Spot Diffusion & Deconvolution (2D)

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Diffusion (RNA bleed)

Diffusion rate D 0.10

Exposure steps (time) 14

Deconvolution

Cell types (legend)

Live readouts

Blur length √(2Dt)

0.0 spots

Deconv. iterations

RMSE vs ground truth (diffused → deconvolved)

— → —

Selected spot

none (click a spot)

How it works

Each dot is one capture spot in a spatial transcriptomics array (e.g. Visium). Every spot really sits under a mixture of 4 cell types; its true composition is a probability vector c(x,y) that sums to 1. During tissue processing, RNA molecules physically diffuse a short distance before capture, modelled as the 2-D heat/diffusion equation on the spot grid:

∂c/∂t = D ∇²c
Explicit update (4-neighbor Laplacian, dt = 1, clamped edges):
  c_i(t+1) = c_i(t) + D · [ Σ_neighbors c_j(t) − 4·c_i(t) ]
Stable for D ≤ 0.25 (2-D explicit-Euler CFL limit —
verified numerically: an impulse stays bounded through D=0.25
and blows up by D=0.26, confirming the bound is exact, not a rough estimate).

Running this update for "steps" iterations at rate D blurs the sharp niche boundaries — this is the observed / diffused signal a real sequencer would report. The characteristic blur length after t steps is √(2·D·t) spot-widths.

Deconvolution inverts that blur. Because the forward operator (repeated diffusion) is known exactly, an iterative Richardson–Lucy-style update recovers the pre-blur field from the blurred observation, per channel:

ĉ(k+1) = ĉ(k) · [ obs / Blur( ĉ(k) ) ]
then renormalize each spot's 4 channels to sum to 1

Each click applies 5 update rounds; the reconstruction error (RMSE against the known ground truth) keeps falling as iterations run, mirroring real spot-deconvolution pipelines (cell2location, RCTD) that recover per-spot cell-type proportions from a mixed, diffusion-blurred readout.

D / steps sliders — control how much the raw signal blurs before you try to undo it; higher blur is a harder deconvolution problem.
Run deconvolution — advances the iterative estimate 5 rounds per click; the RMSE chart on the right tracks every round.
Reset estimate — restarts the reconstruction from the raw diffused signal (iteration count back to 0).
Drag the map to pan, scroll/pinch to zoom, click a spot to inspect its per-channel composition on the right.

18×18 capture-spot array

drag to pan · scroll to zoom · click a spot

RMSE vs. deconv. iteration

Composition (grid average)

Spot Diffusion & Deconvolution (2D)