The reconstruction problem is usually formulated as a variational problem in which one searches for that image that minimizes a so called prior (image model) while insisting on certain image features to be preserved. When the prior can be described by a norm induced by some inner product on a Hilbert space, the exact solution to the variational problem can be found by orthogonal projection. In previous work we considered the image as compactly supported in $\mathbb{L}_{2}(\mathbb{R}^{2})$ and we used Sobolev norms on the unbounded domain including a smoothing parameter γ>0 to tune the smoothness of the reconstructed image. Due to the assumption of compact support of the original image, components of the reconstructed image near the image boundary are too much penalized. Therefore, in this work we minimize Sobolev norms only on the actual image domain, yielding much better reconstructions (especially for γ≫0). As an example we apply our method to the reconstruction of singular points that are present in the scale space representation of an image.