Triangular Prism Surface Area Research Articles

Scale Dependence of Waviness and Unevenness of Natural Rock Joints through Fractal Analysis

The scale dependence of surface roughness is critical in characterising the hydromechanical properties of field-scale rock joints but is still not well understood, particularly when different orders of roughness are considered. We experimentally reveal the scale dependence of two-order roughness, i.e., waviness and unevenness through fractal parameters using the triangular prism surface area method (TPM). The surfaces of three natural joints of granite with the same dimension of 1000 mm×1000 mm are digitised using a 3D laser scanner at three different measurement resolutions. Waviness and unevenness are quantitatively separated by considering the area variation of joint surface as grid size changes. The corresponding fractal dimensions of waviness and unevenness in sampling window sizes ranging from 100 mm×100 mm to 1000 mm×1000 mm at an interval of 100 mm×100 mm are determined. We find that both the fractal dimensions of waviness and unevenness vary as the window size increases. No obvious stationarity threshold has been found for the three rock joint samples, indicating the surface roughness of natural rock joints should be quantified at the scale of the rock mass in the field.

Computation of direction dependent joint surface parameters through the algorithm of triangular prism surface area method: A theoretical and experimental study

Grasselli’s (2001) methodology on the quantification of the “shear-induced potential contact zones” is fundamentally very strong at properly dealing with the physics of the shear phenomenon. However, the high precision measurement system and the triangulation algorithm he used may not be easily accessible for everyone due to its industrial characteristics. Therefore, a simple technique called “the modified shear-induced potential contact zones” is introduced in the present work. By executing this technique on the surface coordinates of a large number of rock joints digitized through a specially developed mechatronic surface scanning device, “maximum possible and total potential contact areas” (A0 and Aθ*) and “directional surface parameters” (θ*max: maximum apparent dip angle, c2 : shape parameter, θ*max/c2 : change of angularity) are calculated in a specified direction. Ratio A0/c2 is proposed for a new directional roughness parameter. Surface roughness is also characterized by fractal dimension (Dtp), alternatively. Using a specially developed shear box, shear tests are performed in the direction of parameter calculation. Then, by a series of quantitative comparisons between the directional surface parameters and both the fractal dimensions and the shear strengths, the ability of the parameters to relate with the surface roughness and the shear strength is examined. In general, the results are satisfactory for the reliability of the modified technique. The main advantage of the introduced technique is its algorithmic simplicity facilitating direct applicability for basic 2D data sets, [z = f (x, y)], which can be practically acquired via easily accessible and cost-effective surface measurement systems. Additionally, another simple recipe is also introduced predicting contact zones visually prior to shearing. Then, total potential contact areas predicted in the shear direction are visually compared with the actual images of contacts observed in the tests. The best match is obtained when the threshold apparent inclination (θ*cr) is chosen equal to the experimental dilation angle (id) unique for the applied normal load. This clearly proves the validity of Grasselli’s (2001) “threshold apparent dip angle” concept.

Classification of Mammograms into Normal, Benign and Malignant based on Fractal Features

Modern life style of women has made them more vulnerable to breast cancer and it is considered as the largest cause of mortality among women. This paper presents a novel method to classify mammograms into normal ones, with benign and malignant microcalcifications, and with malignant and benign tumors using fractal features derived from fractal dimension. Here, three fractal dimension estimation methods such as Differential Box Counting (DBC), Triangular Prism Surface Area (TPSA) and Blanket methods are used for computing the six fractal features utilized for the classification. The new fractal feature f6 obtained using TPSA method is found to be the best with 100% classification accuracy. The average value of f6 is found to be 0.1110, 0.2875, 0.4743, 0.5271 and 0.8558, for normal, benign masses, benign and malignant microcalcifications and malignant masses respectively. The classification performance of the different features was analyzed using the Receiver Operating Characteristics (ROC).

Fractal texture analysis of the healing process after bone loss

Radiological assessment of treatment effectiveness of guided bone regeneration (GBR) method in postresectal and postcystal bone loss cases, observed for one year. Group of 25 patients (17 females and 8 males) who underwent root resection with cystectomy were evaluated. The following combination therapy of intraosseous deficits was used, consisting of bone augmentation with xenogenic material together with covering regenerative membranes and tight wound closure. The bone regeneration process was estimated, comparing the images taken on the day of the surgery and 12 months later, by means of Kodak RVG 6100 digital radiography set. The interpretation of the radiovisiographic image depends on the evaluation ability of the eye looking at it, which leaves a large margin of uncertainty. So, several texture analysis techniques were developed and used sequentially on the radiographic image. For each method, the results were the mean from the 25 images. These methods compute the fractal dimension (D), each one having its own theoretic basis. We used five techniques for calculating fractal dimension: power spectral density method, triangular prism surface area method, blanket method, intensity difference scaling method and variogram analysis.Our study showed a decrease of fractal dimension during the healing process after bone loss. We also found evidence that various methods of calculating fractal dimension give different results.During the healing process after bone loss, the surfaces of radiographic images became smooth. The result obtained show that our findings may be of great importance for diagnostic purpose.

A NEW METHOD FOR CALCULATING THE FRACTAL DIMENSION OF SURFACE TOPOGRAPHY

A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass–Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.

Multi class Abnormal Breast Tis sue segmentation using Texture Features and Analyzing the Growth Factor Using Power Law

This paper motivated to design and develop an automatic model for multi-class breast tissue segmentation and find the growth of the cancer in breast mammogram images. Various breast tissues are categorized by a novel texture features such as PTPSA[Piecewise Triangular Prism Surface Area], intensity difference and regular-intensity in mammogram images. Using CRF-[Classical Random Forest] method segmentation and classification of the features can be obtained in mammogram images. The input image feature values are compared to the ground-truth values for confirming the true positive rate of the proposed approach. Efficacy of abnormal breast tissue segmentation is evaluated using publicly available MIAS training dataset.In this paper, we investigate the consequences of an option, yet just as conceivable, suspicion of tumor growth, to be specific power growth law, acknowledged in direct expand in tumor breadth. We exhibit a simple model for tumor growth, whose global flow demonstrates power law growth of the tumor, much under boundless supplement supply. For corroboration, it is carried out and examined one-, two- and three-dimensional tumor growth tests both in vitro, in MCF-7 cells (breast malignancy cell line) and in vivo, in mouse xenografts.After successful tissue segmentation, the growth of the tissue is analyzed using Power Law. Performance evaluation of the proposed approach can be obtained by comparing the simulation output with the ground truth data. The accuracy of the proposed approach reaches up to 97% for MIAS database in term of tumor detection. Also, simulating radiotherapy under power law, Gompertz and exponential tumor growth, it is indicated that the power law model predicts profoundly diverse conclusions for the usually used treatment. This shows the significance of utilizing the proper tumor growth model when computing ideal measurement fractionation plan for radiotherapy.

Fractal characteristics and microstructure evolution of magnetron sputtering Cu thin films

How to describe surface morphology characteristic and microstructure evolution are the hottest researches of current thin film researches. But in traditional characterization of surface morphology, the roughness parameters are scale related. And the microstructure evolution of thin film during post-treatment is usually not considered in detail. To give a better understanding of the roughness of thin films topography, fractal method is carried out. In addition, microstructure evolution of thin films is analyzed based on the crystallography and energy theory. Cu thin films are deposited on Si (100) substrates by magnetron sputtering, and then annealed at different temperatures. Surface topography is characterized by atomic force microscope (AFM). Triangular prism surface area (TPSA) algorithm is used to calculate the fractal dimension of the AFM images. Apparent scale effect exists between the surface morphology roughness and film thickness. Relationship between the fractal dimension and roughness is analyzed by linear regression method and linear relationship exists between fractal dimension and surface roughness root mean square (RMS). Fractal dimension can be characterized as a scale independence parameter to represent the complex degree and roughness level of surface. With the increase of annealing temperature, surface roughness and fractal dimension decrease. But when the annealing temperature exceeds the recrystallization temperature, due to the agglomeration and coalescence of Cu grain, surface roughness and fractal dimension increase. Scale effect and changing regularity of grain growth and shape evolution for different film thickness under different annealing temperatures are analyzed. Based on minimum total free energy, regularity of grain growth and changing is proposed. The proposed research has some theory significance and applicative value of Cu interconnect process and development of MEMS.

Quantification of Soil Surface Roughness Evolution under Simulated Rainfall

<abstract><title><italic>Abstract.</italic></title> Soil surface roughness is commonly identified as one of the dominant factors governing runoff and interrill erosion. The objective of this study was to compare several existing soil surface roughness indices and to test the use of the revised triangular prism surface area method (RTPM) to calculate the fractal dimension as a roughness index. A silty clay loam soil was sampled, sieved to four aggregate sizes, and each size was packed in soil trays in order to derive four different soil surface roughness classes. Rainfall simulations using an oscillating nozzle simulator were conducted for 90Â min at 50.2 mm h^-1 average intensity. The surface microtopography was digitized by an instantaneous profile laser scanner before and after the rainfall application. Calculated roughness indices included random roughness, variogram sill and range, fractal dimension and fractal length using a fractional Brownian motion (fBm) model, variance and correlation length according to a Markov-Gaussian (MG) model, and fractal dimension using the RTPM. Random roughness is shown to be the best estimator to significantly distinguish soil surface roughness classes. When taking spatial dependency into account, the variogram sill was the best alternative. The fractal dimension calculated from the fBm model did not yield good results, as only short-range variations were incorporated. The MG variance described the large-scale roughness better than the parameters of the fBm model did. The fractal dimension from the RTPM performed well, although it could not significantly discriminate between all roughness classes. Since it covered a greater range of scales, we believe that it is a good estimator of the overall roughness.

Advanced fractal approach for unsupervised classification of SAR images

Unsupervised classification of Synthetic Aperture Radar (SAR) images is the alternative approach when no or minimum apriori information about the image is available. Therefore, an attempt has been made to develop an unsupervised classification scheme for SAR images based on textural information in present paper. For extraction of textural features two properties are used viz. fractal dimension D and Moran’s I. Using these indices an algorithm is proposed for contextual classification of SAR images. The novelty of the algorithm is that it implements the textural information available in SAR image with the help of two texture measures viz. D and I. For estimation of D, the Two Dimensional Variation Method (2DVM) has been revised and implemented whose performance is compared with another method, i.e., Triangular Prism Surface Area Method (TPSAM). It is also necessary to check the classification accuracy for various window sizes and optimize the window size for best classification. This exercise has been carried out to know the effect of window size on classification accuracy. The algorithm is applied on four SAR images of Hardwar region, India and classification accuracy has been computed. A comparison of the proposed algorithm using both fractal dimension estimation methods with the K-Means algorithm is discussed. The maximum overall classification accuracy with K-Means comes to be 53.26% whereas overall classification accuracy with proposed algorithm is 66.16% for TPSAM and 61.26% for 2DVM.

Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock

A direct determination of the fractal dimensions of a fracture surface is essential for a better understanding of its complete topographic characteristics. In this paper, a laser profilometer is employed to measure the topography of a rock's fracture surface. With the use of the triangular prism surface area and projective covering methods, the resultant data set enables us to directly determine the fractal dimensions of a rock's fracture surface. Moreover, a new method, referred to as the cubic covering method, is proposed. The theoretical issues of fractal dimension estimation are also discussed.

The fractal estimation of wood color variation by the triangular prism surface area method

Color variations of the surfaces of fifteen wood species were characterized by fractal dimension of the triangular prism surface area method. Softwood and hardwood indicated apparently different in the mean fractal dimensions of red and green colors. Red color behaved steadier in softwood than in hardwood and green color varied comparatively stronger in hardwood than in softwood. No evident differences between softwood and hardwood were found in the variation of blue color of all the specimens. Following the low-to-high value order of the mean fractal dimensions, three types of combination of red (R), green (G) and blue (B) colors were found: RGB, RBG and BGR. There also existed six types of fractal dimension distribution; namely, plane, included plane, concave, convex, zigzag, and hilly distributions. Fractal dimensions across the grain changed greatly whereas those along the grain became relatively steady. The characteristic difference of color variation was defined for each species, which was inferred to characterize its own unique appearance of surface color. For color matching of wood parts, fractal dimension quantitatively furnishes essential information of color variation in local and overall features. Such evaluation can be efficiently carried out with few measurements along the grain and by detecting a single color (red, green or blue) only.

Spatial methods for characterising land cover and detecting land-cover changes for the tropics

Characterising land cover and detecting land-cover changes using spatial methods is an area of research that has been attracting increasing attention recently. We compare performances of selected pattern recognition methods for characterising different land covers using unclassified Landsat Thematic Mapper (TM) data for a lowland site in north-eastern Costa Rica. Two spatial statistics (fractal dimension, using the isarithm and triangular prism surface area (TPSA) methods, and Moran's I index of spatial autocorrelation) and selected landscape indices (Shannon's diversity index, contagion, and fractal dimension from perimeter/area) were investigated. Mean values of each metric for each cover type were calculated for subset areas representing forest, agriculture, pasture, and scrub, for all seven Landsat-TM bands and the Normalized Difference Vegetation Index (NDVI). Fractal dimension ( D TPSA ) and Moran's I were found to be useful for characterising spatial complexity of Landsat-TM data, whereas the standard landscape indices were not. Values of D TPSA decreased along a gradient of increasing human disturbance: old-growth forest-scrub-pasture-agriculture. These results can be further applied locally using moving windows for change detection in global environmental change studies. Moreover, in this information era characterized by increasingly abundant imagery, these spatial statistics could serve as metadata for content-based data mining of imagery.

A revisitation of the triangular prism surface area method for estimating the fractal dimension of fractal surfaces

Fractal dimension is widely used to give a measure of variability and roughness of curves, signals, objects, sta- tistical distributions, and so on, We found that an often used method, the so-called triangular prism surface- area method, for estimating the tractal dimension of fractal surfaces possesses some intrinsic mistake., in appli- cation. This note describes the misintel"pretation and suggests the proper application, that we calI Revised Tri- angular Prism Method (RTPM). To show its feasibility we apply RTPM to some synthetic Euclidean and frac- tal surfaces of known dimension.

Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method

A new method is proposed to calculate the fractional (fractal) dimension of topographic surfaces. The method uses a three-dimensional geometric equivalent of the “walking dividers” method in two dimensions. that is it takes elevation values at the corners of squares, interpolates a center value, divides the square into four triangles, and then computes the surface areas of the tops of the prisms which result from raising the triangles to their given elevations. By repeating this calculation for different size squares, the relationship between the total area of the topographic surface and the spacing of the squares is established, and used to estimate the fractal dimension. A computer program in the C programming language is presented to read digital terrain matrices and compute the fractal dimension of the surface using the method. Some test data sets are provided along with their computed fractal dimensions. Finally, the advantages and disadvantages of the method are discussed.