In this paper, we present a new adaptive approach to the restoration of noisy blurred images which is particularly effective at producing sharp deconvolution while suppressing the noise in the flat regions of an image. This is accomplished through a multiscale Kalman smoothing filter applied to a prefiltered observed image in the discrete, separable, 2D wavelet domain. The prefiltering step involves a constrained least squares filter based on a very small choice for the regularization parameter, producing an under-regularized restored image. This leads to a reduction in the support of the required state vectors in the wavelet domain, and an improvement in the computational efficiency of the multiscale filter. The proposed method has the benefit that the majority of the regularization, or noise suppression, of the restoration is accomplished by the efficient multiscale filtering of wavelet detail coefficients ordered on quadtrees. Not only does this permit adaptivity to the local edge information in the image, but it leads to potential parallel implementation schemes. In particular, this method changes filter parameters depending on scale, local signal-to-noise ratio, and orientation. Because the wavelet detail coefficients are a manifestation of the multiscale edge information in an image, this algorithm may be viewed as an `edge-adaptive' multiscale restoration approach.