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Real-world 3D models are primarily acquired as large unorganized discrete point sets with attributes. In this paper, we show how to reconstruct multi-scale implicit surfaces with attributes, given those discrete point sets with attributes. In a preprocess, we first subdivide the global domain into overlapping local subdomains by computing a perfectly balanced binary tree. Second, we compute subsets of the points for the inner nodes of the tree for the intermediate resolutions bottom-up by using data thinning algorithms. Third, we reconstruct the surface parts in all nodes of the binary tree from the non-disjunct subsets of the points by using variational techniques with radial basis functions. For the evaluation of the defining function of the implicit surface at the desired scale, we hierarchically blend together the surface parts of the inner nodes by using a family of functions called partition of unity. Our new reconstruction method is particularly robust since the number of data points in the partition of unity blending zones can be specified explicitly. We show how our reconstruction method can also be applied to reconstruct continuous functions for the surface's attributes. In a short discussion, we evaluate the advantages and drawbacks of our reconstruction method compared to existing reconstruction methods for implicit surfaces.