1. Abstract We present a model of curvilinear grouping using piecewise linear representations of contours and a conditional random field to capture continuity and the frequency of different junction types. Potential completions are generated by building a constrained Delaunay triangulation (CDT) over the set of contours found by a local edge detector (Pb). Maximum likelihood parameters for the model are learned from human labeled ground-truth. Using held out test data, we measure how the model, by incorporating continuity structure, improves boundary detection over the local edge detector. We also compare performance with a baseline local classifier that operates on pairs of edgels. Both algorithms consistently dominate the low-level boundary detector at all thresholds. To our knowledge, this is the first time that curvilinear continuity has been shown quantitatively useful for a large variety of natural images. Better boundary detection has immediate application in the problem of object detection and recognition. Use P human the soft ground-truth label defined on CDT graphs: precision close to 100% Pb averaged over CDT edges: no worse than the original Pb Increase in asymptotic recall rate: completion of gradientless contours ImagePbLocalGlobal hierarchy of parts viewing distance Desired Properties: 1)Scale invariance. -Supported by natural image statistics (e.g. power law distributions) 2) Avoid too many spurious completions - Output should be better than the input! Scale-Invariant Contour Completion Using Condition Random Fields Xiaofeng Ren, Charless Fowlkes and Jitendra Malik, UC Berkeley “Bi-gram” model:  contrast + continuity  binary classification (0,0) vs (1,1) logistic classifier “Tri-gram” model: 11 22 LL  Pb L = Xe Pb > 0.2 CDT CDT edges capture most of the image boundaries Curvilinear continuity improves boundary detection A Local Classifier A Global Random Field: Contour Completion in Natural Scenes Our Solution: Trace detected edges, recursively split contours based on angle and generate potential completions using constrained Delaunay triangulation (CDT). Scale-invariant construction with a small number of potential completions. We evaluate the performance of local and global models on three different segmentation datasets. We project each CDT edge back down onto the pixel grid and measure the tradeoff between precision and recall of human marked boundaries. We find that the local model yields a gain in performance but is dominated by the CRF marginals. Our baseline continuity model uses the average contrast and angle between neighboring edges to estimate a posterior probability for each edge in the CDT graph independently. We also consider a conditional random field (CRF) with a binary random variable X e for each edge in the CDT. Singleton potentials incorporate the average contrast while junction potentials assign an energy to each possible configuration of edges incident on a vertex V. When only two edges are turned on, the junction potential also incorporates the angle between them. Low contrast boundaries Low contrast boundaries are included in potential completions! Maximum likelihood CRF parameters are fit via gradient descent. We use loopy belief propagation to perform inference, in particular estimating edge marginals P(X e ).