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Detecting Curved Symmetric Parts using a Deformable Disc Model

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Detecting Curved Symmetric Parts using a Deformable Disc Model. Tom Sie Ho Lee, University of Toronto Sanja Fidler, TTI Chicago Sven Dickinson, University of Toronto. Overview. Motivation. Robustness to taper. Symmetric part detection as sequence finding.

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Detecting Curved Symmetric Parts using a Deformable Disc Model

Tom Sie Ho Lee, University of Toronto Sanja Fidler, TTI Chicago Sven Dickinson, University of Toronto



Robustness to taper

Symmetric part detection as sequence finding

  • Recovering an object’s generic part structure is a key step in bottom-up object categorization
  • Symmetry has formed the basis of many 2D and 3D generic part representations, e.g. skeletons, shock graphs, generalized cylinders, geons
  • Our goal is to detect 2D symmetric parts in a cluttered image
  • Good symmetry follows a curvilinear axis
  • We find high-affinity sequences of disc hypotheses
  • Optimal sequences are computed using dynamic programming
  • Our sequence formulation avoids branching clusters
  • Tapered parts vary in scale along the axis
  • We allow parts to be composed along the axis from disc hypotheses of different scales

Symmetric parts of objects detected by our method

Robustness to curvature

Related work

  • Symmetric parts are often curved
  • We capture curvature explicitly and recover the part in one piece
  • Classical skeletons [Blum ‘67; Brady ‘84] are inapplicable to cluttered scenes
  • Filter-based approaches require reliable templates
  • Contour-based approaches require quadratic grouping complexity
  • Region-based grouping [Levinshtein et al. ‘09] offers a good alternative
  • We demonstrate an improvement on [Levinshtein et al.] by capturing more shape variability and applying an optimal algorithm


[Levinshtein et al.]


[Levinshtein et al.]

[Levinshtein et al.]


1. Representing symmetric parts

3. Finding sequences


Deformable discs

Weizmann Horse Database (WHD)

Berkeley Segmentation Database (BSDS)

Disc hypothesis graph

  • 81 images of horses
  • Manually annotated symmetric parts as groundtruth regions [Levinshtein et al.]
  • Count a hit when IoU-overlap > 40% between groundtruth and detected regions
  • The medial axis transform [Blum] decomposes a shape into the locus of maximal inscribed discs
  • We define a symmetric part as a sequence of deformable discs
  • Discs deform to the shape’s boundary while remaining compact
  • Organize deformable disc hypotheses into graph
  • Place edges between adjacent or overlapping discs
  • Find disc sequences in the graph with high affinity
  • Source of images of diverse objects on cluttered backgrounds
  • We manually annotated symmetric parts on 36 selected images

Object part

Maximal discs

Superpixel approximation

Multi-scale composition

Superpixel approximation

  • Use compact superpixels as deformable disc hypotheses
  • Superpixels from different scales compose a single part

disc hypothesis graph

Cost of a disc sequence



  • Score a sequence P = (d0, …, dn) in terms of local affinities σ(di-1,di) and σ(di-1,di,di+1)
  • Affinities favor local grouping of adjacent discs
  • Ternary affinities favor curvilinear axis (smoothness)
  • Convert affinities into binary costs {si-1,i = 1 - σ(di-1,di)} and ternary costs {ti-1,i,i+1 = 1 - σ(di-1,di,di+1)}

2. Deformable Disc Affinity

Deformable ellipse

Overview of affinity

  • Define affinity between adjacent disc hypotheses
  • High affinity reflects non-accidental symmetry
  • Adjacent discs occupy a region r on which to extract features
  • Train affinity on region symmetry features

Qualitative results

Ellipse fitted to a region hypothesis

  • Cost is normalized by number of discs in sequence
  • Growth term A favors longer sequences

Deformation-invariant space

Our ellipse is parameterized by bending and tapering, axis scaling, and rigid transformations

  • Evaluate symmetry in a warped space invariant to bending and tapering deformations
  • Determine warp by fitting a deformable ellipse to region
  • Extract spatial histogram of boundary edgels
  • Extract interior color and texture features

Finding the optimal sequence

boundary edgels

  • cost(P) can be globally minimized using dynamic programming
  • We use the algorithm of [Felzenszwalb & McAllester ‘06] to compute the global minimum P*

Find parameters w that locally minimize non-linear least squares:

  • Perform best-first search using priority queue of candidate sequences
  • Dequeue candidate sequences and consider possible extensions
  • Repeat minimization to find multiple symmetric parts

Affinity training

  • Learn to map a region r to its affinity σ(r)
  • Generate positive and negative training regions from annotated dataset
  • Extract features on each training region
  • Fit logistic regressor σ(r) to training examples


Candidate sequence extensions

under consideration

  • Symmetric part detector trained on horse images generalizes to diverse objects
  • Symmetry is a powerful and ubiquitous shape regularity

warp W

spatial histogram