

Title of Thesis
Visualization of Positive Data Using Modified
Quadratic Shepard Method: An Optimum Solution 
Author(s)
Ghulam Mustafa 
Institute/University/Department
Details Department of Computer Science & Engineering /
University of Engineering & Technology, Lahore 
Session 2008 
Subject Computer Science 
Number of Pages 121 
Keywords (Extracted from title, table of contents and
abstract of thesis) Visualization, Positive, Data,
Modified, Quadratic, Shepard, MQS, Method, Optimum, Solution,
business, community 
Abstract Many methods arc available to construct the models from data sets that arc sampled on
scattered grid. For Scattered data, the inverse distance weighted methods are considered
better due to their efficiency, extendibility to higher dimensions, global in nature and ease
in implementation. Modified Quadratic Shepard (MQS) method, being smooth with C'
continuity, is a commonly used method in the field of science and engineering such as
geophysics, astronomy and meteorology. The method is available with many libraries and
visualization environments such as Numerical Analysis Group (NAG) and FORTRAN
and Surfer 8. However, it has problem that being a C' continuous method it interpolates
values extending beyond the inherently limits of the data, which is sometimes
meaningless for scientific and business community. So it is not suitable for applications
where there is some inherent constraint on value of data. Examples of such value
constraints arc non negativity, upper and/or lower bounds, and more general constraint
defined by a function. Such constraints commonly arise in science and engineering
applications. Nonnegativity (for simplicity we use the term positivity to mean non
negativity) is a special case of the generalized bounds preserving problem. Positivity of
data is commonly encountered in science, engineering and business. The significance of
positivity lies in the fact that sometimes it does not make sense to talk of some quantity to
be negative. For example the quantities like mass, volume, density, absolute temperate &
pressure, radiation dose, number of persons, chemical concentration, hydraulic
conductivity and porosity are meaningless when negative.
In this thesis we formulate five methods, based on the MQS method, to preserve the
arbitrary lower and upper bounds of data and fulfilling various requirements of
Visualizations and other applications such as efficiency, accuracy, continuity, case in implementation, extendibility to higher dimensionality and gradient estimation etc. All these constrained interpolation functions are C continuous. An optimum solution having most of the above given required characteristics for different sizes and dimensionality of data has been identified.

