Lookup-Table-Based Gradient Field Reconstruction

Abstract

In computer vision there are many applications where it is advantageous to process an image in the gradient domain and then re-integrate 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 re-integrating a non-integrable field. Artefacts can be diminished, but not removed, by using complex to highly complex re-integration 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) 1D gradient field; for many applications the derived gradient field will be non-integrable. 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, then using the Lookup- Table (LUT) constraint we effectively derive the mapping from the multichannel input to the desired, re-integrated, 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 colour or higher-D image into greyscale. We will show, through extensive experiments and complementary theoretical arguments, that our straightforward method preserves the target contrast as well as do complex previous re-integration methods but without artefacts, and with a 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.1

Colour Image Gradient Regression Reintegration

“…In the first place, the pairs of resulting fields (p, q) in each colour channel are only approximately a gradient: one must decide on a reintegration scheme to go back from the domain of derivatives to that of images. A typical choice would be some form of Poisson solver, or other schemas (see [9]). However a more fundamental problem arises in that the time-complexity of the median operation in [1] is high, typically 30% of the CPU cycles of the complete Weiss algorithm [1] including Poisson reintegration in the Fourier domain [3], with overall clocktime of 60s for a 16-member colour image sequence of resolution 1704 × 2272 (on a 3.40 GHz PC with 8.00 GB and an i7-2600 CPU, in unoptimized Matlab).…”

“…One simple idea might be to run the algorithm on a thumbnail image, but then we would obtain a small and unusable reflectance image. In the next section, we show how to bring an innovative idea to bear (in the same flavour as that in [9]) so as to keep to a thumbnail for the reflectance calculation, but still make the algorithm work properly on a full-size image. The novel approach here is to replace reintegration itself by a much simpler method: regression on gradients.…”

“…A conceptually very different yet straightforward approach is developed in [9]. We briefly recapitulate this method since it inspires the method set out here.…”

“…In a clever trick [9], this also means that using the same regression coefficients found from the regression of gradient components, we also arrive at a regression for the reintegrated grey image g from the regression coefficients times the polynomial colour components themselves, not the gradients. I.e., we arrive at not a gradient but an integrated image R. The result is an output image which successfully avoids artifacts [9]. This simple but powerful reintegration method can be applied in a variety of image fusion scenarios, such as converting satellite imaging to RGB [10].…”

Fisher Information and the Combination of RGB Channels

“…Combining the information from several color channels into one scalar value is one of the basic operations in color processing [1][2][3]. A typical example is the separation of the color information into a scalar-valued intensity and a vectorvalued chromaticity component.…”