Shape Descriptors - I

Shape Analysis Applications

(Image 6 Description): Six panels illustrating different shape analysis applications, with images and descriptions.

• Panel 1: Feature Detection

  • Image: A bicycle.
  • Text: “Find salient feature points”

• Panel 2: Correspondences

  • Image: Two figures interacting.
  • Text: “Find matching points between two shapes”

• Panel 3: Segmentation

  • Image: Human figures with body parts colored.
  • Text: “Break a shape into meaningful parts”
  • Legend:
    • head
    • torso
    • upper
    • lower
    • hand
    • upper
    • lower
    • foot

• Panel 4: Registration

  • Image: Two overlapping face scans.
  • Text: “Bring two or more shapes into pointwise alignment”

• Panel 5: Symmetry Detection

  • Image: A complex, symmetrical 3D model of a creature.
  • Text: “Find dominant symmetries of a shape”

• Panel 6: Labeling

  • Image: Human figures with body parts colored(same as panel 3).
  • Text: “Assign labels (“hand”, “wheel”, “wing”…) to segments”
  • Legend: * head * torso * upper * lower * hand * upper * lower * foot

Shape Analysis Applications

(Image 7 Description): Six panels illustrating different shape analysis applications.

• Panel 1: Retrieval

  • Image: Multiple horse shapes.
  • Text: “Find shapes matching the query shape”

• Panel 2: Classification

  • Image: A decision tree for classifying objects.
  • Text:“Find the category (“human”, “car”, “bird”) of a shape”
  • Category → Instance
    • Cereal → Chex, Bran Flakes
    • Apple
    • Stapler
    • Bowl → Striped Bowl, Blue Bowl

• Panel 3: Recognition

  • Image: various objects in the scene.
  • Text: “Find instances of a given shape in a scene”

• Panel 4: Clustering

  • Image: multiple bikes and cars.
  • Text: “Group shapes by similarity”
  • Text: “parameterized embedding”

Shape Descriptor

(Image/Diagram 8 Description): Illustrations relating to global and local shape descriptors.

• Definition: A set of numbers that describes a shape in a way that is:
  • Concise: A few numbers that capture the “essence” of the shape.
  • Quick to compute
  • Efficient to compare
  • Discriminative:
    • Different shapes have different descriptors.
    • Similar shapes have similar descriptors.
• Vector Space: Typically, the descriptors form a vector space with a meaningful distance metric.
• Global: Examples with hand, horse, pot, and camel silhouette.
• Local: Examples with camel.

Global and Local Descriptors

(Image/Diagram 9 Description): Visual examples and explanations of global and local descriptors.

• Global Descriptor:
  • Captures the structure of the entire shape.
  • Can tell different shapes apart.
  • Useful for retrieval, object recognition, etc.
  • (Image: Two very different vehicle silhouettes and a probability vs. distance graph.)
• Local Descriptor:
  • Captures the shape around a point.
  • Can tell different points apart.
  • Useful for segmentation, point correspondences, etc.
  • (Image: Two teapots and two camels with heatmaps.)
• Relationship: Each motivates the other: can modify any global descriptor to produce a local descriptor, and vice versa.

Local and Global - Applications

(Image 10 Description): Summary of applications, categorized by whether they typically use local or global descriptors.

• Local:
  • Feature detection
  • Correspondences
  • Registration
  • Symmetry detection
  • Segmentation
  • Labeling
• Global:
  • Retrieval
  • Classification
  • Recognition
  • Clustering

Local Descriptors

• Definition: Describes the shape in a neighborhood around a point.
  • Neighborhood may be surface-based or volume-based.
• Descriptors to be discussed:
  • Mean curvature
  • Shape diameter
  • Principal components
  • Average distance
  • Distance histogram

Curvature

(Image 12 Description): Two images of a 3D face, one colored according to Gaussian curvature and the other according to mean curvature.

• Gaussian Curvature: The product of the principal curvatures.
• Mean Curvature: The average of the principal curvatures.

Mean Curvature

• Computation: How can we (approximately) compute the mean curvature at a point?
• Two Approximations:
  • Average Projection: Average projection of neighboring points onto the normal vector. (Image: A curve with points and normal vectors, showing the projection.)
  • Fraction of Unit Ball: Fraction of a unit ball covered by the neighboring volume. (Image: A curve and two circles showing the fraction of area covered, one < 0.5 and one > 0.5.)

Shape Diameter

(Image 14 Description): Several examples of 3D shapes (horses and humans) colored according to their shape diameter function (SDF).

• Shape Diameter Function (SDF): Gives the “local thickness” of a shape at each point.

Shape Diameter

(Image 15 Description): Examples showing how to calculate the SDF, including a hand and several human figures.

• Calculation:
  • Shoot rays randomly sampled from a cone surrounding the inward normal.
  • SDF is the average distance (weighted by inverse angle) to the next intersection with the shape, after removing outliers.

Principal Components

(Image 16 Description): A diagram illustrating principal components on a 2D shape.

• Definition: The principal components (eigenvalues of the covariance matrix) of points in the neighborhood capture the directional variation of the shape.
• Interpretations:
  • One large principal component: line-like.
  • Two large principal components: surface-like.
  • Three large principal components: volume-like.

Distance-Based Descriptors

(Image 17 Description): A rabbit model. The first image shows connections from one point to several others. The second image is a color-mapped (composite plot) version of the rabbit.

• Average Distance: Average (geodesic or Euclidean) distance to all other points on the shape.
• Histogram of Distances: A more discriminative measure: plot a histogram of the distribution of distances.