Learn the Law
Not the Atlas

Toward Compositional UV Learning for Generative Humans

A UV atlas can change while the underlying surface stays the same
This note explores the probability law shared across charts and the conditional detail that preserves a chosen coarse state

Read the paper

What a boundary must share

The same character produces different images under different UV cuts and layouts
This research focuses on the conditional probability structure that can persist across those layouts

Left region L and boundary B and right region R on a 3×3 grid A grid of horizontally and vertically adjacent nodes with the central column highlighted in teal The boundary B makes the left and right regions conditionally independent under nearest-neighbor interactions LBR
A 3×3 grid with nearest-neighbor interactions
The central column B carries the information shared by both regions

The boundary carries shared information

L ⟂ R | B

With global conditions fixed and dependencies captured by B the two interiors can be modeled through separate conditional laws

Longer-range interactions such as bending require a boundary with thickness
Contact between distant parts also belongs in the shared state

Paper page 4 · Section 4

Account for every possible interior

F(q) = −log W(q)
W(q) = ∫ exp[−E(x ; q)] dνq(x)

q is the boundary state and x is the interior state
Integrating the weight of admissible interiors against the specified reference measure determines how boundaries should be sampled

Probabilistic gluing counts each shared energy term once and aligns the laws of the boundary and its interiors

Paper pages 5–6 · Section 5

New detail with a fixed coarse state

xf = Hc + Nη
RH = I   RN = 0   Rxf = c

R is a fixed linear operator that restricts the fine state to a coarse state c
Full row rank allows a right inverse H and null-space directions N

Conditional detail in η can be added along N while preserving c
Recovering the distribution also requires learning the variability of the conditional residual

Paper page 7 · Section 6 · Proposition 4

Experiments

Synthetic scalar-field experiments from Section 9
Exact finite-state enumeration and Gaussian matrix calculations examine the components

Boundary choices change the distribution

3×3 grid · Node states −1 / 0 / 1 · Enumeration of all 3⁹ states

< 10−14

TV distance with exact free-energy gluing

The interior conditional laws stay exact while only the boundary sampling method changes

TV is the total variation distance between the reference and assembled distributions
A smaller value means a closer match

Results of the 3×3 finite-state experiment
Boundary samplingTV distanceKL · nats
Free-energy gluing< 10−14< 10−14
Minimum energy0.2793860.319537
Uniform boundary0.4058530.611330
Paper page 10 · Section 9.2 · Table 2

The variation a mean leaves out

17×17 Gaussian scalar field · Refinement degrees of freedom 289 / 81 / 25

74.17%

Total variance lost with mean-only reconstruction

This measures the difference between preserving a mean and recovering a distribution
Restoring the conditional residual recovers the original covariance to numerical precision

Results of the Gaussian refinement experiment
ReconstructionRelative covariance error
Conditional mean only0.651434
Mean and residual6.83 × 10−16

Relative error uses the Frobenius norm
Maximum coarse-state drift in sampled refinement is 0

Paper page 9 · Section 9.1 · Table 1

A learned law on a different grid

Train on 3×3 · Evaluate on 2×5 · Three-parameter model with the same underlying law

Coefficients in a fixed function family are estimated from independent full-grid samples and reused on a grid with a different shape

Each sample count reports the median of eight runs with independent seeds
Evaluation uses KL per grid site

Paper page 10 · Section 9.3 · Table 3
Evaluation KL by training sample count · nats/site
8
0.0165594
32
0.0060978
128
0.0015739

Bar lengths are proportional to median KL from a zero baseline

Toward generative characters

The architecture proposed in Section 7 combines boundary information with probability mass and conditional detail

Boundary information

Choose boundary width and shared variables to capture interactions across regions including bending and nonlocal contact

Boundary probability

Match boundary frequencies with interior conditional laws through a learning structure that connects shared energies and free-energy messages

Persistent detail

Store stable material coordinates and previously generated residuals
The proposed workflow generates new detail in degrees of freedom released by an edit

Paper pages 7–8 · Sections 6–7

Paper and references

The original equations and experimental conditions
Related research on surface representations

AtlasNet

Groueix and colleagues · 2018
Generating 3D surfaces as collections of parametric surface patches