The problem of reconstructing an image from nonuniformly spaced, spatial point measurements is frequently encountered in bioimaging and other scientific disciplines. The most successful class of methods in handling this problem uses the regularization approach involving the minimization of a derivative-based roughness functional. It has been well demonstrated, in the presence of noise, that nonquadratic roughness functionals such as ℓ1 measure yield better performance compared to the quadratic ones in inverse problems in general and in deconvolution in particular. However, for the present problem, all well-evaluated methods use quadratic roughness measures; indeed, ℓ1 performs worse than the quadratic roughness when the sampling density is low. This is due to the fact that the mutual incoherence between the measurement operator (dirac-delta) and the regularization operator (derivative) is low in the present problem. Here we develop a new multiresolution-based roughness functional that performs better than ℓ1 and quadratic functionals under a wide range of sampling densities. We also propose an efficient iterative method for minimizing the resulting cost function. We demonstrate the superiority of the proposed regularization functional in the context of reconstructing full images from nonuniformly undersampled data obtained from a confocal microscope.