Rachid Harba

Fractal Dimension of Trabecular Bone Projection Texture Is Related to Three-Dimensional Microarchitecture

The purpose of this work was to understand how fractal dimension of two‐dimensional (2D) trabecular bone projection images could be related to three‐dimensional (3D) trabecular bone properties such as porosity or connectivity. Two alteration processes were applied to trabecular bone images obtained by magnetic resonance imaging: a trabeculae dilation process and a trabeculae removal process. The trabeculae dilation process was applied from the 3D skeleton graph to the 3D initial structure with constant connectivity. The trabeculae removal process was applied from the initial structure to an altered structure having 99% of porosity, in which both porosity and connectivity were modified during this second process. Gray‐level projection images of each of the altered structures were simply obtained by summation of voxels, and fractal dimension (Df) was calculated. Porosity (φ) and connectivity per unit volume (Cv) were calculated from the 3D structure. Significant relationships were found between Df, φ, and Cv. Df values increased when porosity increased (dilation and removal processes) and when connectivity decreased (only removal process). These variations were in accordance with all previous clinical studies, suggesting that fractal evaluation of trabecular bone projection has real meaning in terms of porosity and connectivity of the 3D architecture. Furthermore, there was a statistically significant linear dependence between Df and Cv when φ remained constant. Porosity is directly related to bone mineral density and fractal dimension can be easily evaluated in clinical routine. These two parameters could be associated to evaluate the connectivity of the structure.

Fractal analysis of trabecular bone texture on radiographs: discriminant value in postmenopausal osteoporosis.

Abstract: Trabecular bone microarchitecture cannot be routinely evaluated. We have developed and validated a fractal analysis of trabecular bone texture on calcaneus radiographs. The aim of this work was to evaluate the ability of the fractal analysis to discriminate a group of 39 postmenopausal women with osteoporotic (OP) vertebral crush fractures (68.0 + 10.8 years) from an age-matched control group of 39 women (68.0 + 10.7 years). The value of the fractal analysis was compared with the value of the femoral neck bone mineral density (FNBMD) and trochanteric bone mineral density (TRBMD). The result is expressed by the parameter Hmean (Hmean= 2 7 fractal dimension). Hmean value was 0.691 + 0.050 in the OP group versus 0.739 + 0.024 in the controls, while FNBMD was 0.598 + 0.113 g/cm2 versus 0.645 + 0.109 g/cm2 and TRBMD was 0.512 + 0.108 g/cm2 versus 0.594 + 0.106 g/cm2 respectively. The statistical significance of the Hmean test (p50.0001) was higher than for FNBMD (p50.05) and for TRBMD (p= 0.0004). We used a receiver operating characteristic (ROC) curve to check this superiority. The area under the ROC curve was 0.824 for Hmean, 0.633 for FNBMD and 0.727 for TRBMD. This superiority of the Hmean ROC curve was statistically significant versus FNBMD, but not versus TRBMD. In a second analysis, we studied the subgroups of OP patients and controls with overlapping FNBMD or TRBMD values to check whether the fractal dimension test could be discriminant in these subgroups. Significant statistical differences were found for Hmean between OP patients and controls in the overlapping subgroup for FNBMD or TRBMD (respectively p= 0.006 and p50.02). These data confirm that the fractal analysis of texture on calcaneus radiographs is able to discriminate OP patients with vertebral crush fracture from controls. This discrimination was stronger than that obtained by FNBMD or TRBMD alone. It was also present when we compared subgroups with overlapping values of FNBMD or TRBMD.

A New Adaptive Switching Median Filter

A new Adaptive Switching Median (ASWM) filter for removing impulse noise from corrupted images is presented. The originality of ASWM is that no a priori Threshold is needed as in the case of a classical Switching Median filter. Instead, Threshold is computed locally from image pixels intensity values in a sliding window. Results show that ASWM provides better performance in terms of PSNR and MAE than many other median filter variants for random-valued impulse noise. In addition it can preserve more image details in a high noise environment.

nth-order fractional Brownian motion and fractional Gaussian noises

A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n-1, n[, where n is any strictly positive integer. They include fBm for the special case n=1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as 1/f/sup /spl alpha// with /spl alpha/ belonging to ]1; +/spl infin/[, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. The covariance function and power spectral densities are calculated. The properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. The results show that the estimator is efficient (unbiased and reaches the Cramer-Rao lower bound) for a large majority of tested values.

Anisotropy changes in post-menopausal osteoporosis: characterization by a new index applied to trabecular bone radiographic images

Bone intrinsic strength is conditioned by several factors, including material property and trabecular micro-architecture. Bone mineral density (BMD) is a good surrogate for material property. Architectural anisotropy is of special interest in mechanics-architecture relations and characterizes the degree of directional organization of a material. We have developed anisotropy indices from the Fast Fourier Transform (FFT) on bone radiographs. We have validated these indices in a cross-sectional uni-center case-control study including 39 postmenopausal women with vertebral fracture and 70 age-matched control cases. BMD was measured at the lumbar spine and femoral neck. A fractal analysis of texture was performed on calcaneus radiographs at three regions of interest (ROIs), and the result was expressed as the H parameter (fractal dimension =H-2). The anisotropy evaluation was based on the FFT spectrum of these three ROIs extracted on calcaneus radiographs. On the FFT spectrum, we have measured the spreading angle of the longitudinal trabeculae called the dispersion longitudinal index (DLI) and the spreading angle of the transversal trabeculae called the dispersion transversal index (DTI). From the measured parameters, an anisotropy index was derived, and the degree of anisotropy (DA) calculated with DLI and DTI. We have compared the results from the vertebral fracture cases and control cases. The best distinction was obtained for the largest ROI located in the great tuberosity of the calcaneus for all parameters ( P <10-4).The DA parameter showed a higher value in vertebral fracture cases (1.746±0.169) than in control cases (1.548±0.136); P <10-4, and the difference persisted after removal of the subjects with hormonal replacement therapy. The analysis of the receiver operating characteristics (ROC) has shown the best results with DA and Hmean: areas under curves (AUCs) respectively of 0.765 and 0.683, while AUCs associated to LS-BMD and FN-BMD were 0.614 and 0.591 lower, respectively. We determined the odds ratios (OR) by uni- and multivariate analysis. Crude ORs were respectively 3.91 (95% CI: 2.22–6.87) and 3.08 (95% CI: 1.72–5.52) for DA and Hmean. Crude ORs were respectively 1.71 (95% CI: 1.15–2.56) and 1.56 (95% CI: 1.05–2.31) for LS-BMD and FN-BMD. All ORs were statistically significant, and those associated to Hmean and anisotropy indices were higher than those of BMD measurements. From a multivariate analysis including anisotropy indices, Hmean, age and FN-BMD, the remaining significant ORs were respectively 6.33 (95% CI: 2.80–14.30) and 3.08 (95% CI: 1.48–6.37) for DA and Hmean. These data have shown that anisotropy indices on calcaneus radiographs can distinguish vertebral fracture cases from control cases. This analysis provides complementary information concerning the BMD and fractal parameter. These data suggest that we can improve the fracture risk evaluation by adding information related to the directional organization of trabecular bone derived from the FFT spectrum on conventional radiographic images.

A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform

BackgroundThe degree of anisotropy (DA) on radiographs is related to bone structure, we present a new index to assess DA.MethodsIn a region of interest from calcaneus radiographs, we applied a Fast Fourier Transform (FFT). All the FFT spectra involve the horizontal and vertical components corresponding respectively to longitudinal and transversal trabeculae. By visual inspection, we measured the spreading angles: Dispersion Longitudinal Index (DLI) and Dispersion Transverse Index (DTI) and calculated DA = 180/(DLI+DTI). To test the reliability of DA assessment, we synthesized images simulating radiological projections of periodic structures with elements more or less disoriented.ResultsFirstly, we tested synthetic images which comprised a large variety of structures from highly anisotropic structure to the almost isotropic, DA was ranging from 1.3 to 3.8 respectively. The analysis of the FFT spectra was performed by two observers, the Coefficients of Variation were 1.5% and 3.1 % for intra-and inter-observer reproducibility, respectively. In 22 post-menopausal women with osteoporotic fracture cases and 44 age-matched controls, DA values were respectively 1.87 ± 0.15 versus 1.72 ± 0.18 (p = 0.001). From the ROC analysis, the Area Under Curve (AUC) were respectively 0.65, 0.62, 0.64, 0.77 for lumbar spine, femoral neck, total femoral BMD and DA.ConclusionThe highest DA values in fracture cases suggest that the structure is more anisotropic in osteoporosis due to preferential deletion of trabeculae in some directions.

Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises

In this letter, it is shown that fast and exact fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) signals can be synthesized by the circulant embedding method (CEM). CEM consists in embedding the N/spl times/N covariance matrix of the stationary fGn process in a larger 2M/spl times/2M circulant matrix such that M /spl ges/N-1. CEM is exact, since second-order statistics of the generated data are those of the Gaussian fGn. CEM is fast, since the optimal case M=N-1 can be reached. Fast and exact fBm sequences can be easily recovered from fGn ones.

Long‐Term Corticosteroid Therapy Induces Mild Changes in Trabecular Bone Texture

The relative roles of bone mineral density (BMD) decrease and of microarchitectural changes in corticosteroid‐induced osteoporosis (CIOP) are debated. Our objective has been to evaluate both bone microarchitecture (by a fractal analysis of texture on radiographs) and BMD in corticosteroid (CS)‐treated patients. In this study, 60 patients from a rheumatology unit with a mean age of 60.6 ± 14.8 years taking CS therapy for more than 6 months and a cumulative dose of prednisone over 1 g and 57 controls among age‐matched patients and hospital staff were recruited. Bone diseases and bone‐modifying drugs (except calcium, vitamin D, and hormonal replacement therapy [HRT]) were considered as exclusion criteria. A fractal analysis of trabecular bone texture was performed on calcaneus radiographs after an oriented analysis in 18 directions. The fractal analysis was based on the fractional Brownian motion model Results were expressed by H parameter (H = 2 – fractal dimension) in each direction, Hmean being the average of 18 directions, Hmini the minimum, and Hmaxi the maximum. BMD was measured by double‐energy X‐ray absorptiometry (DEXA) at the femoral neck (FN) and lumbar spine (LS). The odds ratios (OR) were calculated for a variation of 1 SD. The mean duration and dose of CS therapy was 5.6 ± 6.6 years and 16.9 ± 19.7 g. CS therapy was significantly correlated to a decrease in FN or LSBMD: OR = 1.95,95% confidence interval (CI, 1.29–2.97) and OR = 3.19 (CI, 1.80–5.66), respectively. The Hmean and Hmaxi were significantly lower in the cases than in the controls: P = 0.03 and P = 0.02; OR = 1.67 (CI, 1.10–2.54) and OR = 1.75 (CI, 1.05–2.37). A similar trend was observed with Hmini but the difference did not reach the level of statistical significance: P = 0.06, OR = 1.57 (CI, 1.05–2.37). This study was repeated among cases and controls who had never taken HRT (respectively, n = 40 and n = 39). The results were similar. Among patients taking CS therapy, the presence of nontraumatic fractures was inversely related to BMD values but not to texture parameters. These data have shown that long‐term CS therapy induces both BMD decrease and trabecular bone texture changes. The effect of CS therapy was much stronger on BMD than on the fractal H parameter. These results are in accordance with previous studies showing a lower effect of CS therapy on bone microarchitecture than on bone mass. These results can be contrasted with those observed in women with postmenopausal osteoporosis and vertebral crush fractures in which the variations in the fractal parameters are more significant than the BMD variations.

Changes in Fractal Dimension of Trabecular Bone in Osteoporosis: A Preliminary Study

The trabecular bone network represents a complex structure, either in a macroscopic examination, in a microscopic histological study, or in a radiological projection [1, 2]. Particularly for radiological images, euclidean geometry does not lend itself to the description of such a complex structure. Fractal geometry is much better suited to this analysis, allowing a characterization of this type of rough irregular texture, or of its projection [3,4].

Low bias histogram-based estimation of mutual information for feature selection

This paper presents a low bias histogram-based estimation of mutual information and its application to feature selection problems. By canceling the first order bias, the estimation avoids the bias accumulation problem that affects classical methods. As a consequence, on a synthetic feature selection problem, only the proposed method results in the exact number of features to be chosen in the Gaussian case when compared to four other approaches. In a speech recognition application, the proposed method and the Sturges method are the only ones that lead to a correct number of selected features in the noise free case. In the reduced data case, only the proposed method points out the optimal number of features to select. Finally, in the noisy case, only the proposed method leads to results of high quality; other methods show severely underestimated numbers of selected features.