Intrinsic Images by Entropy Minimization

Graham D. Finlayson (1), Mark S. Drew (2), and Cheng Lu (2)
(1) School of Information Systems,
The University of East Anglia
Norwich, England NR4 7TJ

(2) School of Computing Science,
Simon Fraser University,
Vancouver, B.C. Canada V5A 1S6
{mark, clu}

Table of Contents

    Full text [.pdf]

    Video [.avi MPEG-4 codec V2]

Extra Images


A method was recently devised for the recovery of an invariant image from a 3-band colour image. The invariant image, originally 1D greyscale but here derived as a 2D chromaticity, is independent of lighting, and also has shading removed: it forms an intrinsic image that may be used as a guide in recovering colour images that are independent of illumination conditions. Invariance to illuminant colour and intensity means that such images are free of shadows, as well, to a good degree. The method devised finds an intrinsic reflectivity image based on assumptions of Lambertian reflectance, approximately Planckian lighting, and fairly narrowband camera sensors. Nevertheless, the method works well when these assumptions do not hold. A crucial piece of information is the angle for an "invariant direction" in a log-chromaticity space. To date, we have gleaned this information via a preliminary calibration routine, using the camera involved to capture images of a colour target under different lights. In this paper, we show that we can in fact dispense with the calibration step, by recognizing a simple but important fact: the correct projection is that which minimizes entropy in the resulting invariant image. To show that this must be the case we first consider synthetic images, and then apply the method to real images. We show that not only does a correct shadow-free image emerge, but also that the angle found agrees with that recovered from a calibration. As a result, we can find shadow-free images for images with unknown camera, and the method is applied successfully to remove shadows from unsourced imagery.


In this document, we present results generated using the min-entropy method outlined in the paper. The main result is the realization itself that entropy minimization can be used as a guiding principle for determining the correct illumination invariant characteristic direction. With this principle, the shadow-removal algorithm developed previously is extended to unsourced imagery, without any need to calibrate a camera.

In the file     extraimages.pdf, the table shows the input image, and then its L_1-based chromaticity 2D colour version. To find the min-entropy direction, we project as discussed in the paper into a 1D greyscale image. Calculating the entropy using the algorithm presented, which takes into account the nature of the data, we find the correct direction, which matches that found from an actual calibration in the case of a known camera.

For re-integrating into a full-colour image, two steps are required: finding a shadow-edge map and then growing partial-derivative edges across the shadow-edge followed by another derivative and re-integration in the Fourier domain.