journal = {The European Physical Journal B} You can also select a web site from the following list. This work is supported by the UE EFRR ZPORR project Z/2.32/ I/1.3.1/267/05 Szczecin University of Technology Research and Education Center of Modern Multimedia Technologies (Poland). However, this is not to say that the relationship between lacunarity and fractal dimension is fixed. Claudio D, Anna C, Patrizia M, Luciano S, Emanuela S, et al. Before It may be noted that the binarized image (figure 2D) has gaps (black pixels) within the vessel surface (white pixels). 4, Grauslund J, Green A, Kawasaki R, Hodgson L, Sjlie AK, Wong TY. In our study the measurement of the box counting fractal dimension D0 is performed using a MATLAB program available free online through the MATLAB central database (37). This was used in conjunction with a gliding box algorithm as described in the methods section to quantify the lacunarity of images of the vessel structures. url = {https://doi.org/10.1115%2F1.4042579}, Multifractal analysis and lacunarity are valuable additions to classical measures of vascular morphology and will have utility in future studies of normal, developing and pathological tissues. }, Chen, Y., & Yang, H. (2016). The vascular system and the cost of blood volume. The homogeneity of a structure can be determined by the proximity of its data point to the 1:1 line on the (0) D0 plot and the symmetry of a multifractal spectrum can be assessed by the proximity of its widths to the 1:1 line on the (0) (qi,+) versus (qi,) -(0) plot. Although theoretical structures with monofractal behavior can be generated, such as Kochs snowflake, numerical models of naturally occurring snowflakes in fact exhibit multifractal behavior (54). The higher asymmetry in the spectrum of P0 is partly due to the presence of more vessels and thus higher complexity in the image at P0. Furthermore, the generalized fractal dimensions decreased from P0 to P10, while lacunarity increased (parameter b decreased) (table 2) as the regressing vessels occupied less space. MathWorks is the leading developer of mathematical computing software for engineers and scientists. The generalized fractal dimensions (D0,D1,D2) that we measured using our program all matched this value as we expected for a monofractal (figure 3C,G). This is also likely to be true for many vessel structures, as vessels of the retina were first assumed to follow a simple fractal pattern (30) but have more recently been shown to be multifractal in nature (47). It is important to note that we avoided the use of erosion or skeletonization methods, in order to maintain intensity variations which are likely to be biologically relevant. B shows the asymmetry of the multifractal spectra of images from different tissue types. Arterial adaptations to chronic changes in haemodynamic function: coupling vasomotor tone to structural remodelling. The data points for cortex, skin, and muscle were closer to the 1:1 line than the data point for kidney vessels. Fibroblast Growth Factor-2 Selectively Stimulates Angiogenesis of Small Vessels in Arterial Tree. Doubal FN, MacGillivray TJ, Patton N, Dhillon B, Dennis MS, Wardlaw JM. A graph of lacunarity values L() versus the box widths () for various tissues is presented here. There is no significant difference in vessel diameter between the other images of vessels in the hyaloid vasculature (P>0.05) (table 2; figure 9A). A shows the average diameters of the vessels in the images 8 IL. Joint Multifractal and Lacunarity Analysis of Image Profiles for Manufacturing Quality Control. author = {Yun Chen and Hui Yang}, Lacunarity was an effective measure for analyzing . 4148 (2009), Mandelbrot, B.: The Fractal Geometry of the Nature. Retinal vessels have been shown to have a multifractal organization (47). Morphological closing algorithms (10) may serve to fill the gaps; however this intensity variation is likely to be biologically relevant, and as little manipulation as possible of these images provides the clearest picture of true vessel morphology. Matsushita M, Sano M, Hayakawa Y, Honjo H, Sawada Y. Fractal Structures of Zinc Metal Leaves Grown by Electrodeposition. % p 044501-7, 2019. Select the China site (in Chinese or English) for best site performance. 486493. Create scripts with code, output, and formatted text in a single executable document. MathWorks is the leading developer of mathematical computing software for engineers and scientists. Finally, we further processed the images using a series of MATLAB programs that are freely available from the MATLAB central exchange (see references). A membrane associated mCherry fluorescent reporter line for studying vascular remodeling and cardiac function during murine embryonic development. journal = {Journal of Manufacturing Science and Engineering} Figure 5 shows the lacunarity values measured from the confocal images of various tissues. 456463. 141, No. Tissue Engineering and Organ Structure: A Vascularized Approach to Liver and Lung. PubMedGoogle Scholar. This is as we might expect, since when vessel branches are more dense, there is less available space between vessels, limiting the size distribution of the lacunae. Chen, Yun, and Hui Yang. The data points from the images of the tissue vasculature do not lie on the 1:1 line; in fact, they all deviate from this line (figure 7A). Find the treasures in MATLAB Central and discover how the community can help you! The utility of fractal geometry in landscape design. We then went further to determine whether the vessel structures in these images had a multifractal or monofractal character. 8671, pp. AD are grayscale confocal microscope images of hyaloid vasculature in the retina of a Flk1-myr::mCherry mouse taken from P0 to P10. The b values for the various images of vessels in the tissues are summarized in table 1. In: Chmielewski, L.J., Kozera, R., Shin, B.-S., Wojciechowski, K. By construction, i(q,) [0,1] for each value of q. Springer-Verlag, Heidelberg (1992), CrossRef number = {8}, Thus, the fractal dimension is a single value that reflects the minimum dimension of the space required to embed the object. Wornell, G.: Signal Processing with Fractals: A Wavelet-Based Approach. The fractal dimension, D0 (also known as the Hausdorff-Besicovitch dimension) is (29) used to describe the fractal properties of a structure and is most commonly determined with the box counting algorithm (see figure 1). We address this issue in the section on multifractal analysis. These parameters have also been used to. 4754. }, Imani, F., Yao, B., Chen, R., Rao, P., & Yang, H. (2019). journal = {Journal of Manufacturing Science and Engineering} The simple box-counting fractal dimension method only counts the number of boxes that contain signal pixels, ignoring the density of signal pixels (representing intensity fluctuation) in each box at all box widths (). In fact, very few examples exist of structures with perfect monofractal geometries. Lacunarity is a measure of how data fills space. Inclusion in an NLM database does not imply endorsement of, or agreement with, title = {Joint Multifractal and Lacunarity Analysis of Image Profiles for Manufacturing Quality Control}, Errors were estimated as in table 1. Regions in yellow show the overlap between the two images (A and D) in E. Importantly the vessel diameters from the original and the processed images do not differ significantly, thus the processing steps preserve critical information about the original vessel beds for analysis and comparison. A significant variation in signal intensity was not detected from the top to the bottom of the stack; therefore, no correction was applied along the z-stack before taking the maximum intensity projection. If the data points lie on the 1:1 line, the pattern is homogeneous and can be characterized by D0 alone. Springer, Heidelberg (2013). official website and that any information you provide is encrypted Math. This toolbox reads a 3d surface or 2d image and then compute the multifractal spectrum. As a library, NLM provides access to scientific literature. The first one corresponds to the linear fit to all points and the second one corresponds to the linear fit to the data excluding the lowest and highest values. the variation of certain speculative prices. Vessel regression was studied by analyzing images from the hyaloid vasculature in the eyes of neonatal mouse pups. Zhu M, Madigan MC, van Driel D, Maslim J, Billson FA, et al. number = {3}, - 163.172.100.119. The lowest value corresponds to the width of the pixel in the image and the largest value corresponds to the bounding size of the image. E is the symmetry comparison of the spectra of P0P10 hyaloid vessels. Choose a web site to get translated content where available and see local events and offers. Multifractal analysis takes intensity variations like those seen in membrane labeled vessels (figure 2) into account by measuring pixel density within a box (43). month = {jan}, We present an automated extraction tool with a processing pipeline to characterize 2D representations of 3D microvasculature, using multifractal analysis and lacunarity. Taken together, the data in the table suggest that generalized dimensions D0, D1, and D2 as well as the parameter b can be used to characterize the difference in the morphology of the beds even when differences between mean diameters of the vessels are less pronounced. An efficient implementation of the gliding box lacunarity algorithm. Microvessel Density as a Prognostic Factor in Women with Breast Cancer. number = {3}, Meakin P, Stanley HE, Coniglio A, Witten TA. 140, no. TPCG. https://doi.org/10.1007/978-3-319-46418-3_16, DOI: https://doi.org/10.1007/978-3-319-46418-3_16, eBook Packages: Computer ScienceComputer Science (R0). doi = {10.1115/1.4037891}, LNCS, vol. MathWorks is the leading developer of mathematical computing software for engineers and scientists. The generalized fractal dimensions measured from the spectra show no difference between the original image and the rotated image. Electr. Accelerating the pace of engineering and science. Vascular remodeling of the mouse yolk sac requires hemodynamic force. Tegy Vadakkan (2023). In the case of a fractal N() is related to by the relation N() ~D0. % and lacunarity analysis of image profiles for manufacturing quality control. Joint Multifractal and Lacunarity Analysis of Image Profiles for Manufacturing Quality Control. In images of membrane labeled vessels, the number of signal pixels is different from one box to another for any given box width (figure 2), thus simple box counting analysis is insufficient to fully characterize these complex vessel structures. MATH The images from cortex and skin are hard to compare, and this analysis allowed us to show they have similar morphology based on their space filling properties. Tolle CR, McJunkin TR, Gorsich DJ. Fractal Dimension of Polymer Sorbents. In all cases we were able to fit the data for box widths in the range from 8 to 512 with the hyperbolic function listed above (L() = b/a + c) with R2 > 0.98. Prentice Hall, Newjersy (1996), Williams, G.: Chaos Theory Tamed. The physiological principle of minimum work: I. Thus for the vessel images used in this manuscript projection analysis is similar to the slice by slice analysis in detecting trends between the vessel images. However, we confirmed that the data were not influenced by noise or finite size effect for each data set by comparing the slope over the entire range with the slope obtained from the middle set of values (464 pixels). Efficient box-counting determination of generalized fractal dimensions. R.88(10b), 261263 (2012), Oszutowska-Mazurek, D., Mazurek, P., Sycz, K., Wjciuk, G.W. We have found that indeed all of the vessel networks analyzed have multifractal geometries. The higher asymmetry in the spectrum of P0 is partly due to the presence of more vessels and thus higher complexity in the image at P0. Next, projected images were exported to Adobe Photoshop CS2 and the unsharp mask filter with a radius of 12 pixels was applied to improve image quality by intensifying the vascular patterns while decreasing background noise in nearby regions. % nonlinear spatiotemporal dynamics on fractal surfaces for the whole-heart modeling Efron B. Bootstrap Methods: Another Look at the Jackknife. The lacunarity at a box size is defined as L() = (N() Q2)/Q12. DOI: 10.1115/1.4042579. This is an example of intensity variation inherent in our membrane based label. Google Scholar, Plotnick, R., Gardner, R., Hargrove, W., Prestegaard, K., Perlmutter, M.: Lacunarity analysis: a general technique for the analysis of spatial patterns. In the vascular beds discussed in this paper, the thickness of the z-stacks of the tissue was 17.5 micrometers. The only significant differences between vessel diameters are between the P0 and P10 stage (P=0.006) and between the P7 and P10 stage (P=0.02). The vessel density measurements from images of vessels were indistinguishable between the image of skin and the image of cortex, but both of these differed significantly from the kidney and muscle images. Q1 = i p(i,) and the sum of the square of the number of signal pixels in each box Q2 = i p(i,)2. url = {https://doi.org/10.1115%2F1.4037891}, For multifractal analysis, one must consider the density of signal pixels in each box (6) as shown by Posadas et al. IL shows the average value of the points on the various multifractal spectra derived from each individual slice. Error estimates of vessel diameters are the standard deviations of the measured vessel diameter segments in each image. The numerical values of Dq corresponding to different values of q are the generalized fractal dimensions of a structure. 24(1), 4963 (2014), CrossRef pp Lecture Notes in Computer Science(), vol 9972. Hui Yang (2023). The second limitation of simple fractal analysis is that it assumes that vessels can be described by a single fractal dimension alone, when in reality, complex structures exist with subsets of regions having different scaling properties. Phys. 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