Foundations
Introduction to Image Processing
Explore how arrays become images, how convolution removes fine detail, and how derivatives reveal boundaries.
Open notebooks · Mathematical data science
Learn by computing.
Explore the mathematics of images, learning, and geometry.
Read an experiment, run the code, and follow the ideas.
58Python tours
from foundations to research
from foundations to research
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The Python collection
58 tours
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.
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