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Scale-Invariant Contour Completion Using Condition Random Fields. Xiaofeng Ren, Charless Fowlkes and Jitendra Malik, UC Berkeley. Abstract

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Scale-Invariant Contour Completion Using Condition Random Fields

Xiaofeng Ren, Charless Fowlkes and Jitendra Malik, UC Berkeley


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.

CDT edges capture most of the image boundaries

Contour Completion in Natural Scenes

  • Desired Properties:
  • 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!

Use Phuman the soft ground-truth label defined on CDT graphs: precision close to 100%

viewing distance

hierarchy of parts

Pb averaged over CDT edges: no worse than the original Pb

Increase in asymptotic recall rate: completion of gradientless contours

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.

Curvilinear continuity improves boundary detection

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.

A Local Classifier

“Bi-gram” model:

“Tri-gram” model:

binary classification

(0,0) vs (1,1)

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.










contrast + continuity 



logistic classifier

Low contrast


A Global Random Field:

We also consider a conditional random field (CRF) with a binary random variable Xe 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.

Pb > 0.2

Maximum likelihood CRF parameters are fit via gradient descent. We use loopy belief propagation to perform inference, in particular estimating edge marginals P(Xe).


Low contrast boundaries are included in potential completions!