D1. Knowledge and understanding

Students will learn what are images, and how they are acquired, stored, modified and analysed. They will also acquire knowledge on a selection of mathematical, physical and numerical concepts and methods frequently used in imaging applications.

D2. Applying knowledge and understanding

By the end of the course, students will be able to apply the methods learned in class to a broad class of imaging problems.

D3. Making judgments

By the end of the course, students will be able to identify the techniques and concepts best adapted to a given imaging problem. Students will also have built an awareness of the limitations of these techniques, and thus apply critical thinking in the interpretation of imaging applications.

D4. Communication skills

By the end of the course, students will be able to articulate and describe imaging methods and processes with appropriate terminology, and discuss these methods and processes with experts in the field across many disciplines.

D5. Learning skills

By the end of the course, students will have developed problem solving and critical thinking abilities for rigorous interpretation, analysis, and processing of imaging procedures. Students will also be able to deepen independently their knowledge and competences on more advanced imaging concepts and techniques.

Basic mathematical proficiency in linear algebra, Fourier analysis, functional analysis and differential equations. Previous experience with computer programming is useful but not essential.

IMAGE PROCESSING IN SPATIAL DOMAIN. Sampling, quantization, field of view, dynamic range; affine coordinate transforms, basic intensity transforms; spatial filtering, convolution, correlation; morphological operations, segmentation. IMAGE PROCESSING IN FOURIER DOMAIN. properties of Fourier transforms (linearity, scaling, shifting, convolution, .); special functions (Gaussian, Dirac delta, Dirac comb, rect, sinc); Fourier transform, Fourier series, discrete Fourier transform; sampling, aliasing, ringing, Nyquist criterion; frequency filtering & edge detection, connection to spatial filtering via convolution theorem. SAMPLING, INTERPOLATION AND PIXEL REPRESENTATIONS. windowing, zero-padding; up/down-sampling, aliasing; sampling patterns and pixel basis functions; interpolation via convolution, interpolation condition; nearest neighbour, linear, splines, sinc interpolation kernels; Whittaker-Shannon interpolation and ideal signal reconstruction. IMAGE REPRESENTATIONS. Orthonormal bases; windowed Fourier transform; window functions, uncertainty, leakage, tiling in time-frequency plane; Wavelet transform, mother wavelets, multi-resolution analysis; Dictionaries & atoms. CHARACTERIZATION OF DETECTION SYSTEMS. Linear Systems, impulse response, PSF; frequency formulation, OTF, MTF, PTF; resolution, numerical aperture, Rayleigh limit, FWHM, Strehl ratio; random variables, density functions; correlation vs. convolution; NPS, Noise colours; signal power, noise power; deconvolution problem, Wiener deconvolution. IMAGING SYSTEMS AND WAVE PROPAGATION. Helmholtz equation, diffraction integral; impulse response, Huygens principle; near-field/Fresnel & far-field/Fraunhofer - approximations; Fresnel diffraction and propagation using convolution; Fraunhofer diffraction from an aperture; wavefront aberrations & Zernike polynomials. INTERFEROMETRIC IMAGING AND IMAGING OF FAR-FIELD FOURIER AMPLITUDES. Interferometry with reference wave, on-axis Holography, twin image problem, off-axis holography, Fourier holography; fringe interferometry, structured light, fringe scanning/phase-stepping techniques; X-ray grating interferometry; phase wrapping and unwrapping; wavefront sensing, phase problem; phase retrieval; coherent diffraction imaging, ptychography; crystallography. TOMOGRAPHY. Reconstruction from projections, radon transform, sinogram; Fourier slice theorem, filtered back-projection; algebraic formulation, system matrix; inverse problems, constraint sets, iterative solutions. LEAST SQUARES OPTIMIZATION. Estimation and least squares principle; cost function, iterative gradient-based solution, convex optimization; weighted least squares; iterative methods (algebraic reconstruction technique). CONSTRAINED OPTIMIZATION & MAXIMUM LIKELIHOOD OPTIMIZATION. cost function, Lagrange multipliers, data consistency, regularizations; denoising using Tikhonov regularization; denoising using total variation regularization; the Lp-norm and convex optimization; probability vs. likelihood, log-likelihood, maximum likelihood optimization; Bayes theorem, prior, posterior; image classification.

●Gonzalez, R. C. & Woods, R. E. (2007) Digital Image Processing (3rd Edition). Prentice Hall. ISBN 978-0131-68728-8
● Chityala, R. & Pudipeddi, S. (2020) Image Processing and Acquisition using Python. Chapman and Hall. ISBN 978-0367-19808-4
● Various online resources

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New concepts and methods will be introduced during class lectures, along with demonstrations using python. Group discussions and real-time quizzes will be used as immediate feedback on student understanding. Weekly reading and exercise assignments will be provided to consolidate material seen in class, and to help develop image processing skills.

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Assessment includes an oral test in which open-ended questions are offered on the main topics of the course. This final assessment also includes an approximately 10-minute presentation in which the student will explain the results of a scientific publication selected from a list provided a few weeks earlier. During the open-ended portion of the exam, the student will be expected to demonstrate the ability to explain and articulate the key concepts of the course, and to know how to apply them in specific cases. The presentation will be evaluated on the clarity and correctness of the topic.

The overall score for the examination is given by a grade expressed in thirtieths. To pass the exam (18/30), the student must demonstrate sufficient knowledge of the topics listed in the content section. On the other hand, to achieve the maximum score (30/30 cum laude), the student must demonstrate excellent knowledge of all the topics covered in the course and demonstrate an excellent ability to apply the acquisitions in new problems.