# ⚛  L1 Principle — Compressed Ultrafast Photography (CUP) — trillion fps imaging

**ID:** `L1-151` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Ultrafast Imaging — *Coded-aperture + streak camera 1 Tfps imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D xyt video
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554229

---

## 🧠 1. Introduction

**Compressed Ultrafast Photography (CUP) — trillion fps imaging** is a **linear inverse problem** whose unknown lives in **3D xyt video** space, within the **Coded-aperture + streak camera 1 Tfps imaging** sub-domain of **Ultrafast Imaging**.

Measurements consist of photons collected by an optical detector via a **compressed ultrafast** sensing mechanism.

The forward operator applies, in order: element-wise multiplication by a binary mask; S · temporal · coded operator; L · streak sweep operator; phase is lost; only intensity is measured; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D xyt video is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 28); dmd_mask_drift dominates the stability cliff; streak_nonlinearity and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → Binary coded aperture → S · temporal · coded → L · streak sweep → Magnitude-squared → Temporal integration → **y** (detector).

```
y = ∫_t dt |·|² `L.streak_sweep` `S.temporal.coded` M ⊙ x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.diag.binary` | Element-wise multiplication by a binary mask |
| `S.temporal.coded` | S · temporal · coded operator |
| `L.streak_sweep` | L · streak sweep operator |
| `L.modulus_squared` | Phase is lost; only intensity is measured |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Ultrafast Imaging |
| Sub domain | Coded-aperture + streak camera 1 Tfps imaging |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 3D_xyt_video |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 10 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 3D xyt video is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 28); dmd_mask_drift dominates the stability cliff; streak_nonlinearity and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `cup` · **Forward operator:** `cup_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| N frames | — | 100 |
| Cross talk | — | 0 |
| Photon count | — | 200 |
| Dmd mask drift | — | 0 |
| Low photon regime | — | 0 |
| Streak nonlinearity | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| N frames | — | 20 – 500 |
| Cross talk | — | 0.0 – 0.3 |
| Photon count | — | 10 – 5000 |
| Dmd mask drift | — | 0.0 – 0.3 |
| Low photon regime | — | 0.0 – 1.0 |
| Streak nonlinearity | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `560` and δ = `10`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xe4a745062413162527d0bf4d3f2652023b62cf6fb94ef132d303fc7767462e9a`
- **Chain tx hash:** `0x1c8081115525d0fec2c04ea5317cc8d3b1bc761180cd3234c0bfb841218f29f5`
- **Chain block:** `41554229`

---

## File Mapping

This bundle consists of: `L1-151.md`, `L1-151.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-151.md` | Source of truth — edit this | Human or LLM |
| `L1-151.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-151`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._