Python tours · Numerical Tours

Foundations

Introduction to Image Processing

Explore how arrays become images, how convolution removes fine detail, and how derivatives reveal boundaries.

Foundations

Fourier and Wavelet Approximation

Represent the same image in Fourier and wavelet coordinates, then reconstruct it from a limited number of coefficients.

Foundations

Explore Fourier and Wavelets Interactively

Use interactive controls to vary how many transform coefficients an image keeps.

Foundations

Wavelet and DCT Image Compression

Compare wavelet and discrete cosine representations for image approximation.

Foundations

Le traitement numérique des images

Découvrez comment représenter une image par une matrice, modifier son contraste et construire des filtres simples.

Wavelets & compression

Two-Dimensional Daubechies Wavelets

Build a separable orthogonal wavelet transform from its analysis filters and reconstruct the image with the corresponding synthesis filters.

Wavelets & compression

Volumetric Wavelet Processing

Extend multiscale image processing to a three-dimensional volume.

Wavelets & compression

Image Approximation in Orthogonal Bases

Compare linear and nonlinear approximations in several orthogonal bases.

Wavelets & compression

Entropy Coding and Compression

Construct Huffman codes from symbol probabilities and check that encoding followed by decoding recovers the data exactly.

Image restoration

Linear Image Denoising

Study denoising as a tradeoff between suppressing random fluctuations and retaining useful detail.

Image restoration

Wavelet Image Denoising

Remove noise by shrinking wavelet coefficients and reconstructing the image.

Image restoration

Wavelet Block Thresholding

Process neighboring wavelet coefficients together instead of thresholding each coefficient independently.

Image restoration

Nonlocal Means Denoising

Find similar image patches and average their center pixels using similarity weights.

Image restoration

Rank Filters and Mathematical Morphology

Replace local pixel averages with order statistics such as medians and quantiles.

Image restoration

Stein Unbiased Risk Estimation

Choose denoising parameters by estimating reconstruction risk from the noisy observation itself.

Inverse problems

Variational Image Deconvolution

Recover an image from a blurred observation by balancing data fidelity with a regularity penalty.

Inverse problems

Sparse Image Inpainting

Reconstruct missing pixels by promoting sparsity in a wavelet representation.

Inverse problems

Matrix Completion with the Nuclear Norm

Recover a low-rank matrix from a subset of its entries.

Inverse problems

Sparse Recovery with ℓ¹ Minimization

Investigate when a sparse vector can be recovered from fewer measurements than unknowns.

Inverse problems

Learning Sparse Dictionaries

Learn an overcomplete collection of image atoms by alternating sparse coding and dictionary updates.

Optimization

Gradient Descent

Follow gradient descent on smooth objectives and apply it to image restoration.

Optimization

Gradient and Projected Gradient Methods

Develop gradient descent for quadratic objectives, image denoising, and constrained inpainting.

Optimization

Forward–Backward Splitting

Recover a sparse signal by combining a smooth data-fidelity gradient with soft thresholding.

Optimization

Forward–Backward Splitting on the Dual

Solve total-variation denoising through its dual formulation.

Optimization

Douglas–Rachford Splitting

Recover a sparse signal under exact linear constraints by alternating two proximity operators.

Optimization

Chambolle–Pock Primal–Dual Splitting

Implement a primal–dual algorithm for total-variation inpainting and denoising.

Optimization

Interior-Point Methods

Approach constrained optimization from the interior of the feasible set using a logarithmic barrier.

Optimization

Smooth Bilevel Sparse Regularization

Reformulate a sparse regression objective through a smooth factorization and an inner linear solve.

Optimal transport

Optimal Transport by Linear Programming

Compute a minimum-cost coupling between discrete probability distributions.

Optimal transport

Entropic Optimal Transport

Replace a linear transport problem with a smooth entropically regularized objective.

Optimal transport

Entropic Transport and Applications

Use Sinkhorn scaling to compare and transform discrete distributions.

Optimal transport

Advanced Sinkhorn Algorithms

Explore refinements of entropic transport beyond basic alternating scaling.

Optimal transport

Semi-Discrete Optimal Transport

Transport a continuous distribution toward a finite set of weighted points.

Optimal transport

Unbalanced Optimal Transport

Allow transport to create or remove mass when exact marginal matching is inappropriate.

Geometry & segmentation

Image Edge Detection

Locate image boundaries using derivatives at several scales.

Geometry & segmentation

Parametric Active Contours

Evolve a closed curve under smoothing and image-dependent forces.

Geometry & segmentation

Level-Set Active Contours

Represent a moving contour as the zero level of a function and evolve that function on a grid.

Geometry & segmentation

Dijkstra and Fast Marching

Propagate a distance front through a weighted domain, first with a graph update and then with an eikonal update.

Geometry & segmentation

Manifold Learning with Isomap

Recover a low-dimensional representation of a curved point cloud using neighborhood-graph distances.

Geometry & segmentation

Mesh Parameterization

Map a triangulated surface into the plane by fixing its boundary and solving for interior coordinates.

Geometry & segmentation

Mesh Denoising

Smooth noisy vertex positions using discrete differential operators on a triangulated surface.

Signals & graphics

Audio Source Separation with Sparsity

Separate several audio sources from a smaller number of mixtures using sparse time-frequency structure.

Signals & graphics

Gaussian Texture Synthesis

Generate random textures with prescribed second-order statistics using Fourier-domain operations.

Signals & graphics

Fluid Dynamics

Advect a velocity field and an image through a two-dimensional domain.

Machine learning

PCA, Nearest Neighbors, and Clustering

Explore a labeled dataset through principal components, neighborhood classifiers, and clustering.

Machine learning

Linear Regression, Ridge, and Lasso

Fit and compare linear regression models with quadratic and sparse penalties.

Machine learning

Logistic and Kernel Classification

Turn linear scores into class probabilities and optimize a logistic loss.

Machine learning

Stochastic Gradient Descent

Replace full gradients with random sample updates and compare the resulting trajectories.

Machine learning

Multilayer Perceptrons

Build a small neural network and differentiate its loss through successive layers.

Machine learning

Neural Networks for Function Approximation

Train a multilayer network to approximate a nonlinear function.

Machine learning

Deep Neural Networks and Automatic Differentiation

Implement forward and backward propagation, then compare manual derivatives with automatic differentiation.

Machine learning

Simplex Regression with Trend Filtering

Estimate a structured signal under simplex constraints and a trend-filtering penalty.

Machine learning

Deep Texture Synthesis

Synthesize a texture by matching Gram matrices of pretrained convolutional features.

Machine learning

Graphical Lasso

Infer sparse conditional dependencies by estimating a Gaussian precision matrix.

Machine learning

Interacting Particle Systems

Evolve a cloud of particles under pairwise kernel interactions and compare analytic and automatic gradients.

Machine learning

Conformal Prediction

Construct prediction regions by comparing a candidate observation with observed conformity scores.

Machine learning

Denoising Diffusion Models

Study forward noising and reverse sampling on a one-dimensional Gaussian mixture.

Machine learning

Diffusion Models with JAX

Implement Gaussian-mixture diffusion and score learning with JAX and Flax.