# Estimation of the Hurst exponent of noisy or blurred fractal textures. Application to computer-aided mammogram analysis

Barbara Pascal, Claire Launay, Hermine Biermé

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Barbara Pascal, Claire Launay, Hermine Biermé. Estimation of the Hurst exponent of noisy or blurred fractal textures. Application to computer-aided mammogram analysis. 2025. hal-05271258

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# ESTIMATION OF THE HURST EXPONENT OF NOISY OR BLURRED FRACTAL TEXTURES. APPLICATION TO COMPUTER-AIDED MAMMOGRAM ANALYSIS.

Barbara Pascal<fnref n="1" />, Claire Launay<fnref n="2" />, Hermine Biermé<fnref n="3" />

## ABSTRACT

One over eight women will be diagnosed with breast cancer in her lifetime. As early detection is key to the survival of the patient, recent works aiming at improving the prognosis of breast cancer focused on upstream assessment of cancer risk, so that women at higher risk are prescribed more frequent screenings. A promising research avenue relies on the characterization of mammographic textures through their fractal attributes. However, the presence of noise or blur in mammograms, inherent to real data acquired in a clinical context, might significantly impairs both the accuracy and the robustness of fractal attribute estimators. The aim of the present work is twofold. First, the sensitivity to noise and blur of two state-of-the-art estimators of the Hurst exponent, a widely used fractal attribute, is quantified on different synthetic texture models. Second, these estimators, respectively based on wavelet leaders and monogenic wavelets, are applied to real mammograms extracted from the VinDr-Mammo public dataset. The comparison of the estimated Hurst exponents to the values in the literature shows an overestimation of the Hurst exponents of dense tissues from old datasets, suspected to originate from a model mismatch or from a lack of robustness to changes in the screened population.

**Index Terms—** Computer-aided mammogram analysis, Fractal textures, Hurst exponent, Wavelet leaders, Monogenic signal

## 1. INTRODUCTION

**Context.** Breast cancer is an important cause of mortality amongst women [1]. Detecting the tumor at its earliest stage has soon been identified as a key factor improving the survival of the patient [2]. Thus upstream breast cancer risk assessment is crucial and has triggered massive research efforts in the past decades [3–7]. A major bottleneck is the difficulty to interpret images acquired from X-ray examination, as the visual aspect of tissues similar in nature might differ significantly across the mammography [8]. This motivated the design of computer-aided tools [9]. To help radiologists to read biomedical images, fractal features have been widely used, not only for mammogram analysis [3, 6, 7, 10], but also far beyond, to detect bone osteoporosis [11] or probe white matter in the brain [12].

**Related works.** Biological tissue properties have been shown to correlate strongly with the Hurst exponent of the mammographic image, denoted by *H*, reflecting the regularity and scale-invariance properties of the observed textures [3, 6, 7]. Notably, the authors of the seminal work [8] showed that for fatty tissues the Hurst exponent was $H_f = 0.30 \pm 0.10$, while for healthy dense tissues, it was $H_d = 0.65 \pm 0.10$. This classification has then been enriched with another category corresponding to disrupted dense tissues in [6, 7], charac-

terized by $H_r = 0.50 \pm 0.05$, which are suspected to be at risk of letting cancerous lesions emerge and grow. These promising findings motivated a systematic analysis of the detectability of patches of disrupted tissues embedded in fatty or dense microenvironments [13]. Though, these values are not consistently retrieved in other studies, e.g., in [14] the Hurst exponent is estimated using generalized quadratic variations in dense tissue regions and found to belong to the interval $[0.18, 0.42]$, far smaller than $H_d = 0.65 \pm 0.10$, which severely departs from the estimates obtained using the Wavelet Modulus Maxima Method [6–8]. Although it has long been documented for radiographs that the presence of quantum noise or of modulation of the transfer function deteriorates the accuracy of fractal dimension estimators [15], the authors found no such systematic analysis for X-ray mammograms, at the notable exception of [10] documenting the influence of subsampling on the estimation of the fractal dimension, and [16] proposing a topology and Hurst functions estimator robust to additive white Gaussian noise. Interestingly, the estimation bias in [15] depends on the texture characteristics and on the estimator.

**Goals, contributions and outline.** The motivation of this work is to elucidate the discrepancy between the different estimations of Hurst exponents of mammographic textures reported in the literature. Three directions are investigated: the degradation of the textures during data acquisition, a mismatch between real data and mathematical texture models, a lack of robustness of the Hurst exponent estimators. To that aim, the effect of noise and blur on the estimation of the Hurst exponent of synthetic textures, mimicking real mammographic textures through three different fractal field models, is systematically explored. These investigations include the comparison of two state-of-the-art estimators, based on wavelet leaders and monogenic wavelets respectively. Intensive numerical simulations are conducted on simulated fractal fields and real mammographic textures extracted from the recent VinDr-Mammo⁶ public dataset [17] and compared. In the case of dense mammographic textures, a significant gap between the estimated Hurst exponent and the values reported in the literature, which cannot be explained neither by noise nor by blur in the data as it is not observed for fatty tissues, strongly suggests either a model mismatch or a poor robustness to change in the screened population. Solid theoretical ground for the observed behavior of the estimated Hurst exponent under additive Gaussian noise or convolutional blur at extreme degradation levels is provided, supporting the consistency of the results. The texture models compared are defined in Sec. 2 and their main statistical and geometrical properties are presented. Then, Sec. 3 provides a self-contained description of the two Hurst exponent estimators. Numerical experiments on both synthetic and real mammographic textures are described and their results are thoroughly discussed in Sec. 4. Conclusion and research perspectives are provided in Sec. 5.

<footnote n="1">¹ Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France [barbara.pascal@cnrs.fr](mailto:barbara.pascal@cnrs.fr)</footnote>

<footnote n="2">² Université Bretagne Sud, CNRS LMBA, UMR 6205, F-56000, Vannes, France [claire.launay@univ-ubs.fr](mailto:claire.launay@univ-ubs.fr)</footnote>

<footnote n="3">³ Université de Tours, Université d'Orléans, CNRS, IDP, UMR 7013, F-37200, Tours, France [hermine.bierme@univ-tours.fr](mailto:hermine.bierme@univ-tours.fr)</footnote>
## 2. ISOTROPIC TEXTURE MODELS

From a mathematical perspective, a gray-scale image is often represented as the restriction to a compact rectangular domain $\Omega \subseteq \mathbb{R}^2$ of a real-valued field $F: \mathbb{R}^2 \to \mathbb{R}$. In particular, textured images, which present structured random fluctuations of the light intensity, are well-described by stochastic fields. Notably, self-similar Gaussian fields and their extensions [18] have been widely used to model natural, e.g., biomedical [6, 8, 19], images. The visual aspect of random fields is strongly controlled by their correlation structure. Let $F$ be a zero-mean random field with finite second moments and for any $\delta \in \mathbb{R}^2$ define the field $F^\delta(\underline{x}) = F(\underline{x} + \delta) - F(\underline{x})$. Then if there exist a constant $c > 0$ and $H \in \mathbb{R}$ such that, in the large lag $\underline{h}$ limit:

$$ \mathbb{E} \left[ F^{\delta}(\underline{x} + \underline{h}) F^{\delta}(\underline{x}) \right] \underset{\|\underline{h}\| \to \infty}{=} c \|\underline{h}\|^{2H-2} + o \left( \|\underline{h}\|^{2H} \right) \quad (1) $$

the field $F$ has a Hurst exponent of $H$, which characterizes long-range statistical dependency of the increments of the field [20, 21].

**Fractional Brownian field.** Among the most common self-similar Gaussian fields, the fractional Brownian field $B_\alpha$ of parameter $\alpha \in (0, 1)$ [22] has the following harmonizable representation:

$$ B_{\alpha}(\underline{x}) = \int_{\mathbb{R}^2} \frac{e^{-i\underline{x}\cdot\underline{\omega}} - 1}{\|\underline{\omega}\|^{\alpha+1}} d\tilde{W}(\underline{\omega}) \quad (2) $$

where $\underline{x} \cdot \underline{\omega} = x_1\omega_1 + x_2\omega_2$ denotes the Euclidean product in $\mathbb{R}^2$, $\|\underline{\omega}\| = \sqrt{\omega_1^2 + \omega_2^2}$ the associated norm and $\tilde{W}$ the Fourier transform of a two-dimensional Wiener process. The covariance function of the field $B_\alpha$ writes for any $\underline{x}, \underline{y} \in \mathbb{R}^2$:

$$ \mathbb{E} [B_{\alpha}(\underline{x})B_{\alpha}(\underline{y})] = \frac{1}{2} (\|\underline{x}\|^{2\alpha} + \|\underline{y}\|^{2\alpha} - \|\underline{x} - \underline{y}\|^{2\alpha}). \quad (3) $$

It can be derived from (3) that $B_\alpha$ satisfies (1) with a Hurst exponent of $H = \alpha$. Two examples of fractional Brownian fields are provided in Figure 1a with respective parameters $\alpha_f = 0.3$ (top), yielding a rather irregular texture, and $\alpha_d = 0.65$ (bottom), corresponding to a smoother texture. Additionally, using Eq. (2), it can be shown that $F = B_\alpha$ satisfies the H-self-similarity property for $H = \alpha$:

$$ \forall c > 0, \quad \{F(c\underline{x}); \underline{x} \in \mathbb{R}^2\} \stackrel{(\text{law})}{=} c^H \{F(\underline{x}); \underline{x} \in \mathbb{R}^2\} \quad (4) $$

hence justifying the term fractal field to refer to the fractional Brownian field (2). As can be deduced from its covariance function (3), the fractional Brownian field is isotropic but not stationary, which impairs its ability to model real textures which are homogeneous [13, 23]. This motivated the design of new fractal texture models.

**Fractional Gaussian field.** Originally introduced in [23] to model images of a gas-liquid flow through a porous media and recently proved relevant for mammogram images [13], the fractional Gaussian field $G_\alpha$ combines horizontal and vertical increments of a fractional Brownian field $B_\alpha$ to construct a stationary, asymptotically isotropic and H-self-similar [18, Def. 4.2.3] Gaussian random field:

$$ G_{\alpha}(\underline{x}) = \frac{1}{2} (B_{\alpha}(\underline{x} + \underline{e}_1) + B_{\alpha}(\underline{x} + \underline{e}_2) - 2B_{\alpha}(\underline{x})) \quad (5) $$

where $\underline{e}_1 = (1, 0)$ and $\underline{e}_2 = (0, 1)$. The expression of the covariance of $G_\alpha$ provided in [23, Prop. 4] shows that the fractional Gaussian field has a Hurst exponent of $H = \alpha - 1$. Examples of fractional Gaussian fields are displayed in Fig. 1b, with Hurst exponent $\alpha_f = 0.3$ (top) and $\alpha_d = 0.65$ (bottom). Inheriting from the fractional Brownian field, the smaller $\alpha$, the more irregular the texture.

**Filtered fractional Brownian field.** Inspired from filters used in monogenic wavelet analysis in [24], an exactly isotropic, asymptotically H-self-similar Gaussian field has been introduced in [13]. This

<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Three models of synthetic textures. Top: irregular, fatty tissue-like, random fields, of parameter $\alpha_f = 0.3$. Bottom: smoother, dense tissue-like, random fields of parameter $\alpha_d = 0.65$.</figcaption></figure>

so-called filtered fractional Brownian field is defined as

$$ C_{\alpha}(\underline{x}) = \langle B_{\alpha}, u_{\underline{x}} \rangle, \quad \langle \cdot, \cdot \rangle \text{ the scalar product in } L^2(\mathbb{R}^2), \quad (6) $$

and consists in an undecimated isotropic filtering of the fractional Brownian field (2) of parameter $\alpha$ with a smooth radial function $u$ which satisfies $\int_{\mathbb{R}^2} u(\underline{x})d\underline{x} = 0$. To ensure asymptotic H-self-similarity [18, Def. 4.2.3] and the desired stationarity of the random field $C_\alpha$, it has been proposed in [13] to choose $u$ as the high-pass filter proposed in [24, Sec. 4.2] at scale $2^3$. The filtered fractional Brownian field can be viewed as an isotropic generalization of the fractional Gaussian field. As is, it has a Hurst exponent of $H = \alpha - 1$ [13]. Examples of filtered fractional Brownian fields are displayed in Fig. 1c, with parameter $\alpha_f = 0.3$ (top) and $\alpha_d = 0.65$ (bottom). Again, the smaller $\alpha$, the more irregular the texture. It is worth noting that the visual aspects of the filtered fractional Brownian field, Fig. 1c, and the fractional Gaussian field, Fig. 1b differs significantly, although they have the same Hurst exponent. This shows their complementarity in modeling real fractal textures.

## 3. ESTIMATION OF THE HURST EXPONENT

Numerous estimators of the Hurst exponent of rough signals and textures have been proposed [25, 26]. A first class of popular methods relies on quadratic variations and their generalizations [27, 28]. The present work focuses on a second category of estimators which resort to multiscale analysis techniques, leveraging the tight connection between the Hurst exponent of a texture and its self-similarity properties. These estimators benefit from both a small estimation variance and a low computational load [29], two crucial properties to accurately estimate Hurst exponent of large-size images in biomedical applications [30]. Two estimators, introduced in [30] and [24] and described in a self-contained manner below, are compared, based on wavelet leaders [31] and on monogenic wavelets [32] respectively.

**Wavelet leader formalism.** From a scaling function $\phi$ and a mother wavelet $\psi$, with $N_\psi \ge 1$ vanishing moments, one can define a proper one-dimensional multiscale analysis [33] which extends to a two-dimensional analysis by defining the two-dimensional scaling function $\psi^{(0)}$ and the three associated directional wavelets $\psi^{(m)}$ as

$$ \left\{ \begin{array}{ll} \psi^{(0)}(\underline{x}) = \phi(x_1)\phi(x_2), & \psi^{(1)}(\underline{x}) = \psi(x_1)\phi(x_2), \\ \psi^{(2)}(\underline{x}) = \phi(x_1)\psi(x_2), & \psi^{(3)}(\underline{x}) = \psi(x_1)\psi(x_2), \end{array} \right. \quad (7) $$

where $\underline{x} = (x_1, x_2) \in \mathbb{R}^2$ and $m \in \{1, 2, 3\}$. Then, the wavelet coefficients of a, possibly random, field $F: \Omega \subseteq \mathbb{R}^2 \to \mathbb{R}$ write $Y_F^{(m)}(j, \underline{k}) = 2^{-j} \langle F, \psi_{j, \underline{k}}^{(m)} \rangle$ where $\langle \cdot, \cdot \rangle$ denotes the scalar product in $L^2(\mathbb{R}^2)$, and $\psi_{j, \underline{k}}^{(m)}$ corresponds to the m wavelet $\psi^{(m)}$ dilated to scale $a = 2^j$ and translated to location $\underline{x} = 2^j \underline{k}$, with $\underline{k}$ belonging to a dyadic lattice. The factor $2^{-j}$ corresponds to the choice of an $L^1$-normalization of the wavelet transform. Assuming furthermore
<figure><img src="image_3.png" /><figcaption>**Fig. 2:** Noisy filtered fractional Brownian fields (6): additive Gaussian noise of variance $\sigma^2$, corresponding to (12) with $\delta = 1$ pixel. Top row: fatty tissue-like texture of parameter $\alpha_f = 0.3$. Bottom row: dense tissue-like texture with $\alpha_d = 0.65$. From left to right: the increasing noise level leads to a decrease of regularity.</figcaption></figure>

that the support of the wavelet $\psi$ is compact, one can define the wavelet leader coefficients $\mathcal{L}_{j,\underline{k}}$, at scale $2^j$ and location $\underline{x} = 2^j \underline{k}$ as the local supremum of modulus of wavelet coefficients in a small neighborhood across all finer scales $j' \le j$ [30, 34, 35]

$$ \mathcal{L}_{j, \underline{k}} = \sup_{\substack{\lambda_{j', \underline{k}'} \subset 3\lambda_{j, \underline{k}} \\ m \in \{1, 2, 3\}}} |2^j Y_F^{(m)}(j', \underline{k}')|, \quad (8) $$

with $\lambda_{j,\underline{k}} = [\underline{k}2^j, (\underline{k}+1)2^j)$ and $3\lambda_{j,\underline{k}} = \bigcup_{\underline{p} \in \{-1,0,1\}^2} \lambda_{j,\underline{k}+\underline{p}}$. For the three Gaussian random fields presented in Sec. 2, it can be shown

that, at small scales, the wavelet leaders exhibit a power low behavior fully characterized by the Hurst exponent $H^1$ of the analyzed field: $\mathcal{L}_{j,\underline{k}} \simeq \eta(\underline{x})2^j H$ as $2^j \to 0$, where $\eta(\underline{x}) > 0$ does not depend on $j$. To get a robust estimator of the Hurst exponent, the structure functions have been defined in [30, 31] as the spatial averages of the $q$-th order of the leaders at each scale $2^j$, $S_{j,q} = K_j^{-1} \sum_{\underline{k} \in \underline{Z}} 2^j \underline{k} \in \underline{\Omega} (\mathcal{L}_{j,\underline{k}})^q$ where $K_j$ denotes the number of points of the $2^j$-dyadic lattice falling in the image domain $\Omega$. Inheriting from the scaling behavior of the leaders, the structure function of order $q$ behaves at smaller scales as $S_{j,q} \propto 2^{jqH}$. Then, the Hurst exponent can be estimated through a linear regression on the logarithm of the structure function with respect to the octave $j$, leading to the family of unbiased wavelet leaders-based Hurst estimators

$$ \hat{H}_{q}^{\text {lead }}=\frac{\nu_{1} \sum_{j=j_{1}}^{j_{2}} \log S_{j,q}-\nu_{q,0} \sum_{j_{1} j=2}^{j_{2}} j q \log S_{j,q}}{q\left(\nu_{q,1}^{2}-\nu_{q,2} \nu_{q,0}\right)} \quad (9) $$

where $\nu_{q,p} = \sum_{j=j_1}^{j_2} (jq)^p$ for $p \in \{0, 1, 2\}$.

**Multiscale monogenic representation.** Leveraging the monogenic representation of images which has been shown to capture very efficiently local geometric structure [36], a multiscale monogenic wavelet representation has been proposed [37–39]. This representation relies on the Riesz transform [40] of the field $F$, $\mathcal{R}F = (\mathcal{R}_1 F, \mathcal{R}_2 F)$, defined in the Fourier domain as $\mathcal{F}(\mathcal{R}_1 F)(\underline{\xi}) = -i \frac{\xi_1}{\|\underline{\xi}\|} \mathcal{F}(F)(\underline{\xi})$ and $\mathcal{F}(\mathcal{R}_2 F)(\underline{\xi}) = -i \frac{\xi_2}{\|\underline{\xi}\|} \mathcal{F}(F)(\underline{\xi})$, $\forall \underline{\xi} = (\xi_1, \xi_2) \in \mathbb{R}^2$, with $\mathcal{F}$ the Fourier transform. As proposed in [24, 32], the multiscale decomposition of the field $F$ is given by the undecimated filtering of the field by a radial function $u$. At octave 0, $u$ is the high-pass filter defined in [24, Sec. 4.2], leveraged in Sec. 2 to define filtered fractional Gaussian fields, and for $j \ge 1$, $u_{j,\underline{n}}(\underline{x}) = 2^{-j} u(2^{-j}(\underline{x} - \underline{n}))$. At octave $j$, the monogenic signal of a random field $F$ through the radial function $u_{j,\underline{n}}$ is the following vectorial random field with values in $\mathbb{R}^3$:

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Blurry filtered fractional Brownian fields (6): convolution with a Gaussian kernel of length $\delta$, corresponding to (12) with $\sigma^2 = 0$. Top row: fatty tissue-like texture of parameter $\alpha_f = 0.3$. Bottom row: dense tissue-like texture with $\alpha_d = 0.65$. From left to right: the increasing blur length leads to an increase of regularity.</figcaption></figure>

$\mathcal{M}F(u_{j,\underline{n}}) = (\langle F, u_{j,\underline{n}} \rangle, \langle \mathcal{R}_1 F, u_{j,\underline{n}} \rangle, \langle \mathcal{R}_2 F, u_{j,\underline{n}} \rangle)$. If $F$ is $H$-self-similar, the squared amplitude of the associated monogenic signal $A_{j,\underline{n}}^2 = \|\mathcal{M}F(u_{j,\underline{n}})\|^2$ verifies, for two octaves $j_1 \le j_2$, $\frac{\mathbb{E}(A_{j_2,\underline{n}}^2)}{\mathbb{E}(A_{j_1,\underline{n}}^2)} = 2^{2(j_2-j_1)(H+1)}$. Moreover, at a given octave $j$, an unbiased consistent estimator of the squared amplitude is given by

$$ \hat{A}_{j}^{2}=N^{-1} \sum_{\underline{n} \in \Omega \cap \underline{Z}^{2}} \|\mathcal{M} F\left(u_{j, \underline{n}}\right)\|^{2}, \quad (10) $$

where $N$ is the number of pixels in the image domain, and $\mathcal{M}F(u_{j,\underline{n}})$ is the monogenic signal computed at octave $j$ and at pixel $\underline{n}$. The resulting monogenic wavelet-based Hurst exponent estimator writes

$$ \hat{H}^{\text {monog }}=\frac{1}{2 \log (2)\left(j_{2}-j_{1}\right)} \log \left(\frac{\hat{A}_{j_{2}}^{2}}{\hat{A}_{j_{1}}^{2}}\right)-1. \quad (11) $$

## 4. NUMERICAL EXPERIMENTS

**Algorithmic setup.** The generation of the synthetic textures corresponding to the three models described in Sec. 2 is performed using the turning-band method [41]. Both synthetic and real textures are normalized to be of zero mean and unit variance. For the wavelet leader-based estimator (9), the Daubechies wavelets with a number of vanishing moments $n_\psi = 2$ are used and the structure function of order $q = 2$ is computed from the WLBMF toolbox.<fnref n="2" /> Furthermore, a fractional integration of parameter $\gamma_{\text{int}} = 2$ is made to estimate not only positive but also negative Hurst exponents $H \ge -2$. Finally, the linear regression of (9) is performed between octaves $j_1 = 2$, discarding the first scale which is biased [30, 31], and $j_2 = 5$. The monogenic wavelet-based estimator is computed between octaves $j_1 = 3$ and $j_2 = 4$ using the image monogenic analysis toolbox<fnref n="3" />.

**Robustness of Hurst exponent estimators.** To compare the accuracy and robustness of the Hurst exponent estimators presented in Sec. 3, synthetic textures of known Hurst exponents are generated with the models presented in Sec. 2. These textures are first corrupted by additive Gaussian noise with variance increasing from $\sigma^2 = 10^{-4}$ to $10^2$. Then, they are blurred by convolution with Gaussian kernels of lengths ranging from $\delta = 10^{-3}$ to $10^4$. In practice, at each pixel $\underline{n}$ of the image domain, the degraded texture $\tilde{F}$ satisfies

$$ \tilde{F}(\underline{n})=\left(g_{\delta} \star F\right)(\underline{n})+\sigma \xi_{\underline{n}}, \quad (12) $$

where $F$, the original texture, can be either $B_\alpha(2)$, $G_\alpha(5)$ or $C_\alpha(6)$ and $(\xi_\alpha)_{\underline{n} \in \Omega \cap \underline{Z}^2}$ consists in independent identically distributed zero-

<footnote n="1">¹From Sec. 2, the Hurst exponent equals $H = \alpha$ for the stationary fractional Brownian field (2) and $H = \alpha - 1$ for the fractional Gaussian (1b) and filtered fractional Brownian (6) fields.</footnote>

<footnote n="2">²Wavelet Leader and Bootstrap based MultiFractal analysis toolbox</footnote>

<footnote n="3">³Image Monogenic Analysis toolbox</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 4:** Hurst exponent of degraded synthetic textures. Mean estimated Hurst exponents, from either the wavelet leader-based estimator (9) (circle marks, dashed lines) or the monogenic wavelet-based estimator (11) (triangle marks, solid lines), over 20 realizations of the synthetic random fields (2), (5) and (6) are reported, together with 95% Gaussian confidence intervals. *Top:* Textures with additive Gaussian noise of varied variance $\sigma^2$, (12) with $\delta = 0$. *Bottom:* Blurred textures with varied blur length $\delta$, (12) with $\sigma^2 = 0$.</figcaption></figure>

mean unit-variance Gaussian variables.<fnref n="4" /> For each texture model, two parameters are explored, $\alpha_f = 0.3$ and $\alpha_d = 0.65$, to mimic fatty and dense tissues respectively. Fig. 4 reports the Hurst exponents estimated with either the wavelet leader-based estimator (9) (circle marks) or with the monogenic wavelet estimator (11) (triangle marks), when the noise (top plots) or the blur (bottom plots) are varied, for each texture model and type of imitated tissues, averaged over 20 texture realizations together with 95% Gaussian confidence intervals. Theoretical Hurst exponent values are indicated as dashed gray lines. First of all, the two estimators systematically agree very precisely, for all texture models and regularities, noise levels and blur lengths. At small noise (resp. blur) level, they both estimate $H$ correctly and with negligible fluctuations across texture realizations. Then, as the noise level increases, the textures gets more irregular and, as expected, their Hurst exponent decreases until it reaches $H_{wn} = -1$ corresponding the Hurst exponent of the two-dimensional white noise [42]. On the contrary, as the blur length increases the textures get smoother, and their estimated Hurst exponent are larger. One can note the unanimous limit of $H_{smooth} = 2$ reached at very large blur lengths, which is fully determined by the number of vanishing moments of the analyzing wavelet $n_\psi$ capping the estimated Hurst exponent [33, Sec. 6.1.3].<fnref n="5" /> Tab. 1 illustrates the good coincidence of $H_{smooth}$ and $n_\psi$ for the three different texture models presented in Sec. 2 with parameter $\alpha_f = 0.65$ corrupted by a Gaussian blur of typical length $\delta = 10^4$. Overall, for both estimators, the presence of noise (resp. blur) in the analyzed images induces a systematic underestimation (resp. overestimation) of their Hurst exponent, the bias increasing with the degradation level.

**Real mammograms analysis.** The effect of noise and blur on the estimation of the Hurst exponent of real mammographic textures is illustrated on $P = 20$ patches of 512 × 512 pixels extracted from $P = 20$ different BI-RADS A (resp. D), fatty (resp. dense), mammograms reported in the VinDr-Mammo dataset [17].<fnref n="6" /> The Hurst exponents estimated with the monogenic wavelet-based estimator (11), averaged across the $P$ patches, are compared to those estimated on the three synthetic texture models, averaged over 20 realizations, in Fig. 5. At extreme noise (resp. blur) levels, the real mammographic texture have similar Hurst exponents of $H_{wn} = -1$ (resp. $H_{smooth} = 2$) as the synthetic textures for both breast densities. For

<figure><img src="image_6.png" /><figcaption>(a) Fatty BI-RADS A mammograms (b) Dense BI-RADS D mammograms</figcaption></figure>

<table><caption>**Fig. 5:** Effect of noise and blur on the estimation of the Hurst exponent of synthetic fractal vs. real mammographic textures. Mean monogenic wavelet-based estimate of the Hurst exponent over 20 realizations of the synthetic random fields (2), (5) and (6) and $P = 20$ mammographic patches from VinDr-Mammo dataset<fnref n="6" /> are reported, together with 95% Gaussian confidence intervals. For each tissue density, fatty (a) and dense (b), *Top:* additive Gaussian noise of varied variance $\sigma^2$. *Bottom:* blurred with varied blur length $\delta$.</caption><thead><tr><th>n<sub>ψ</sub></th><th>B<sub>α</sub></th><th>G<sub>α</sub></th><th>C<sub>α</sub></th></tr></thead><tbody><tr><td>H<sup>lead</sup> (9)</td><td>2.09 ± 0.01</td><td>2.10 ± 0.02</td><td>1.83 ± 0.02</td></tr></tbody></table>

<caption>**Table 1:** Extreme blur limit matching $H_{smooth} = n_\psi$. Estimated Hurst exponent of a fractal fields of parameter $\alpha_f = 0.65$ degraded with a Gaussian blur of typical length $\delta = 10^4$ in (12).</caption>

small and intermediate degradation levels, the Hurst exponent of real fatty mammographic textures, Fig. 5a, behaves similarly as for the fractional Brownian field with Hurst exponent $H = \alpha_f$, which confirms the results reported in the literature [6–8]. However, for dense mammograms in Fig 5b, the Hurst exponents estimated on real X-ray images departs from the Hurst exponents measured on all the synthetic texture models of parameter $\alpha_d$ presented in Sec. 2. The estimated Hurst exponents of degraded real textures are consistently smaller than those of the fractional Brownian field. More importantly, the estimated Hurst exponent is the same for fatty, Fig. 5a, and dense, Fig 5b, mammographic tissues, contradicting the claim that the Hurst exponent is a universal signature of tissue density [6–8].

## 5. CONCLUSIONS AND PERSPECTIVES

The present work quantified on synthetic textures how the presence of noise (resp. blur) significantly impairs the estimation of their original Hurst exponent by causing a dramatic underestimation (resp. overestimated) of the texture regularity. Experiments on the effect of noise an blur on real mammographic textures showed a mismatch between the Hurst exponents estimated on mammograms from the recent VinDr-Mammo dataset and the values reported in the literature [6–8]. This discrepancy concerning only dense tissues but not fatty tissues, these results seriously question the universality of isotropic fractal field models for describing mammographic textures and emphasize the need for an enrichment of the mathematical models of biological textures, accounting for example for anisotropy [14, 24, 43] and to explore the robustness of the Hurst exponent-based procedures to change in the screened population. The mammograms of both the recent VinDr-Mammo and the old DDSM<fnref n="7" /> databases will thus be analyzed with the recent promising tools suited to the fine characterization of anisotropic textures developed in [24]. Going further, segmenting mammograms based on both their Hurst exponent and anisotropy properties, e.g., following [43], could greatly enhance computer-aided risk assessment tools.

<footnote n="4">⁴Note that the effects of noise or blur are studied *independently*, so that in (12), either $\sigma^2 > 0$ and then $\delta = 0$, or $\delta > 0$ and then $\sigma^2 = 0$.</footnote>

<footnote n="5">⁵The choice of the function $u$ from [24, Sec. 4.2] makes (10) akin to a multiscale analysis with a $n_\psi = 2$-vanishing moments Gaussian wavelet.</footnote>

<footnote n="6">⁶www.physionet.org/content/vindr-mammo/1.0.0/</footnote>

<footnote n="7">⁷www.cancerimagingarchive.net/collection/cbis-ddsm/</footnote>
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