Indexing the archive…
Your Universe of Digital Possibilities
A camera with one pixel. A micromirror array lays a random ±1 pattern over a hidden scene and a single photodiode records one number; do it 4,915 times for 16 384 pixels and the equations are fewer than the unknowns — infinitely many pictures fit. Pick the one with the least energy and you get mush. Pick the sparsest (Candès, Romberg & Tao; Donoho, 2006) and, when the world is sparse, the picture comes back. The rebuild never sees the scene. You may turn around once it is done.
With fewer measurements than unknowns, infinitely many images fit y. Ask for the one whose coefficients in a basis Ψ have the smallest ℓ₁ norm: the corners of the ℓ₁ ball pick out sparse solutions, and if the scene really is sparse in Ψ, it is the scene. The prior is the basis — name it.
A gradient step on the misfit, then soft-thresholding: every coefficient shrinks toward zero by λ/L, and the small ones land on it exactly. FISTA adds a momentum term and converges as 1/k² rather than 1/k (the plain iteration is Daubechies, Defrise & De Mol, 2004).
A k-sparse signal of length N is recovered from on the order of k log(N/k) random measurements — the scaling, not the constant. The Donoho–Tanner phase diagram draws where the bet actually pays.
Uniform sampling needs twice the highest frequency, whatever the signal. Compressed sensing asks a different question — not how fast, but how sparse — and for a sparse scene, random masks need far fewer numbers than pixels. Nyquist is not broken; it answers for every signal, and sparsity is a promise about this one.
The Orfeo Edition’s second shadow trades the gantry for a mirror array, and the projection for a dot product — but the grammar is the same: the scene stays veiled, the evidence is one number per mask, the rebuild sees only those numbers, and the misfit is the only honest number before turning around. What is new is the prior. With fewer equations than unknowns the null space is huge, and the min-norm answer simply leaves it empty; ℓ₁ fills it with the sparsest guess, and the guess is named on the pane. When it is wrong (pixels for shapes, anything for a Siemens hub) the residual says where. The Strobe is the rule being sidestepped; The Scan is the same evidence with lines instead of masks; The Code is what redundancy buys when the channel is noisy.