Digital Image Processing, Computer Sciencer Science

Fourier Descriptor And Moments, Computer Science

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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