A Box-Counting Method with Adaptable Box Height for Measuring the Fractal Feature of Images

Most of the existing box-counting methods for measuring fractal features are only applicable to square images or images with each dimension equal to the power of 2 and require that the box at the top of the box stack of each image block is of the same height as that of other boxes in the same stack, which gives rise to inaccurate estimation of fractal dimension. In this paper, we propose a more accurate box-counting method for images of arbitrary size, which allows the height of the box at the top of each grid block to be adaptable to the maximum and minimum gray-scales of that block so as to circumvent the common limitations of existing box-counting methods

Application of Fractal Dimension for Quantifying Noise Texture in Computed Tomography Images

Purpose Evaluation of noise texture information in CT images is important for assessing image quality. Noise texture is often quantified by the noise power spectrum (NPS), which requires numerous image realizations to estimate. This study evaluated fractal dimension for quantifying noise texture as a scalar metric that can potentially be estimated using one image realization. Methods The American College of Radiology CT accreditation phantom (ACR) was scanned on a clinical scanner (Discovery CT750, GE Healthcare) at 120 kV and 25 and 90 mAs. Images were reconstructed using filtered back projection (FBP/ASIR 0%) with varying reconstruction kernels: Soft, Standard, Detail, Chest, Lung, Bone, and Edge. For each kernel, images were also reconstructed using ASIR 50% and ASIR 100% iterative reconstruction (IR) methods. Fractal dimension was estimated using the differential box‐counting algorithm applied to images of the uniform section of ACR phantom. The two‐dimensional Noise Power Spectrum (NPS) and one‐dimensional‐radially averaged NPS were estimated using established techniques. By changing the radiation dose, the effect of noise magnitude on fractal dimension was evaluated. The Spearman correlation between the fractal dimension and the frequency of the NPS peak was calculated. The number of images required to reliably estimate fractal dimension was determined and compared to the number of images required to estimate the NPS‐peak frequency. The effect of Region of Interest (ROI) size on fractal dimension estimation was evaluated. Feasibility of estimating fractal dimension in an anthropomorphic phantom and clinical image was also investigated, with the resulting fractal dimension compared to that estimated within the uniform section of the ACR phantom. Results Fractal dimension was strongly correlated with the frequency of the peak of the radially averaged NPS curve, having a Spearman rank‐order coefficient of 0.98 (P‐value \u3c 0.01) for ASIR 0%. The mean fractal dimension at ASIR 0% was 2.49 (Soft), 2.51 (Standard), 2.52 (Detail), 2.57 (Chest), 2.61 (Lung), 2.66 (Bone), and 2.7 (Edge). A reduction in fractal dimension was observed with increasing ASIR levels for all investigated reconstruction kernels. Fractal dimension was found to be independent of noise magnitude. Fractal dimension was successfully estimated from four ROIs of size 64 × 64 pixels or one ROI of 128 × 128 pixels. Fractal dimension was found to be sensitive to non‐noise structures in the image, such as ring artifacts and anatomical structure. Fractal dimension estimated within a uniform region of an anthropomorphic phantom and clinical head image matched that estimated within the ACR phantom for filtered back projection reconstruction. Conclusions Fractal dimension correlated with the NPS‐peak frequency and was independent of noise magnitude, suggesting that the scalar metric of fractal dimension can be used to quantify the change in noise texture across reconstruction approaches. Results demonstrated that fractal dimension can be estimated from four, 64 × 64‐pixel ROIs or one 128 × 128 ROI within a head CT image, which may make it amenable for quantifying noise texture within clinical images

Medical image processing using fractal functions

In this paper, a comparison was made between a modified methods for repeated engineering modeling in order to increase the accuracy of medical images. A comparison was made between different types in terms of classification accuracy. The lacuinartiy feature has also been used to reduce the noise ratio in the received images. The results showed the importance of fractal IFS in medical pulse compression, where a ratio of (98%) was obtained in reducing noise and a ratio of (0.421) in the gap coefficient was obtained. It separated the diseased tissues from the healthy tissues by applying several multi-fractal factors. Fractal image compression is dependent on subjective similarity, with one part of the image being the same as the other part of a similar image. The partial coding is constantly linked to the grayscale images by dividing a color RGB image into three channels - red, green and blue, and is compressed independently by considering each color segment as a specific gray scale image. Based on the smart neural network, the patterns are distinguished for the medical images used by a few learning time and positive error 0.22%

Magnetic nanocomposites based on shape memory polyurethanes

Shape memory composites based on a commercial segmented polyurethane and magnetite (Fe3O4) nanoparticles (MNPs) were prepared by a simple suspension casting method. The average sizes of individual magnetic particles/clusters were determined by TEM microscopy and corroborated from SAXS patterns. The magnetization properties of selected samples were evaluated using zero field cooling/field cooling (ZFC/FC) measurements and magnetization loops obtained at different temperatures. The results showed that magnetization at high field (20 k Oe) and coercitivity measured at 5 K increase with magnetite content and that all the composite films exhibit superparamagnetic behavior at 300 K. The specific absorption rate (SAR) of the nanocomposites was calculated by experimentally determining both the specific heat capacity and the heating rate of the films exposed to an alternant magnetic field. All nanocomposites were able to increase their temperature when exposed to an alternant magnetic field, although the final temperature reached resulted dependent of the MNPs concentration. What is more, a fast and almost complete recovery of the original shape of the nanocomposites containing more than 3 nominal wt.% MNP was obtained by this remote activation applied to the previously deformed samples.Instituto de Física La Plat

Magnetic nanocomposites based on shape memory polyurethanes

Shape memory composites based on a commercial segmented polyurethane and magnetite (Fe3O4) nanoparticles (MNPs) were prepared by a simple suspension casting method. The average sizes of individual magnetic particles/clusters were determined by TEM microscopy and corroborated from SAXS patterns. The magnetization properties of selected samples were evaluated using zero field cooling/field cooling (ZFC/FC) measurements and magnetization loops obtained at different temperatures. The results showed that magnetization at high field (20 k Oe) and coercitivity measured at 5 K increase with magnetite content and that all the composite films exhibit superparamagnetic behavior at 300 K. The specific absorption rate (SAR) of the nanocomposites was calculated by experimentally determining both the specific heat capacity and the heating rate of the films exposed to an alternant magnetic field. All nanocomposites were able to increase their temperature when exposed to an alternant magnetic field, although the final temperature reached resulted dependent of the MNPs concentration. What is more, a fast and almost complete recovery of the original shape of the nanocomposites containing more than 3 nominal wt.% MNP was obtained by this remote activation applied to the previously deformed samples.Instituto de Física La Plat

Enhanced BC Algorithm Incorporating a Novel Sampling Step and a Fractional Box Count

The Box-Counting (BC) method is one of the most commonly used algorithms for fractal dimension calculation of binary images in the fields of Engineering, Science, Medical Science, Geology and so on due to its simplicity and reliability. One of the issues related to fractal dimension is data sampling that involves a process where a certain size of box is taken from a given image and it has a direct effect on the precision of the fractal dimension estimation. The Geometric Step (GS) method, arithmetic step method, and divisor step method are the representative methods. The GS method is mainly used because of its efficiency. However, the GS method has some drawbacks in nature. If the image size is large, it provides insufficient data for regression analysis. It can be applied to the image of pixel size for 100 [%] pixel utilization. Application of the GS method to an image of may waste pixels in the calculation and degrade the estimation accuracy. In this thesis, a novel sampling method is proposed in order to resolve the shortcomings of the GS method on the basis of the intuitive observation that an estimate may have a higher degree of precision if more pixels are utilized in each step and a sufficiently large number of fitting data are guaranteed. The proposed sampling method is an improved version of the conventional GS method, called the modified GS (MGS) method. The MGS method selects some additional step sizes with higher pixel utilization rate among the middle values between the integer powers of 2 to constitute the overall step set with the GS method. Not all sampling methods including the MGS method can guarantee 100 [%] pixel utilization when the BC method is applied to images of an arbitrary size. This study suggests a novel fractional counting method to resolve the problem of pixel waste. The proposed counting method counts pixels of fractal within a discarded box (not of size) and adds its fractional count normalized by both the average pixel number of all boxes with size and step size to integer count. The performance of the enhanced BC method incorporating the MGS method and fractional counting method is verified on a set of deterministic fractal images whose theoretical dimensions are well known and compared it with those of the existing BC methods. The experimental results show that the proposed method outperforms the conventional BC method and triangle BC method.Contents List of Tables ⅲ List of Figures ⅳ Abstract ⅵ Chapter 1. Introduction 1.1 Motivation 1 1.2 Research objectives 3 1.3 Organization of the thesis 3 Chapter 2. Overview of Fractal Theory 2.1 Definition of fractal 5 2.2 Fractal dimension 7 2.3 Fractal geometry 9 2.3.1 Mandelbrot set and Julia set 10 2.3.2 Koch snowflake (Opened) 11 2.3.3 Apollonian gasket 12 2.3.4 Vicsek fractal 13 2.3.5 Sierpinski triangle 14 2.3.6 Rand cantor 15 2.3.7 Koch curve 85° 16 2.3.8 Sierpinski carpet 17 2.3.9 Hilbert curve 18 Chapter 3. Existing Box-Counting Methods 3.1 Conventional BC method 20 3.2 Triangle BC method 25 Chapter 4. Enhanced BC method 4.1 Existing sampling methods and their drawbacks 27 4.1.1 Sampling methods 27 4.1.2 Pixel utilization 30 4.1.3 Drawbacks of existing sampling methods 30 4.2 New sampling method 32 4.3 Fractional box count 35 4.4 Procedure of the enhanced BC method 38 Chapter 5. Experiments and Review 5.1 Experiments on deterministic fractal image 41 5.1.1 test image 41 5.1.2 Determination of 43 5.1.3 Experiment with images of pixels 44 5.1.4 Experiments on rotated image 45 5.1.5 Experiment with images of pixels 46 5.2 Experiments on non-deterministic fractal images 51 5.2.1 Converting color images to binary images 51 5.2.2 Coastline images 52 Chapter 6. Conclusion 56 References 58 Appendix 61Maste