Use of the compactly supported B-spline wavelet of Chui and Wang is hindered by loss of accuracy on decomposition, through truncation of weight sequences which are countably infinite. Adaptations to finite intervals often encounter significant problems with error near boundaries, called edge effects. For multiresolution analysis on a finite interval which employ the piecewise linear B-wavelet the present research provides a frontal approach to decomposition which avoids truncation of weight sequences, experiences no error at boundaries, and which exhibits a factor of three increase in computational efficiency, over the usual approach characterized by truncation of infinite weight sequences. As a further modest contribution, a simple derivation of the piecewise linear B-spline wavelet for $L\sb2(R)$ is given. The simple technique is then applied to the derivation of supplementary boundary wavelets, which are necessary in order to complete the piecewise linear B-wavelet basis on a finite interval. There is also presented a modification to the Chui and Quak piecewise-cubic spline multiresolution analysis for the finite interval. The modification is intended to simplify implementation. Boundary scaling functions with multiple nodes at interval endpoints are rejected, in favor of the classical B-spline scaling function restricted to the interval. This necessitates derivation of revised boundary wavelets. In addition, a direct method of decomposition results in significant bandwidth reduction on solving an associated linear systems. Image distortion is reduced by employing natural spline projection. Finally, a hybrid projection scheme is proposed, which particularly for large systems further lowers operation count. Numerical experiments which try the algorithm are performed: The problems of edge detection, data compression, and data smoothing by thresholding in the wavelet transform domain are examined. The cubic B-spline wavelet yields compression ratios as high as 40 to 1 in the numerical experiments.