Distance Sets for Shape Filtersand Shape Recognition

Cosmin Grigorescu, Student Member, IEEE,

andNicolai Petkov

INTRODUCTION

When only the contours or even parts of the contours of the objects are left, like in Fig. 1(b) and (c), we are still able to discriminate between the two objects.

A. Distance Sets

Let be a set of perceptually significant points in an image, which we will call feature points.

Let be the distance between point and its-nearest neighbor from S, . We call the local descriptor :

the distance set, more precisely the N-distance set, of point to its first nearest neighbors within .

A. Distance Sets

The feature points in an image can be of different types.

In the character “ C”, for instance, one can consider the points at the extremities of the contour to be of an “end-of-line” type and all other points to be of another, “line” or “contour” type.

As a matter of fact, similar, evidently perceptually significant features are extracted in the visual cortex by so-called simple and complex cells (for lines and edges) , and end stopped cells.

A. Distance Sets

Given two points and from two images and their associated distance sets and , we define the relative difference between the i-neighbor and j-neighbor distances of and , respectively, as

A. Distance Sets

In this way, the quantity indicates how dissimilar two points and are with respect to their distances to feature points in their respective neighborhoods.

A. Distance Sets

A. Distance Sets -Example 1

A. Distance Sets -Example 2

A weaker requirement , can be imposed on the points of

B. Labeled Distance Sets

Let be the set of possible feature labels and let be one such label.

We define the labeled distance subset of a point p to its first neighbor feature points of type as follows:

B. Labeled Distance Sets

Let be the dissimilarity between two labeled distance subsets of type computed according to (3).

B. Labeled Distance Sets

for each label type we can assign a different weight and define the dissimilarity between two labeled distance sets and associated with and as follows:

B. Labeled Distance Sets – Example 1

We assign two types of labels: a regular point, associated with a pixel coming from the contours of the letters, and an end-of-line point at the extremities of the contours.

SETS OF (LABELED) DISTANCE SETS (Definition)

SETS OF (LABELED) DISTANCE SETS -Example

Dissimilarity Between Sets of (Labeled) Distance Sets

Given two sets of points Let be a one-to-one mapping from

and let be the set of all such mappings. Let be a positive constant.

Dissimilarity Between Sets of (Labeled) Distance Sets

Finally, we define the dissimilarity between the sets of distance sets S1 and S2 as follows:

Dissimilarity Between Sets of (Labeled) Distance Sets

Similarly, in the case of sets of labeled distance sets,and the cost of a mapping , is defined as

Dissimilarity Between Sets of (Labeled) Distance Sets

Orientation and Scale Invariance

Unless orientation is involved in the definition and extraction of features, a set of distance sets does not depend in any way on the orientation of an object in the 2-D plane:only distances matter.

scale invariance is obtained by resizing all objects to the same bounding box.

DISTANCE SET SHAPE FILTERS (Definition)

Let S1 be the set of all possible subsets of a reference object.

We define the operator associated with the set S1 and having as parameters the number of neighbors taken into account , and a threshold value as follows:

Example of Printed Character Recognition

DISTANCE SET SHAPE FILTERS (Definition)

If sets of labeled distance sets are used with being the set of possible labels, the operator is defined as follows:

Example of Printed Character Recognition

Application to Continuous Handwritten Text

Application to Continuous Handwritten Text

Finding Objects in Complex Scenes

We assigned a different label to each of the four orientations, and a labeled distance set was built for each point of the combined edge map by computing the distances to the first nearest neighbor points occurring in each oriented edge map, k=1…4 .

Finding Objects in Complex Scenes

SHAPE COMPARISON

SHAPE COMPARISON

For computing the cost of mappings, the value of the constant in (11) was chosenas the average pair-wise dissimilarity of two points:

SHAPE COMPARISON –Application to Handwritten Character Recognition

In a more elaborate example, a database of 286 characters, 11 characters for each of the 26 letters, handwritten by one person was used, Fig. 13.

SHAPE COMPARISON –Application to Object Recognition Based on 2-D Views

In the following, we evaluate the shape comparison procedure for recognition of objects based on their 2-D appearances.

Each object from this database is photographed from 72 different views around the object, corresponding to equally spaced 5 angular offsets

The number of neighbors taken into account within a distance set is , where are the numbers of feature points in two contour maps that are compared.

SHAPE COMPARISON –Application to MPEG-7 Shape Database Retrieval

The time needed for computing the dissimilarity between two objects, each being described by approximately 250 points and having associated distance sets of 100 distances, is about 0.7 s on a regular Pentium III/667 MHz computer.

With this experimental setup, we achieved an overall retrieval rate of 78.38%. Previous studies report 78.17% [21], 76.51% [20], 76.45% [62] and 75.44% [19].