Digital image processing is a vast field of applied mathematics that covers those processes whose inputs and outputs are images and those that extract attributes and patterns from images. In my thesis two different subcategories of digital image processing are investigated: pattern recognition and feature extraction, in particular the recognition of algebraic curves in images and edge detection techniques, and image compression, with particular attention to map-aided techniques. Patter recognition is the study of semi-automated and automated methods for the recognition of pattern and regularities in data. In the first part of my thesis, I present a novel method for the recognition of curvilinear profiles in digital images. The proposed method, semi-automatic for both closed and open planar profiles, is essentially based on a piecewise application of the Hough transform technique. The Hough transform is a known technique used in image analysis and digital image processing to recognize shapes in images. One of the drawbacks of this technique is the need to identify a potentially approximating family of curves before the recognition algorithm can be successfully applied. Thus, we developed an innovative procedure for the recognition of both closed and open curvilinear profiles in 2D digital images, without knowing neither a family of predefined curves nor a predefined look-up table of a prototypal shape. Our method provides a G1 continuous spline curve – eventually containing C0 junctions where cusps occur – which approximates the sought profile. Edge detection is a widely used tool in image processing with the aim of identifying abrupt changes or discontinuities in a digital image. In the second part of my thesis, I present two original edge detection methods, based on Radial Basis Functions interpolation. For the detection of jump discontinuities in 1D problems, we developed an iterative method based on interpolation with Variably Scaled Kernels (VSKs). This is shown to outperform an existing iterative edge detection method based on multiquadric radial basis function interpolation. To extend our purely one-dimensional edge detector to any dimension, we then introduce an innovative non iterative technique that detects edges by identifying the local maxima of the normalized absolute values of the RBF interpolant coefficients. The RBF interpolant is built-upon the compactly supported C2 Wendland function and exploits its advantageous properties to provide a robust and low-cost method. Numerical examples in 1D and 2D are included to illustrate its effectiveness and efficiency. Image compression is a specific type of data compression with the aim of reducing the amount of data necessary for image storage and transmission. Image compression has an increasingly important role in diverse applications, such as remote sensing, videoconferencing, medical imaging and many more. One of the classical approaches to image compression are multi-scale wavelet based methods. They do not always lead to fully satisfactory results as they do not adapt to the local structure of images, such as edges. Techniques to solve this drawback have been developed in recent works. Because of the need to locally adapt the compression methods to the geometry of image, feature extraction plays a significant role also in this case. In the last part of thesis, I present two original multi-scale image compression algorithms that are map-aided, to ensure a better faithfulness of the reconstruction to the original image. These methods use a prediction step with a multiquadric radial basis function interpolant and WENO scheme to determine the shape parameter. For the first method an edge detection procedure is applied to the original image, from this we obtain an edge map that determines the local prediction step. For the second method, instead, we compute different local reconstructions and we use a map to save the best one.

Digital image processing is a vast field of applied mathematics that covers those processes whose inputs and outputs are images and those that extract attributes and patterns from images. In my thesis two different subcategories of digital image processing are investigated: pattern recognition and feature extraction, in particular the recognition of algebraic curves in images and edge detection techniques, and image compression, with particular attention to map-aided techniques. Patter recognition is the study of semi-automated and automated methods for the recognition of pattern and regularities in data. In the first part of my thesis, I present a novel method for the recognition of curvilinear profiles in digital images. The proposed method, semi-automatic for both closed and open planar profiles, is essentially based on a piecewise application of the Hough transform technique. The Hough transform is a known technique used in image analysis and digital image processing to recognize shapes in images. One of the drawbacks of this technique is the need to identify a potentially approximating family of curves before the recognition algorithm can be successfully applied. Thus, we developed an innovative procedure for the recognition of both closed and open curvilinear profiles in 2D digital images, without knowing neither a family of predefined curves nor a predefined look-up table of a prototypal shape. Our method provides a G1 continuous spline curve – eventually containing C0 junctions where cusps occur – which approximates the sought profile. Edge detection is a widely used tool in image processing with the aim of identifying abrupt changes or discontinuities in a digital image. In the second part of my thesis, I present two original edge detection methods, based on Radial Basis Functions interpolation. For the detection of jump discontinuities in 1D problems, we developed an iterative method based on interpolation with Variably Scaled Kernels (VSKs). This is shown to outperform an existing iterative edge detection method based on multiquadric radial basis function interpolation. To extend our purely one-dimensional edge detector to any dimension, we then introduce an innovative non iterative technique that detects edges by identifying the local maxima of the normalized absolute values of the RBF interpolant coefficients. The RBF interpolant is built-upon the compactly supported C2 Wendland function and exploits its advantageous properties to provide a robust and low-cost method. Numerical examples in 1D and 2D are included to illustrate its effectiveness and efficiency. Image compression is a specific type of data compression with the aim of reducing the amount of data necessary for image storage and transmission. Image compression has an increasingly important role in diverse applications, such as remote sensing, videoconferencing, medical imaging and many more. One of the classical approaches to image compression are multi-scale wavelet based methods. They do not always lead to fully satisfactory results as they do not adapt to the local structure of images, such as edges. Techniques to solve this drawback have been developed in recent works. Because of the need to locally adapt the compression methods to the geometry of image, feature extraction plays a significant role also in this case. In the last part of thesis, I present two original multi-scale image compression algorithms that are map-aided, to ensure a better faithfulness of the reconstruction to the original image. These methods use a prediction step with a multiquadric radial basis function interpolant and WENO scheme to determine the shape parameter. For the first method an edge detection procedure is applied to the original image, from this we obtain an edge map that determines the local prediction step. For the second method, instead, we compute different local reconstructions and we use a map to save the best one.

Mathematical tools for images

SCHENONE, DANIELA
2018-12-14

Abstract

Digital image processing is a vast field of applied mathematics that covers those processes whose inputs and outputs are images and those that extract attributes and patterns from images. In my thesis two different subcategories of digital image processing are investigated: pattern recognition and feature extraction, in particular the recognition of algebraic curves in images and edge detection techniques, and image compression, with particular attention to map-aided techniques. Patter recognition is the study of semi-automated and automated methods for the recognition of pattern and regularities in data. In the first part of my thesis, I present a novel method for the recognition of curvilinear profiles in digital images. The proposed method, semi-automatic for both closed and open planar profiles, is essentially based on a piecewise application of the Hough transform technique. The Hough transform is a known technique used in image analysis and digital image processing to recognize shapes in images. One of the drawbacks of this technique is the need to identify a potentially approximating family of curves before the recognition algorithm can be successfully applied. Thus, we developed an innovative procedure for the recognition of both closed and open curvilinear profiles in 2D digital images, without knowing neither a family of predefined curves nor a predefined look-up table of a prototypal shape. Our method provides a G1 continuous spline curve – eventually containing C0 junctions where cusps occur – which approximates the sought profile. Edge detection is a widely used tool in image processing with the aim of identifying abrupt changes or discontinuities in a digital image. In the second part of my thesis, I present two original edge detection methods, based on Radial Basis Functions interpolation. For the detection of jump discontinuities in 1D problems, we developed an iterative method based on interpolation with Variably Scaled Kernels (VSKs). This is shown to outperform an existing iterative edge detection method based on multiquadric radial basis function interpolation. To extend our purely one-dimensional edge detector to any dimension, we then introduce an innovative non iterative technique that detects edges by identifying the local maxima of the normalized absolute values of the RBF interpolant coefficients. The RBF interpolant is built-upon the compactly supported C2 Wendland function and exploits its advantageous properties to provide a robust and low-cost method. Numerical examples in 1D and 2D are included to illustrate its effectiveness and efficiency. Image compression is a specific type of data compression with the aim of reducing the amount of data necessary for image storage and transmission. Image compression has an increasingly important role in diverse applications, such as remote sensing, videoconferencing, medical imaging and many more. One of the classical approaches to image compression are multi-scale wavelet based methods. They do not always lead to fully satisfactory results as they do not adapt to the local structure of images, such as edges. Techniques to solve this drawback have been developed in recent works. Because of the need to locally adapt the compression methods to the geometry of image, feature extraction plays a significant role also in this case. In the last part of thesis, I present two original multi-scale image compression algorithms that are map-aided, to ensure a better faithfulness of the reconstruction to the original image. These methods use a prediction step with a multiquadric radial basis function interpolant and WENO scheme to determine the shape parameter. For the first method an edge detection procedure is applied to the original image, from this we obtain an edge map that determines the local prediction step. For the second method, instead, we compute different local reconstructions and we use a map to save the best one.
14-dic-2018
Digital image processing is a vast field of applied mathematics that covers those processes whose inputs and outputs are images and those that extract attributes and patterns from images. In my thesis two different subcategories of digital image processing are investigated: pattern recognition and feature extraction, in particular the recognition of algebraic curves in images and edge detection techniques, and image compression, with particular attention to map-aided techniques. Patter recognition is the study of semi-automated and automated methods for the recognition of pattern and regularities in data. In the first part of my thesis, I present a novel method for the recognition of curvilinear profiles in digital images. The proposed method, semi-automatic for both closed and open planar profiles, is essentially based on a piecewise application of the Hough transform technique. The Hough transform is a known technique used in image analysis and digital image processing to recognize shapes in images. One of the drawbacks of this technique is the need to identify a potentially approximating family of curves before the recognition algorithm can be successfully applied. Thus, we developed an innovative procedure for the recognition of both closed and open curvilinear profiles in 2D digital images, without knowing neither a family of predefined curves nor a predefined look-up table of a prototypal shape. Our method provides a G1 continuous spline curve – eventually containing C0 junctions where cusps occur – which approximates the sought profile. Edge detection is a widely used tool in image processing with the aim of identifying abrupt changes or discontinuities in a digital image. In the second part of my thesis, I present two original edge detection methods, based on Radial Basis Functions interpolation. For the detection of jump discontinuities in 1D problems, we developed an iterative method based on interpolation with Variably Scaled Kernels (VSKs). This is shown to outperform an existing iterative edge detection method based on multiquadric radial basis function interpolation. To extend our purely one-dimensional edge detector to any dimension, we then introduce an innovative non iterative technique that detects edges by identifying the local maxima of the normalized absolute values of the RBF interpolant coefficients. The RBF interpolant is built-upon the compactly supported C2 Wendland function and exploits its advantageous properties to provide a robust and low-cost method. Numerical examples in 1D and 2D are included to illustrate its effectiveness and efficiency. Image compression is a specific type of data compression with the aim of reducing the amount of data necessary for image storage and transmission. Image compression has an increasingly important role in diverse applications, such as remote sensing, videoconferencing, medical imaging and many more. One of the classical approaches to image compression are multi-scale wavelet based methods. They do not always lead to fully satisfactory results as they do not adapt to the local structure of images, such as edges. Techniques to solve this drawback have been developed in recent works. Because of the need to locally adapt the compression methods to the geometry of image, feature extraction plays a significant role also in this case. In the last part of thesis, I present two original multi-scale image compression algorithms that are map-aided, to ensure a better faithfulness of the reconstruction to the original image. These methods use a prediction step with a multiquadric radial basis function interpolant and WENO scheme to determine the shape parameter. For the first method an edge detection procedure is applied to the original image, from this we obtain an edge map that determines the local prediction step. For the second method, instead, we compute different local reconstructions and we use a map to save the best one.
File in questo prodotto:
File Dimensione Formato  
schenone_thesis_onesided.pdf

accesso aperto

Descrizione: tesi di dottorato
Dimensione 11.29 MB
Formato Adobe PDF
11.29 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/1228782
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact