MATHEMATICAL IMAGE AND SIGNAL PROCESSING
3D GRAPHICS LAB (1st teaching unit of VIRTUAL REALITY PROGRAMMING)
Prof. Matteo Manganelli
(INF/01, 5 CFU)
1. Programming languages
- Review of C
- Introduction to Object Oriented Programmino with C++
- Data structures for Computer Graphics
- Examples and programming exercises.
2. OpenGL
- Fundamentals of Computer Graphics
- The graphics rendering pipeline
- Introduction to OpenGL
- Comparison of OpenGL and Direct3D
- The OpenGL Utility Toolkit (GLUT)
- Examples of OpenGL applications using C/C++
VIRTUAL REALITY (2nd teaching unit of VIRTUAL REALITY PROGRAMMING)
Prof. Matteo Manganelli
(INF/01, 5 CFU)
This course will teach computer graphics programming using the Open Source 3D engine OGRE:
- Review of Object Oriented Programming with C++
- Introduction to 3D engines
- OGRE design overview
- OGRE first steps
- Scene management
- Materials
- Resources management
- Animation
- Billboard and particles
- Ogre Tools
SIGNAL AND IMAGE CODING (1st teaching unit of IMAGE CODING AND
COMPRESSION)
Prof. Domenico Vitulano
(INF/01, 5 CFU)
-
Definition of Entropy
Lossless and lossy coding
Variable length codes: taxonomy of codes.
Noiseless source theorem
Fano Shannon code
Huffman code and its optimality
-
Variants of Huffman code
Arithmetic coding
Overview of universal coding: FGK scheme
Dictionary coding: LZ77, LZ78 and LZW
Lossy coding: transform based schemes
Theory of high bit rate coding
Quantization results and compandors
Audio signals: A-law and mu-law
Run-length 1-D and 2-D
Some results of the low bit rate coding
The problem of the optimal basis: Karhunen-Loeve
JPEG
Wavelets and zero-trees
Matlab implementation of Huffman and JPEG
FRACTAL AND IMAGE COMPRESSION (2nd teaching unit of IMAGE CODING
AND COMPRESSION)
Prof. Domenico Vitulano
(INF/01, 5 CFU)
-
Overview on fractals
Isometries on the real plane
Metric spaces, Cauchy and convergent sequences
Completeness of a metric space
Lipschitz and contractive functions and their properties
Banach theorem
Hausdorff distance, fractals space and theorem relative its completness
Theory of IFS
The Collage theorem
Some elements of statistics
Populations and samples
Propagation of an error on a derived of calculated value
Likelihood Method
Best fit: a general theory
Matlab implementation of the best fit
Linear prediction
Optimal predictor design
Delta modulation
DPCM
Fractal coding: PIFS
Matlab implementation of isometries, contractive functions and IFS, generations of some
fractals (Sierpinski triangle etc.)
COMPUTER VISION (1st teaching unit of MATHEMATICAL ANALYSIS FOR
IMAGES)
Prof. Riccardo March
(MAT/05, 5 CFU)
The problem of surface reconstruction from images. Optic and photometric principles of image
formation. Reflective properties of surfaces. Lambertian surfaces. Shape from shading. Smoothness
constrain on the reconstructed surface.
Boundary conditions: occlusions, stereographic projection. Mathematical model and algorithm by
Horn and Ikeuchi. Principles of stereo vision. The problem of identifying correspondent points.
Binocular disparity and epipolar geometry. Description of a mathematical model and an algorithm
for stereo reconstruction. The problem of optical flow: estimate of the velocity vector field of
moving objects from temporal sequences of images. Smoothness constraint on the optical flow.
Mathematical model and algorithm by Horn and Schunck. Unitary mathematical framework:
computer vision problems as ill-posed inverse problems. An outline of Tikhonov regularization
theory. Regularization with discontinuities: meaning of discontinuous solutions. Link between
occlusions and discontinuities. Blake and Zisserman’s approach. Detailed description of algorithms
for the numerical solution of the following problems: shape from shading, visible surface
reconstruction from pairs of stereo images, computation of optical flow.
3D ANIMATION (2nd teaching unit of 3D MODELING AND ANIMATION)
Prof. Alessandro Pandolfi
(INF/01, 5 CFU)
-
History of Animation  From stone age to 3D
Traditional 2D animation techniques
Occidental vs Japan animation
Comparison between 2D and 3D animation
Brief introduction to 3D software (Maya) • Basic interface
- Viewport navigation instruments
- Basic Animation tools
- Introduction to the Graph Editor\Dope Sheet
- Fcurve's manipulation
- Animation Workflow • Animator's role
- Planning animation
- Organized Key framing
• Animation Techniques • 12 Principles of Animation
- Squash & Stretch
- Anticipation
- Staging
- Straight Ahead Action vs Pose to Pose Action
- Follow Through and Overlapping Action
- Slow In and Slow Out or Eases
- Arcs
- Secondary Action
- Timing
- Exaggeration
- Solid Drawing
- Appeal
• Animating Forces , Internal and External Forces
- Inertia
- Weight
-
Energy level
Force Transmitted through Flexible Joint
Jointed Limbs
Progressive Break of Joints
• Posing & Shape
- Line of Action (Flow Lines)
- Balance
- Sense of Weight
- Defining the Character (Outer expression of Inner realities)
• Facial Animation  The Golden Triangle
- Facial Expressions at the posing stage
- Eyes movement
- Blinks
- Eye Darts
- Lip-Sync
• Basic Acting for Animators
WAVELETS AND SPLINES (1st teaching unit of WAVELETS: this unit course
covers, in alternate years, splines and/or wavelets.)
Prof. Carla Manni
(MAT/08, 7 CFU)
The course covers, in alternate years, splines and/or wavelets.
As for splines, it presents an introduction to construction and main properties
of splines and wavelets with special emphasis to their application in
imaging processing.
Topics: splines and B-splines, construction and main geometric properties.
The topics covered on wavelets are:
Wavelet transform. Multiresolution analysis: decomposition and reconstruction
algorithms.
Smooth, orthonormal and compactly supported wavelets. Spline wavelets.
Examples and applications.
FRAMES (2nd teaching unit of WAVELETS)
Prof. Sandra Saliani
(MAT/05, 5 CFU)
Riesz basis. Riesz basis of translates. Generalized multiresolution analysis (GMRA). Dual GMRA.
A discrete transform for biorthogonal Wavelets. Compactly supported biorthogonal Wavelets.
Frames in Hilbert spaces. Dual frames. Tight frames. The frame operator. Redundancy of frames.
Frames of translates. Frames of exponentials. Gabor frames in L2(R). Necessary and sufficient
conditions for Gabor frames. The representation of the Gabor frame operator. The Zak transform.
Applications of Gabor frames. Wavelet frames. Necessary and sufficient conditions for Wavelet
frames. Dyadic Wavelet frames. Frames multiresolution analysis.
Textboos:
-
D. Walnut, An Introduction to Wavelet Analysis, Birkhauser, Basel, 2001
online notes, Analisi di Fourier e trattamento numerico dei segnali
WAVELET LAB
Prof. Vittoria Bruni
(INF/01, 5 CFU)
- Introduction to Matlab
- Discrete Fourier Transform
- Comparison between the Short time Fourier Transform and the Continuous Wavelet Transform
- Subband coding and filter banks: Implementation of the DiscreteWavelet Transform (with and
without the Matlab wavelet toolbox)
-Wavelets and filters: vanishing moments, compact support
-Families of wavelets
-Criteria for the selection of an appropriate wavelet
-Characterization of singularities
Applications: Denoising (thresholding, shrinkage, ...), Compression (zero-tree, entropy coding,...),
Edge Detection (2D-WT and maxima of its modulus)
HARMONIC ANALYSIS
Prof. Massimo Picardello
(MAT/05, 6 CFU)
Review of orthonormal expansions in Hilbert space. Fourier series and their convergence in L2.
Criteria for pointwise and uniform convergence of Fourier series. The rate of decay of Fourier
coefficients. Gibbs phenomenon.
Convergence of approximate identities.
The Fourier transform and its properties. Rate of decay of Fourier transforms. The Plancherel
theorem and the inversion formula.
Convergence induced by families of semi-norms. The Schwartz class.
Outline of duality for Banach and Frechet spaces. Tempered distributions. Convergence in the
distribution sense for sequences and series of tempered distributions. Differentiation and
convolution of tempered distributions. Fourier transform of a tempered distribution.
The sampling process in terms of distributions. The Poisson inversion formula. Whittaker
expansions and the Shannon sampling theorem. Aliasing.
The Discrete Fourier Transform and its properties. The Fast Fourier Transform.
Textbook:
online notes, Analisi di Fourier e trattamento numerico dei segnali
H.Dym, H.P.McKean, Fourier Series and Integrals, Academic Press, New York, 1972
W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, New York, 1991
COMPUTER GRAPHICS (1st teaching unit of 3D RENDERING, for students
without previous experience of elementary computer graphics)
Prof. Massimo Picardello
(MAT/05, 5 CFU)
Color theory.
Perspective transformations: parallel and central projections, vanishing points, 3D hidden lines
removal.
Stereoscopic projections. Euler angles. Rotations in 3D. Quaternions.
Image sampling and aliasing; filtering.
Octrees.
Clipping in 3D. Image and object precision. Coherence. Hidden area removal: z-buffer. BSP. Scan
line. Warnock method. Atherton-Weiler method. Ray Tracing.
Rendering of 3D scenes: lights and euristic models of illumination. Lambert diffusion. Phong
model. Shading and interpolation. Physics-based models for surface diffusion and reflection.
Rendering with mapping: texture map, bump map, reflection map.
Methods for shadows: Williams, Atherton-Weiler-Greenberg.
Transparency: double z-buffer, Snell law for refraction.
Recursive Ray Tracing and methods for acceleration.
Radiosity. Form factors and their computation. Substructuring. Iterative solutions: Jacobi, GaussSeidel and Southwell relaxation. Progressing refinement. Hints of multi-pass methods.
Textbooks:
J.Foley, A. van Dam, S.Feiner, J.Hughes, Computer Graphics: Principles and Practice, 2nd
edition in C, Addison-Wesley, Reading (Mass.), 1996
online notes, Algoritmi e metodi analitici, numerici e statistici in Computer Graphics
GLOBAL ILLUMINATION (2nd teaching unit of 3D RENDERING)
Prof. Massimo Picardello
(MAT/05, 5 CFU)
Radiometry, photometry, bidirection reflectance distribution function, flux, radiance, the rendering
equation: hemisphere and area formulation.
Probabilistic methods for computing integrals: Monte-Carlo, samples with assigned distribution,
variance and variance-minimizing probability distribution. Variance reduction, stratified sampling.
Importance function and the Green kernel of the rendering operator: Global reflectance distribution
function.
Stochastic Ray Tracing and its variance.
Stochastic radiosity: stochastic Jacobi relaxation, discrete random walk methods, continuous
random walks (photon tracing), variance of all these methods.
Outline of multipass methods: final gathering, Metropolis light transport, Irradiance caching.
Photon mapping, photon density and radiance gathering via nearest neighbour photons, light
transports used in gathering
Textbooks:
Ph.Dupré, Ph.Bekaert, K.Bala, Advanced Global Illumination, A.K.Peters, Natick (Mass.),
2003
online notes, Algoritmi e metodi analitici, numerici e statistici in Computer Graphics
PHOTON MAP (in alternative to GLOBAL ILLUMINATION: this course deals with
the photon map method only, but requires autonomous computer coding, normally in
C++)
(MAT/08, 5 CFU)
Prof. Massimo Picardello
Radiometry, photometry, bidirection reflectance distribution function, flux, radiance, the rendering
equation: hemisphere and area formulation.
Probabilistic methods for computing integrals: Monte-Carlo, samples with assigned distribution,
variance and variance-minimizing probability distribution. Variance reduction, stratified sampling.
Photon mapping, photon density and radiance gathering via nearest neighbour photons, light
transports used in gathering.
Development of working computer code for rendering with the photon map method.
Textbook: H.W.Jensen, Photorealistic Image Rendering with Photon Map, A.K.Peters, Natick
(Mass.), 2002
TOMOGAPHY (1st teaching unit of IMAGE RECOGNITION AND
RECONSTRUCTION)
(MAT/05, 5 CFU)
Prof. Massimo Picardello
Review of Fourier transform in L2, Shannon sampling theorem, tempered distributions,
holomorphic functions.
Outline of the Hilbert transform.
Outline of Bessel functions and applications to diffraction tomology.
The Radon transform and the Fourier slice theorem. The dual operator: back projection. Examples
of inversion theorems of the Radon transform: the use of the Hilbert transform.
Textbooks:
A.C.KaK, M.Slaney, Principles of Computerized Tomographic Imaging, SIAM 2001
(chapters 3,4,6)
G.T. Herman, Image Reconstruction from Projections, Academic Press, New York, 1980
(chapter 8 and chapter 16 sect.6)
course notes by the instructor
ELEMENTS OF FUNCTIONAL ANALYSIS
(MAT/05, 6 CFU)
Prof. Lucio Damascelli
(Warning: this course, shared with the Faculty of Engineering, starts approximately
ten days before the normal starting date, that is around the 21st of September)
Review of metric spaces, normed spaces, inner product spaces, separability, completeness,
connectedness, compactness, uniform continuity and the contractive fixed point theorem
and its applications
- Spaces of continuous functions: theorems of Ascoli-Arzela' and Stone-Weierstrass
- Bounded linear operators on Banach spaces
- The Lebesgue integral on Rn; Lp spaces
- Hilbert spaces, orthogonal projections and best approximation, the Riesz representation theorem
- Orthogonal systems, Fourier series in Hilbert spaces, Fourier series in L2, sketch of criteria for
pointwise or uniform convergence
- Linear operators on Hilbert spaces, compact self-adjoint operators on separable Hilbert spaces and
their spectral decomposition
- Short outline on the Uniform Boundedness theorem and weak topology
- Some outline on the Fourier transform on L1 and L2
- Elements of function theory in one complex variable, with emphasis on calculus of residues
Textbook: course notes by the instructor
RENDERING LAB
Prof. Massimo Picardello
(INF/01, 8 CFU)
This course is aimed to the development of a complete physically based photorealistic renderer and
shader. The full aim can not be reached in one year, but only putting together the material covered
in two or three subsequent editions of the course. The students should be familiar with the C++
programming language, and possibly skilled programmers, because they will not be assisted in
computer coding, only in mathematical and algorithmic aspects. The mathematical aspects cover
geometry, linear transformations and perspective transformations, shapes and meshing, sampling
(including anti-aliasing), probabilistic sampling (including stratified and low discrepancy), filters,
MonteCarlo integration, reduction of variance. The algorithmic aspects include kd-trees,
radiometry, models for light reflection and diffusion, BSDF, bump mapping, texture construction
and texture mapping, volume scattering and phase function, light transport, the rendering equation,
path tracing, irradiance caching, photon mapping, volume rendering with participating media. All
these topics are covered in prerequisite courses (Computer Graphics, Global Illumination), but here
the student will be asked to develop code to implement these ideas in the renderer.
Textbook: M.Pharr, G.Humphreys, Physically Based Rendering, from Theory to Implementation,
Morgan Kaufmann (Elsevier), S.Francisco, 2004
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
AND IMAGE RESTORATION
Prof. Daniele Bertaccini
Finite difference approximations
Boundary Value Problems and steady state
Elliptic equations
The initial value problems for ODEs
Diffusion equations and parabolic problems
Advection equations and hyperbolic systems
Mixed-type partial differential equations (PDEs)
Smoothing PDEs
Nonlinear diffusion
Smoothing-Enhancing PDEs
Chemotaxis PDEs
References
[1] R. J. LeVeque -- Finite Difference Methods for ODEs and PDEs, Steady State and Time
Dependent Problems. SIAM, Philadelphia, 2007
[2] G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing. Partial Differential
Equations and the Calculus of Variation, Springer, 2006