Three-dimensional tumor perfusion reconstruction using fractal interpolation functions

2001 ◽  
Vol 48 (4) ◽  
pp. 462-473 ◽  
Author(s):  
O.I. Craciunescu ◽  
S.K. Das ◽  
J.M. Poulson ◽  
T.V. Samulski
Keyword(s):  
Three Dimensional ◽  
Fractal Interpolation ◽  
Tumor Perfusion ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

3D Fractal Interpolation Functions

2020 ◽  
Vol 12 (1) ◽  
pp. 120-123
Author(s):  
Rashad A. Al-Jawfi
Keyword(s):  
Data Analysis ◽  
Three Dimensional ◽  
Iterated Function Systems ◽  
Data Reconstruction ◽  
Fractal Interpolation ◽  
Scalar Data ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Interpolation Functions

For visualizing one, two- and/or three-dimensional data, reconstruction mechanism is generally considered. In the present study, I have examined the application of iterated function systems in interpolation. I have also presented new fractal interpolation derivations for three-dimensional scalar data. The interpolations can indicate uncertainty of the data, allow tenability from the data, represent the data statistically at different scales, and may also more accurately facilitate data analysis.

Hyperthermia Induced 3D Temperature Distribution in a Human Sarcoma With Tumor Perfusion Reconstructed Using Fractal Interpolation Functions

2000 ◽  
Author(s):  
Oana I. Craciunescu ◽  
Shiva K. Das ◽  
Robert L. McCauley ◽  
Thaddeus V. Samulski
Keyword(s):  
Temperature Distribution ◽  
Medical Center ◽  
Blood Perfusion ◽  
Temperature Reconstruction ◽  
Linear Interpolation ◽  
Fractal Interpolation ◽  
Tumor Perfusion ◽  
Fractal Interpolation Functions ◽  
Duke University ◽  
Interpolation Functions

Abstract Essential to the success of optimized thermal treatment during hyperthermia is accurate modeling. Advection of energy due to blood perfusion significantly perturbs the temperature and without accurate estimates of the magnitude of the local tissue blood perfusion it is unlikely that accurate estimates of the temperature distribution can be made. It is shown here that the blood mass flow rate per unit volume of tissue in the Pennes’ bio-heat equation can be modeled using a relative perfusion index (RPI) determined with dynamic-enhanced magnetic resonance imaging (DE-MRI). The existing technology limits the DE-MRI perfusion data to be acquired in a limited number of slices. Consequently, the tumor perfusion data is interpolated using fractal interpolation functions (FIFs), as it has been shown that the RPI data is fractal, and that fractal interpolation is superior to linear interpolation when a 3D fractal-like scaling exists. For illustration, a patient treated with hyperthermia at Duke University Medical Center for a high-grade leg tissue sarcoma is modeled. For control, the resultant temperatures are compared to non-invasively measured temperatures using the MR thermometry technique. Strong correlation is found between the DE-MRI perfusion images, the MR chemical shift images during heating, and the numerical simulation of the temperature field, emphasizing the relation between the DE-MRI measured values and advective heat loss in tissue. The fractal interpolation of DE-MRI data to obtain the 3D perfusion gives a more accurate temperature distribution compared to linear interpolation. For even a better temperature reconstruction, further models need to include large vessels.

scholarly journals A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 767
Author(s):  
Alexandra Băicoianu ◽  
Cristina Maria Păcurar ◽  
Marius Păun
Keyword(s):  
Approximation Method ◽  
Finite System ◽  
Countable System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Finite Systems ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Finite Set ◽  
Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .

CONSTRUCTING MULTIPLE WAVELET PACKETS WITH FIFS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 165-169
Author(s):  
GANG CHEN ◽  
ZHIGANG FENG
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Scaling Functions ◽  
Wavelet Packet Decomposition ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

By using fractal interpolation functions (FIF), a family of multiple wavelet packets is constructed in this paper. The first part of the paper deals with the equidistant fractal interpolation on interval [0, 1]; next, the proof that scaling functions ϕ1, ϕ2,…,ϕr constructed with FIF can generate a multiresolution analysis of L2(R) is shown; finally, the direct wavelet and wavelet packet decomposition in L2(R) are given.

SURFACE FITTING AND ERROR ANALYSIS USING FRACTAL INTERPOLATION

2012 ◽  
Vol 22 (08) ◽  
pp. 1250194 ◽  
Author(s):  
HONG-YONG WANG ◽  
JIA-BING JI
Keyword(s):  
Error Analysis ◽  
Iterated Function Systems ◽  
Surface Fitting ◽  
Continuous Functions ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Continuous Surface ◽  
Fractal Interpolation Surface ◽  
Interpolation Functions

The fitting of a given continuous surface defined on a rectangular region in ℝ2 is studied by using a fractal interpolation surface, and the error analysis of fitting is made in this paper. The fractal interpolation functions used in surface fitting are generated by a special class of iterated function systems. Some properties of such fractal interpolation functions are discussed. Moreover, the error problems of fitting are investigated by using an operator defined on the space of continuous functions, and the upper estimates of errors are obtained in the sense of two kinds of metrics. Finally, a specific numerical example to illustrate the application of the procedure is also described.

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