Many of the known approaches on plane detection and segmentation require either: a three-dimensional reconstruction, a rough calibration of the camera, a single plane dominant in the image, or to assume that the plane is mostly textured, constraining the range of application of those methods and failing in many situations [1]. One of the main problems is that planarity constraint alone may be poorly informative, in a statistic point of view, because coplanarity likelihoods are high only at high gradient regions.

In order to cope with this problem, our approach combines color information and planarity: when color information is added, joint planar and color likelihoods describe better the complete planar surfaces. The only required interaction in our method is the user marking some samples of coplanar regions on one of the images. We have chosen an assisted strategy to initialize our algorithm; however, it accepts any other initialization method. Once the coplanar regions are defined, fully automated image registration and segmentation are iteratively done.

For registration, our approach assumes that two-dimensional homographies relate coplanar features between both views and uses the standard constancy hypothesis for color intensities matching. The homography induced by each plane is computed by minimizing a function defined as the sum of squared differences of the image intensities. In our implementation we opted for using a Levenberg-Marquardt optimization technique and, in order to avoid converging to local minima, we use a multi-scale approach.

We estimate the likelihood of a pixel associated to a homography by the probability that the given pixel is translated at the second image to a pixel with similar color information. On the other hand, density distributions for color classes are empirically estimated by using a histogram technique. Then, we estimate the joint likelihood for planar and color information as the product of the homography likelihood and the color likelihood.

Given these likelihoods, the goal at the segmentation stage is to find a probability field indicating which model is supported for every pixel in the image. We use a reliable Bayesian technique named quadratic Markov measure fields (QMMF) [2], suitable for probability fields with low entropy. The optimal estimator for the probability measure field is found as the minimum of a potential function having the advantage of being quadratic, and the Gauss-Seidel method may be used for its minimization.

We selected stereo pair views showing special difficulties (untextured surfaces, different coplanar models with similar texture, or single surfaces containing several textures or colors) in order to show the efficiency of our approach. Although automatic planar segmentation methods have been found in the state of the art, interactivity may help to correct wrong segmentations and to reduce computational time from minutes [3] to seconds.

1

D. Scharstein, R. Szeliski, "A taxonomy and evaluation of dense two-frame stereo correspondence algorithm", International Journal of Computer Vision, 47:7-42, 2002. doi:10.1023/A:1014573219977

2

M. Rivera, O. Ocegueda, J.L. Marroquin, "Entropy-controlled quadratic Markov measure field models for efficient image segmentation", IEEE Trans. on Image Processing, 8(12):3047-3057, 2007. doi:10.1109/TIP.2007.909384

3

M. Lin, "Surfaces with Occlusions from Layered Stereo", Ph.D. thesis, Stanford University, USA, 2003.

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