A wavelet based method for image reconstruction from gradient data with applications

In this paper, an algorithm for image reconstruction from gradient data based on the Haar wavelet decomposition is proposed. The proposed algorithm has two main stages. First, the Haar decomposition of the image to be reconstructed is obtained from the given gradient data set. Then, the Haar wavelet synthesis is employed to produce the image. The proposed algorithm is based on the relationship between the Haar analysis and synthesis filters and the model for the discretized gradient. The approach presented here is based on the one by Hampton et al. (IEEE J Sel Top Signal Process 2(5):781–792, 2008) for wavefront reconstruction in adaptive optics. The main strength of the proposed algorithm lies in its multiresolution nature, which allows efficient processing in the wavelet domain with complexity $${\fancyscript{O}}(N)$$O(N). In addition, obtaining the wavelet decomposition of the image to be reconstructed provides the possibility for further enhancements of the image, such as denoising or smoothing via iterative Poisson solvers at each resolution during Haar synthesis. To evaluate the performance of the proposed algorithm, it is applied to reconstruct ten standard test images. Experiments demonstrate that the algorithm yields results comparable in terms of solution accuracy to those produced by well-known benchmark algorithms. Further, experiments show that the proposed algorithm is suitable to be employed as a final step to reconstruct an image from a gradient data set, in applications such as image stitching or image morphing.