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# Accession Number:

## ADP012041

# Title:

## The (2-5-2) Spline Function

# Descriptive Note:

## Conference paper

# Corporate Author:

## VIRGINIA POLYTECHNIC INST AND STATE UNIV BLACKSBURG DEPT OF ELECTRICAL AND COMPUTER ENGINEERING

# Report Date:

## 2000-01-01

# Pagination or Media Count:

##
9.0

# Abstract:

## Splines have been used extensively in the interpolation of multidimensional data sets. Linear interpolation utilizes second order splines first degree piecewise polynomials and has widespread popularity because of its ease of implementation. Cubic splines are often used when higher degrees of smoothness are required of the interpolation process. Linear interpolation has the advantages of not requiring the solution of an inverse problem the data points are themselves the coefficients of the triangular basis functions and extremely efficient generation of the output sample points. Unfortunately, the linear-interpolating function has only Csup 0 continuity the function is continuous but its derivatives are discontinuous and therefore lacks the required smoothness for many applications. We provide a new algorithm in this paper based on the efficient derivative summation approach to spline rendering. Cubic B-spline interpolation for uniformly spaced data points provides Csup 2 continuity. The interpolation function can be rendered quite efficiently from the basis coefficients and the basis function, using a cascade of four running average filters. Unser et el. have shown a digital filter solution for the inverse problem of obtaining the spline coefficients from the data points. A matrix inversion solution is also commonly used. Both solutions require the use of floating point multiplication and addition, while the forward problem can be implemented utilizing only fixed-point additions. In this paper, we develop a class of spline basis functions which solve the interpolation problem using only simple arithmetic shifts and fixed point additions for solutions to both the forward and inverse problems. The system impulse response for the new interpolators appears to be closer to the ideal interpolator than the B-spline interpolator. We refer to the new splines as 2-5.2 splines

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#