CopubliShed by the ieee CS and the aip R e p r o d u c i b l e r e s e a r c h M assive computation is transforming science, as researchers from numerous fields launch ambitious projects involving large-scale computations. Emblems of our age include data mining for subtle patterns in vast data-• bases; and massive simulations of a physical system's com-• plete evolution repeated numerous times, as simulation parameters vary systematically.The traditional image of the scientist as a solitary person working in a laboratory with beakers and test tubes is long obsolete. The more accurate image-not yet well recognized-depicts a computer jockey working at all hours to launch experiments on computer servers. In fact, today's academic scientist likely has more in common with a large corporation's information technology manager than with a philosophy or English professor at the same university. A rapid transition is now under way-visible particularly over the past two decades-that will finish with computation as absolutely central to scientific enterprise. However, the transition is very far from completion. In fact, we believe that the dominant mode of scientific computing has already brought us to a state of crisis. The prevalence of very relaxed attitudes about communicating experimental details and validating results is causing a large and growing credibility gap. It's impossible to verify most of the results that computational scientists present at conferences and in papers. The crisisTo understand our claim, and the necessary response, we must look at the scientific process more broadly. Originally, there were two scientific methodological branches-deductive (for example, mathematics) and empirical (for example, statistical data analysis of controlled experiments). Many scientists accept computation (for example, large-scale simulation) as the third branch-some believe this shift has already occurred, as one can see in grant proposals, keynote speeches, and newsletter editorials. However, while computation is already

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere S 2 , the special orthogonal group SO(3), the positive definite matrices SP D(n), and the Grassmann manifolds G(n, k). The representations are based on the deployment of Deslauriers-Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the Exp and Log maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as S n−1 , SO(n), G(n, k), where the Exp and Log maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.