissues in the forward scattering range. On the other hand, the lower values of *α* in the back-scattering angles (130° ≤ *θ* ≤ 150°) indicates the predominance of small scale heterogeneity in the dysplastic tissues. It is also pertinent to note that around 60°, the differences are reduced, possibly due to the simultaneous contribution of different sized scatterers in this angular range resulting in “lumped” effects.

Figure 5 shows the variation of the mean values of the power-law scaling exponent *α* as a function of the scale for normal (Fig. 5(a)) and dysplasia (Fig. 5(b)) for forward scattering angles 40° and 60° and for backward scattering angles 120° and 140°. Note that, for this
analysis we have not used *α*-values for lower scales (*s* = 1,2,3) in order to suitably extract
the power law behavior of the fluctuations. The results are thus shown for scales *s* = 4
to 9. Standard deviations for scale 6 (as representative of all scales) are shown by error-bars. This analysis was performed following the method discussed in Sec. 2.3. Apparent differences in the absolute values of *α* for Fourier (Fig. 4) and wavelet (Fig. 5) analysis arises from the fact that, in Fourier analysis, we analyze the signal itself while in wavelet analysis, we obtain the *α* values from the fluctuations which can be thought of as higher derivatives of the signal. We observe that the *α* values show a strong dependence on the scattering angle *θ* and the presence of a broad spectrum of processes such that 0.9 ≤ *α* ≤ 1.3. This arises possibly from the contribution of differently sized scatterers as has been mentioned earlier. Never-the-less, the observed trends in *α* from the wavelet analysis is qualitatively similar to that observed from the Fourier analysis. Here also, the values for *α* of the dysplastic tissues are higher at forward scattering angles, the reverse is the case for backscattering angles. For example, in the forward scattering region, at 40°, we observe that the normal tissues show an average *α _{n}* = 1.123 ± 0.052, while the dysplastic tissues show an average

*α*= 1.143 ± 0.104 over aforementioned scales. Similarly, in the backward scattering region, we obtain

_{c}*α*= 1.191 ± 0.034 and

_{n}*α*= 1.068 ± 0.028 for normal and dysplastic tissues respectively at 140°. In the other regions also, we found differences between normal and dysplastic samples in the

_{c}*α*values averaged over aforementioned scales.

As discussed in Sec. 2.4, we have depicted the mean values of the Hurst exponent (*H* = *h*(*q* = 2)) as a function of the scattering angle *θ* in Fig. 6; the error bars representing standard deviations. Once again the average values for H are found to be higher for dysplastic tissues in the forward scattering angles (0.214 ±0.003 ≤ *H _{c}* ≤ 0.332 ±0.009 and 0.187 ±0.038 ≤

*H*≤ 0.308 ± 0.062). In contrast, the values for

_{n}*H*of the normal tissues are higher in the backscattering angles (0.284 ±0.027 ≤

*H*≤ 0.312 ±0.003 and 0.251 ±0.003 ≤

_{n}*H*≤ 0.264 ±0.032). This is consistent with the results presented in Figs. 4 and 5.

_{c}The singularity spectrum *f*(*β*) is a quantitative indicator of the exact nature of the self-similarity and its width represents the strength of the multi-fractality. In Fig. 8, we have shown the *f*(*β*) as obtained from Eq. (7) derived following Sec. 2.4. The results are shown for a typical normal and dysplastic tissue sample. We observe that the dysplastic sample has a higher multi-fractality than the normal sample, indicated by the width of singularity spectrum. It must be noted that for a mono-fractal, the singularity spectrum is similar to a Gaussian with a very small variance. This is consistent with our observations of Fig. 7, where, the plots of *h*(*q*) vs *q* for the same (as in Fig. 8) normal and dysplastic tissues at a few representative forward and backward scattering angles are shown. For mono-fractals, the *h*(*q*) is independent of *q*. In Fig. 7, we observe that for the normal samples, at angles 50° – 70°, the dependence of *h*(*q*) on *q* is relatively weaker (Δ*h*(*q*) ∼ 0.376 ± 0.114), while in the same angular region (50° – 70°), dysplastic tissues show stronger dependence on *q* (Δ*h*(*q*) ∼ 0.546±0.074). The normal samples thus show a wide range of trends from mono-fractality to multi-fractality. However, dysplastic tissue show strong multi-fractality for all scattering angles. In order to quantify the strength of multi-fractality, we have also determined the width of the singularity spectrum (Δ*β*), defined as *β _{max}* –

*β*for all the samples. The mean and the standard deviations of Δ

_{min}*β*corresponding to scattering angles 50° – 70° (where maximum differences were observed) for the normal and dysplastic tissues were found to be 0.442 ±0.077 and 0.556 ±0.044 respectively.

The spectral correlation matrices are shown for typical normal and dysplastic tissues (same tissues, the results of which are used in Figs. 7 and 8) in Fig. 9, following Sec. 2.5. It is clear from Figs. 9(a) and 9(b) that though the normal tissues do not show any distinct correlation sectors in this domain, other than the expected correlation that would occur at neighboring wavelengths; dysplastic tissues show the presence of three dominant sectors. It must also be noted that the range of correlation increases from 0.70 – 1.00 for normal to 0.97 – 1.00 for dysplasia. This indicates a higher correlated behavior of the dysplastic tissues than the normal tissues in addition to domain formations in the spectral range. This possibly arises due to the fact that during dysplastic progression, the homogeneous cell morphology gives way to a more fragmented and heterogeneous structure.

## 5. Conclusion

In conclusion, we have applied a combined Fourier based and discrete wavelet based analysis on the fluctuations extracted from the elastic light scattering spectra of normal and dysplastic human cervical tissues. This approach clearly revealed otherwise hidden signatures of self-similarity in spectral fluctuation for both normal and dysplastic tissues, with interesting differences in the nature of self-similarity. Wavelet Based Multi-Fractal De-trended Fluctuation Analysis (WB-MFDFA) of these fluctuations indicated the existence of multi-fractal nature. Dysplastic tissues showed marginally higher multi-fractality compared to their normal counterparts. The scaling exponent was observed to have angular dependence, possibly arising from the size distribution of the scatterers present in such complex systems. Interestingly, while the value for the fractal scaling exponent *α* and the Hurst parameter *H* for the dysplastic tissues were higher at forward scattering angles (20° – 60°), the reverse was the trend in backscattering angles (120° – 150°). Initial results of this study thus show that the fractal scaling exponent (*α*), Hurst parameter (*H*) and the strength of multi-fractality of light scattering spectral fluctuations, derived and quantified via the multi-resolution fluctuation analysis, hold promise as potentially useful diagnostic parameters. We should however, note that the initial results presented in this manuscript are based on data acquired from a limited number of tissue samples. A more rigorous study on larger population of samples is currently underway to evaluate the diagnostic potential of this approach. Extraction and quantification of the multi-fractal nature of fluctuations in tissue using such multi-resolution analysis combined with appropriately designed light scattering measurement methods may ultimately lead to the development of non-invasive optical tools for pre-cancer detection.

## Acknowledgments

The authors would like to acknowledge Dr. Asha Agarwal, G. S. V. M. Medical College and Hospital, Kanpur, for providing the tissue slides and for fruitful discussions. SG and JS would like to thank P. Manimaran for helpful discussions. HP would like to thank IIT-K where a part of the work was performed. Corresponding author’s phone: +913473279130 (extn: 206).

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