Representative surface reconstruction algorithms taking a gradient field as input enforce the integrability constraint in a discrete manner. While enforcing integrability allows the subsequent integration to produce surface heights, existing algorithms have one or more of the following disadvantages: They can only handle dense per-pixel gradient fields, smooth out sharp features in a partially integrable field, or produce severe surface distortion in the results. In this paper, we present a method which does not enforce discrete integrability and reconstructs a 3D continuous surface from a gradient or a height field, or a combination of both, which can be dense or sparse. The key to our approach is the use of kernel basis functions, which transfer the continuous surface reconstruction problem into high-dimensional space, where a closed-form solution exists. By using the Gaussian kernel, we can derive a straightforward implementation which is able to produce results better than traditional techniques. In general, an important advantage of our kernel-based method is that the method does not suffer discretization and finite approximation, both of which lead to surface distortion, which is typical of Fourier or wavelet bases widely adopted by previous representative approaches. We perform comparisons with classical and recent methods on benchmark as well as challenging data sets to demonstrate that our method produces accurate surface reconstruction that preserves salient and sharp features. The source code and executable of the system are available for downloading.

Surface gradients are useful to surface reconstruction in single view modeling, shape-from-shading, and photometric stereo. Previous algorithms minimize a complex, nonlinear energy functional, or require dense surface gradients to perform integration to generate 3D locations, or require user-input heights to constrain the solution space, or produce severe distortion and smooth out surface details. Most single-view algorithms output a Monge patch (height-field), which may introduce further surface distortion along object silhouettes and surface orientation discontinuities. Our proposed algorithm operates on a single view of complete or incomplete data. The data can be gradients without 3D locations, or 3D locations without gradients. The output surface, which is not necessarily a height-field, preserves salient depth and orientation discontinuities. Experimental comparisons on both simple and complex data show that our method produces better surfaces with significantly less distortion and more details preserved. The implementation of our closed-form solution is very straightforward.