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A comparison of bilinear interpolation, cubic convolution, and brownian interpolation with least squares matching

ACRS 1997

Poster Session 2

A Comparison of Bilinear Interpolation, Cubic Convolution, and
Brownian Interpolation with Least Squares Matching

Jin-Tsong Hwang and Tian-Yuan Shih
Department of Civil Engineering
National Chiao-Tung University
Hsin-Chu, Taiwan,R.O.C.


Abstract

This study compares two fractal interpolation schemes based on fractional Brownian motion (fBm) with two convectional interpolation schemes: bilinear interpolation and cubic convolution. Numerical experiments are performed with a pair of digitized stereo photographs. The original image is reduced to a lower spatial resolution for interpolation. Both the geometric index based on the positioning accuracy of the image matching and the statistic indices such as the root mean square error, average error, maximum deviation, and correlation coefficient derived by comparing interpolated and the reference images indicate that the convectional interpolation schemes are better than the Brownian schemes.


Introduction

This paper examines four interpolation techniques: bilinear interpolation, cubic convolution, weighted Brownian interpolation (Polidori and Chorowicz, 1993), and the modified weighted Brownian interpolation derived in this study. The global statistic and geometric accuracy indices of the interpolated images generated by the four interpolation techniques are computed. The statistical indices include root mean square error (RMSE), average error Bias), correlation coefficient (), and maximum deviation, while the geometric index is the positioning accuracy derived from image matching. Two sampling schemes are also applies to reduce the test image into a lower spatial resolution. The first is decimation which takes the Nth pixel every N pixel. The other is averaging which takes the average of (N+1) x (N+1) window every N pixel. Numerical experiments are also performed with the SUBURB dataset provided by ISPRS Working Group III/3. This dataset includes a pair of digitized stereo photographs, as well as the DTM generated from this pair.


The Interpolation Schemes

Weighted Brownian Interpolation (polidori and chorowizc, 1993)

According to fBm (Mandelbort, 1983), the expected value of the difference in grayscale value over the distance dx is proportional to (dx)H where H is a constant and lies in the range 0<H<1, with a Gaussian distribution (Yokoya et al., 1989; Polidori and Chorowizc, 1993). The relationship between parameter H and fractal dimension D is in Equation (1).

D=3-H (1)

Parameter H can be determined by fitting a straight line with Equation (2):

Log(E[|Z(X+dx)-Z(X)|])=H.Log(dx)+K (2)

Where both H and K are constant. Equation (2) indicates that a plot of (E[|Z(X+dx)-Z(X)|]) as a function of dx on a log-log plot lies on a straight line and its slope is H (Yokaya et al., 1989).

Let s2 be the variance of grayscale differences over a distance dx that equals one pixel in the input image. If the straight line are extended toward a shorter distance, the variance
s21/2 over a distance dx equal to