The document discusses Fourier descriptors (FDs) and moments in digital image processing, highlighting their roles in shape analysis and object recognition. FDs offer a compact representation of shapes based on their contours, while moments provide statistical descriptions of an object's shape and intensity distribution. Both methods are invariant to transformations like translation, rotation, and scaling, making them valuable for various applications such as image retrieval and classification.

1. DIGITAL IMAGE
PROCESSING
FOURIER DESCRIPTORS and
moments
Submitted by
Malarvizhi V
II MSc.Cs
Nahar Saraswathi College of Arts and
science

2. Introduction
Fourier Descriptors (FDs) are mathematical
representations of the shape of an object, derived
from the Fourier Transform of its boundary.
They are widely used in shape analysis,
object recognition, and pattern matching due to
their ability to succinctly represent complex
shapes and their invariance to transformations
such as scaling, translation, and rotation.

3. How FD’s Work
1. Contour Extraction
The first step in using FDs is extracting the contour or
boundary of the object from the image. This contour is typically
represented as a sequence of coordinate points, ((x_i, y_i)),
where (i) indexes the points along the boundary.
2. Complex Representation
The contour is then converted into a complex sequence (z(i)
= x_i + jy_i), where (j) is the imaginary unit. This complex
sequence allows the boundary to be treated as a signal that can be
analyzed using Fourier Transform.

4. Interesting facts
3. Fourier Transform
The Discrete Fourier Transform (DFT) is applied to the complex sequence (z(i)).
The result is a set of Fourier coefficients (Z(k)), where (k) represents the frequency
components. These coefficients are the Fourier Descriptors (FDs).
4. *Normalization (Invariance):*
To achieve invariance to scale, translation, and rotation, the FDs are often
normalized:
Translation Invariance: The first coefficient (Z(0)), which represents the DC
component or the centroid of the shape, is set to zero.
Scale Invariance: The FDs are divided by the magnitude of the first non-zero
coefficient.
Rotation Invariance: The phase information can be ignored or adjusted.
5. *Shape Reconstruction:*
The inverse Fourier Transform can be applied to a subset of the FDs to
reconstruct an approximation of the original shape. Using only the low-frequency
components provides a smoothed version of the shape, highlighting its general structure.

5. Application
1. Shape Description and Recognition:
FDs provide a compact and efficient representation of shape, which is
useful for comparing and recognizing objects based on their contours.
2. Object Classification:
By comparing the FDs of different objects, it's possible to classify them
into categories based on their shape.
3. Image Retrieval:
In content-based image retrieval, FDs can be used to search for images
containing objects with similar shapes.
4. Shape Matching:
FDs are used in matching algorithms to find objects in an image that
resemble a given template shape.

6. TITTLE 2:
MOMENTS

7. Moments are scalar quantities that provide a
statistical description of an object's shape and intensity
distribution in an image. Moments can be used to
characterize various properties of an object, such as its
area, center of mass, orientation, and eccentricity.
Different types of moments (e.g., geometric
moments, central moments, normalized moments, and
Zernike moments) are used for specific applications in
image analysis.
INTRODUCTION

8. Types Of Moments
1. Geometric Moments:
Geometric moments are the most basic form of moments, defined for an image (I(x, y)) as:
[
M_{pq} = sum_{x} sum_{y} x^p y^q I(x, y)
]
where (p) and (q) are the orders of the moments. The zeroth moment ((M_{00})) represents the
area (or sum of pixel intensities) of the object, while higher-order moments capture more complex features
of the shape.
2. Central Moments:
Central moments are calculated relative to the centroid (center of mass) of the object, which
removes the influence of translation:
[
mu_{pq} = sum_{x} sum_{y} (x - bar{x})^p (y - bar{y})^q I(x, y)
]
where (bar{x}) and (bar{y}) are the coordinates of the centroid. Central moments are invariant to
translation.

9. Continues..
3. Normalized Moments:
Normalized moments are obtained by scaling central
moments, making them invariant to scale and translation:
[
eta_{pq} = frac{mu_{pq}}{mu_{00}^{left(1 + frac{p+q}{2}
right)}}
]
These moments are particularly useful in comparing objects of
different sizes.
4. Zernike Moments:
Zernike moments are a type of orthogonal moments that are
rotation-invariant and defined on the unit disk. They are computed
using Zernike polynomials, which are complex-valued functions.
Zernike moments are used for applications requiring high accuracy in
shape representation, such as in medical imaging.

10. Applications
1.Shape Description and Recognition:
Moments provide a powerful way to describe the shape of
objects, which can be used to recognize and classify them.
2. Object Classification:
By analyzing the moments of different objects, one can classify
them into different categories based on their shape characteristics.
3. Image Segmentation:
Moments can be used to segment images by identifying regions with
specific shape or intensity characteristics.
4. Feature Extraction:
Moments are often used as features in machine learning algorithms
for tasks like object detection and recognition.

11. FS’s(VS)Moments
FDs are particularly suited for boundary-based shape analysis,
focusing on the contour of objects, making them ideal for shape
matching and retrieval.
Moments provide a broader description of the entire object, including
its interior intensity distribution, which is useful for tasks requiring
global shape characteristics.

12. Both FDs and certain moments (central and Zernike
moments) can be made invariant to transformations such
as translation, rotation, and scaling, making them robust
tools in image analysis.
Application Scenarios
FDs are typically used in scenarios where boundary
precision and compact representation are critical.
Moments are used in scenarios requiring detailed
analysis of object shape and intensity distribution, such as
medical imaging and object recognition.
Invariance

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