# 1. Introduction
### Content-based image retrieval (CBIR)

It processes the information contained in image data and creates an abstraction of its content in terms of visual attributes.

#### Efficient space features
* identfiability
* translation, scale and rotation invariance
* affine invariance : even when we apply transformation, scaling or rotation, we preserve the collinearity and lines.
* noise resistance
* occultation invariance
* statistically independent
* reliable

#### Shape descriptor is some set of numbers that are produced to describe a given shape feature.

* the descriptors should be as complete as possible to represent the content of the information items.
* the descriptors should be represented and stored compactly. The size of descriptor vector must not be too large.
* the computation of distance between descriptors should be simple; otherwise the execution time would be too long.

#### Shape feature extraction and representation plays an important role in the following categories of applications:

* shape retrieval: searching for all shapes in a typically large database of shapes that are similar to a query shape
* shape recognition and classification: determining whether a given shape matches a model sufficiently, or which of representative class is the most similar.
* shape alignment and registration: transforming or translating one shape so that it best matches another shape, in whole or in part.
* shape approximation and simplification: constructing a shape of fewer elements (points, segments, triangles, etc.),that is still similar to the original.

# 2. Shape Parameters

Shape-based image retrieval consists of the measuring of similarity between shapes represented by their features.

### Center of Gravity
The center of gravity is also called centroid. Its position should be fixed in relation to the shape, no matter how the points distribution is.

### Axis of least inertia
The axis of least inertia is unique to the shape. It serves as a unique reference line to preserve the orientation of the shape. The axis of least inertia (ALI) of a shape is defined as the line for which the integral of the square of the distances to points on the shape boundary is a minimum.

### Average bending energy
Circle has minimum of it.

### Eccentricity
Eccentricity is the measure of aspect ratio. It is the ratio of the length of major axis to the length of minor axis. It can be calculated by principal axes method or minimum bounding rectangle method.

#### Minimum bounding rectangle
Minimum bounding rectangle is also called minimum bounding box. It is the smallest rectangle that contains every
point in the shape

Elo (Elongation) = 1 − W/L

#### Circularity ratio
Circularity ratio represents how a shape is similar to a circle.

#### Ellipse variance
Ellipse variance Eva is a mapping error of a shape to fit an ellipse that has an equal covariance matrix as the shape:

#### Rectangularity
Rectangularity represents how rectangular a shape is, i.e. how much it fills its minimum bounding rectangle:

#### Convexity
Convexity is defined as the ratio of perimeters of the convex hull O over that of the original contour O.

#### Solidity
Solidity describes the extent to which the shape is convex or concave.

#### Euler number
Euler number describes the relation(by subtracting) between the number of contiguous parts and the number of holes on a shape.

#### Profiles
The profiles are the projection of the shape to x-axis and y-axis on Cartesian coordinate system.

#### Hole area ratio
Hole area ratio HAR is defined as
HAR = Ah/As

# 7. Scale space approaches

A curve is embedded into a continuous family of gradually simplified versions.

Main idea of scale spaces is that the original curve should get more and more simplified, and so small structures should vanish as parameter sigma increases.

This is to separate small details from relevant shape properties as the segments of curves which are neglected contains information which are not that relavent.

So basically it is used for smoothing and elimination of small details.

## Curvature scale-space

The Curvature Scale Space . It is a mapping of the image of the object from three dimansional space to a space which represents each point as a curvature w.r.t. the arclength.

(From a paper)
CSS plot of an image is itself a unique representation of an image . Instead of matching images , matching their CSS plots is more computationally efficient and accurate.

Drawbacks are that only a single object should be on the image , it should form a closed contour and the foreground and background should have a definite intensity difference.Thus we are concentrating on the boundary values of the image only and the internal features are ignored . This resticts the usefulness of the project to some extent.

## Intersection points map
When two or more features intersect in a map, the location of the intersection is often important and may need to be captured. ... To create the points at intersections, there must be a clear point at which the lines intersect other lines or the boundary of a polygon.

Instead of characterizing the curve with its curvature involving 2nd order derivatives, it uses the intersection points between the smoothed curve and the original.

As the standard deviation of the Gaussian kernel increases, the number of the intersection points decreases.

By analyzing these remaining points, features for a pattern can be defined.
Since this method deals only with curve smoothing, it needs only the convolution operation in the smoothing process.

So this method is faster than the CSS one with equivalent performances.

The main weakness of this approach is that it fails to handle occulted contours and
those having undergone a non-rigid deformation.

# 8. Shape transform domains
Methods which are formed by the transform of the detected object or the transform
of the whole image.

Transforms can therefore be used to characterize the appearance of images.

The shape feature is represented by the all or partial coefficients of a transform.

(An image transform can be applied to an image to convert it from one domain to another. Viewing an image in domains such as frequency or Hough space enables the identification of features that may not be as easily detected in the spatial domain)

## Fourier descriptors
Fourier descriptors A method used in object recognition and image processing to represent the boundary shape of a segment in an image.
(The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. ... The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.)

## One-dimensional Fourier descriptors
Fourier descriptor (FD) is obtained by applying Fourier transform on a shape signature that is a one dimensional function which is derived from shape boundary coordinates (cf. Section 3).

The normalized Fourier trans-
formed coefficients are called the Fourier descriptor of the shape.

FD derived from different signatures has signifficant different performance on shape retrieval.

One-dimensional FD has several nice characteristics such as simple derivation, simple normalization and simple to
do matching.

## Region-based Fourier descriptor
The region-based FD is referred to as generic FD (GFD)

Basically, GFD is derived by applying a modified polar Fourier transform (MPFT) on shape image.

In order to apply MPFT, the polar shape image is treated as a normal rectangular image and three steps are repeated till the image is rotated by 360°.

The result of these steps is that an image in polar space plots into Cartesian space.
Figure 36 shows the polar shape image turning into normal rectangular image.

The Fourier transform is acquired by applying a discrete 2D Fourier transform on this shape image.

## Wavelet transform
Using the wavelet transform, a hierarchical planar curve descriptor is developed.

This descriptor decomposes a curve into components of different scales so that the coarsest scale components carry the global approximation information while the finer scale components contain the local detailed information.

(When you reduce an image, you get an image at a coarser scale, while the original is the finer scale: because in the big image there are fine details and in the small image, only coarser details.)

The wavelet descriptor has many desirable properties such as multi-resolution representation, invariance, uniqueness, stability, and spatial localization.

## Angular radial transformation
The angular radial transform (ART) is a moment-based image description method adopted in MPEG-7 as a 2D region-based shape descriptor.

## Shape signature harmonic embedding

## R-Transform
The R-Transform to represent a shape is based on the Radon transform.

Given a large collection of shapes, one R-transform per shape is not efficient to distinguish from the others because the R-transform provides a highly compact shape representation. In this perspective, to improve the description, each shape is projected in the Radon space for different segmentation levels of the Chamfer distance transform.

(The Chamfer distance between U and V is given by the average of distances between each point ui∈U, n(U) = n and its nearest edge in V, Here n is the number of points in U. With the use of a distance function, it is possible to reduce the cost function, so that it can be evaluated in linear time)

## Shapelets descriptor
Shapelets descriptor was proposed to present a model for animate shapes and extracting meaningful parts of objects.

The model assumes that 2D simple closed curves are formed by a linear superposition of a number of
shape bases.

## Conclution
Extracting a shape feature in accordance with human perception is not an easy task. Due to the fact that human vision and perception are an extraordinary complicated system.

In addition, choosing appropriate features for a shape recognition system must consider what kinds of features are suitable for the task.

There exists no general feature which would work best for every kind of images.

## Comparison table for shape transform domain
Reconstructure-

Invariance- The property is unchanged regardless of changes in the conditions of measurement.

Affine Transformation helps to modify the geometric structure of the image, preserving parallelism of lines but not the lengths and angles.