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In computer vision, there are many applications, where it is advantageous to process an image in the gradient domain and then reintegrate the gradient field: important examples include shadow removal, lightness calculation, and data fusion. A serious problem with this approach is that the reconstruction step often introduces artefacts-commonly, smoothed and smeared edges-to the recovered image. This is a result of the inherent ill-posedness of reintegrating a nonintegrable field. Artefacts can be diminished but not removed, by using complex to highly complex reintegration techniques. Here, we present a remarkably simple (and on the face of it naive) algorithm for reconstructing gradient fields. Suppose we start with a multichannel original, and from it derive a (possibly one of many) 1-D gradient field; for many applications, the derived gradient field will be nonintegrable. Here, we propose a lookup-table-based map relating the multichannel original to a reconstructed scalar output image, whose gradient best matches the target gradient field. The idea, at base, is that if we learn how to map the gradients of the multichannel original onto the desired output gradient, and then using the lookup table (LUT) constraint, we effectively derive the mapping from the multichannel input to the desired, reintegrated, image output. While this map could take a variety of forms, here we derive the best map from the multichannel gradient as a (nonlinear) function of the input to each of the target scalar gradients. In this framework, reconstruction is a simple equation-solving exercise of low dimensionality. One obvious application of our method is to the image-fusion problem, e.g., the problem of converting a color or higher-D image into grayscale. We will show, through extensive experiments and complementary theoretical arguments, that our straightforward method preserves the target contrast as well as do complex previous reintegration methods, but without artefacts, and with - substantially cheaper computational cost. Finally, we demonstrate the generality of the method by applying it to gradient field reconstruction in an additional area, the shading recovery problem.