# Quality knee: what does a blurred or small face cost, and can a measure predict it?

VR-012. Companion to the minimum-face-size studies VR-005 and VR-013 (see the
[requirement register](requirements.md)), which located the size floor at 40 px;
this asks the same question for **sharpness**, and asks whether any cheap
measure taken on the aligned crop can be acted on at inference.

Run by
[`scripts/validation/quality_knee.py`](https://REPOLINK/scripts/validation/quality_knee.py)
through the `sae_embed` bindings — detection, the ArcFace warp, the embedder,
the five candidate measures and the Platt calibration are all the shipped C++.

## Protocol

1670 gallery actors with 3 or more mugshots (of 2456 total), one image held out
per actor as a probe, the remaining 10326 embeddings staying in the gallery at
native resolution. Only the probe degrades — reference mugshots are clean and
the face coming out of the video is not.

Each probe passes through a **joint grid**: downscale to *S*×*S* and back to
112 (the sampling loss), then blur at level *L* in canonical pixels. Three blur
families, 36 cells each, 60120 probe-cell records per family:

| family | models | parameter |
|---|---|---|
| Gaussian | soft focus, a generic stand-in | sigma 0 … 3 |
| **Disc** | **real optical defocus** — the circle of confusion | radius 0 … 6 |
| Motion | camera pan or moving subject | length 0 … 21 px |

The three are not interchangeable, and sweeping only the first was the original
design error — one that would have produced a wrong answer, not merely an
incomplete one (Result 3). A defocused lens spreads a point into a **uniform
disc**, whose transfer function is a jinc — `2·J1(x)/x` — that crosses zero and
goes negative, annihilating whole frequency bands and returning the ones beyond
each zero phase-reversed. A Gaussian MTF is strictly positive and monotone and
does neither. More practically: defocus and motion are how a face ends up
**large and useless**, while Gaussian blur as swept here mostly co-occurs with
small faces. That difference decides whether sharpness carries anything the size
filter does not.

Families are compared at matched **per-axis PSF standard deviation** (σ for a
Gaussian, R/2 for a disc, L/√12 for a linear smear), never at equal raw
parameter, which would compare different amounts of damage.

Identification is the pipeline's own decision: per-actor best-of-N cosine →
Platt sigmoid → accept above `prob_threshold` 0.754. Never a raw cosine
(AR-024).

## Result 1 — sharpness is not a sufficient statistic

Sorting the 36 Gaussian cells by `hf_energy_ratio`, the six sigma-3 cells land
at effectively identical measured sharpness:

| size | sigma | hf_energy_ratio | TPI |
|---|---|---|---|
| 16 | 3 | 0.0003 | **15.3%** |
| 24 | 3 | 0.0003 | 63.2% |
| 32 | 3 | 0.0003 | 79.4% |
| 48 | 3 | 0.0003 | 86.6% |
| 64 | 3 | 0.0004 | 88.4% |
| 112 | 3 | 0.0005 | **91.0%** |

Same measured sharpness, a **76-point spread in identification**. It inverts
too: 16 px unblurred measures 0.0033 and scores 23.5%, while 48 px at sigma 2
measures *lower* at 0.0021 and scores 96.6%.

A canonical-frame sharpness scalar cannot separate *attenuated* high
frequencies from *destroyed* spatial sampling. Blur suppresses the high band
while preserving mid-frequency facial geometry exactly; downsampling to 16 px
destroys that geometry outright. Both look alike to any measure keyed on
high-frequency energy.

This is the measured basis for AR-028's rule that the axes are **kept separate
and not collapsed into one scalar**, and it settles the double-counting
question: size and sharpness are not redundant, and neither substitutes for the
other.

## Result 2 — blur is a cliff, and it breaks confidence, not identity

TPI % by size (rows) against Gaussian sigma (columns):

| size | 0 | 0.5 | 1 | 1.5 | 2 | 3 |
|---|---|---|---|---|---|---|
| 16 | 23.5 | 24.0 | 25.0 | 24.6 | 23.9 | 15.3 |
| 24 | 85.7 | 85.1 | 85.6 | 85.9 | 82.6 | 63.2 |
| 32 | 95.9 | 95.9 | 96.0 | 95.5 | 93.7 | 79.4 |
| 48 | 98.7 | 98.7 | 98.4 | 98.1 | 96.6 | 86.6 |
| 64 | 98.6 | 98.6 | 98.8 | 98.4 | 97.5 | 88.4 |
| 112 | 98.9 | 98.9 | 98.8 | 98.6 | 98.0 | 91.0 |

Three regimes: **sigma ≤ 1.5 is free** (every cell moves under 1.5 points, sign
flipping at random — at 16 px it slightly *improves*, smoothing upscale
artifacts); sigma 2 costs 1–3 points; the 2→3 step costs 7–19. A smooth
discount curve is therefore the wrong shape — the response is flat, then falls
off a cliff.

**The cost peaks at the size knee, not at full resolution.** Sigma 3 costs
−22.5 points at 24 px but only −7.9 at 112 px and −8.3 at 16 px. Blur has no
intrinsic cost; it costs in proportion to how close the observation already sits
to the decision boundary. At 112 px there is margin to spare, at 16 px the probe
is already below threshold, and at 24 px it sits exactly on the knee.

**What blur destroys is confidence, not ranking.** Rank-1 barely moves: 99.3% →
99.2% at 112 px across the whole sigma range. The extreme case is 16 px at sigma
3, where rank-1 is **80.2%** while TPI is **15.3%** — 65 points of probes have
the correct actor ranked first and are rejected anyway for falling under the
probability threshold.

That is why **FPI never left 0.1% in any of the 108 cells across all three
families**. Degradation produces TBI, never a wrong name. The calibration
degrades gracefully, which is what SR-002 needs.

## Result 3 — the blur *family* matters more than the blur *amount*

Comparing families by their raw parameter is meaningless — sigma, radius and
length are different units. They are matched here by the **per-axis standard
deviation of the PSF**, which puts them on one scale:

| family | per-axis σ | level giving σ = 3 px |
|---|---|---|
| Gaussian σ | σ | 3 |
| Disc radius R | R/2 | 6 |
| Motion length L | L/√12 | 10.4 |

For reference the ArcFace template places the eyes 35.2 canonical px apart, so
σ = 3 px is 9% of the inter-ocular distance.

TPI at matched severity, interpolated within each family:

| size | σ=3 Gaussian | σ=3 Motion | σ=3 **Defocus** | defocus penalty |
|---|---|---|---|---|
| 16 | 15.3 | 15.1 | 11.0 | +4.3 |
| 24 | 63.2 | 61.1 | 41.4 | +21.9 |
| 32 | 79.4 | 76.1 | 50.4 | +29.0 |
| 48 | 86.6 | 80.9 | 52.6 | +34.0 |
| 64 | 88.4 | 81.7 | 51.0 | +37.4 |
| 112 | 91.0 | 82.1 | **46.9** | **+44.0** |

**Optical defocus is up to 44 points more destructive than a Gaussian of
identical spread**, and the ordering is defocus ≫ motion > Gaussian throughout.
At σ=1 the three families are indistinguishable, and at σ=2 they differ by under
5 points; the divergence appears only when both the blur is severe *and* the face
is large.

That pattern is physically consistent. At 16 px the resampling has already
removed the high frequencies, so the PSF's shape has nothing left to act on and
all three agree. At 112 px the full spectrum is present and shape decides: a
Gaussian MTF rolls off gently and always leaves *some* energy at every
frequency, so the embedder receives a merely attenuated signal, while a disc MTF
is a jinc that **hits exact zeros** — whole frequency bands annihilated rather
than attenuated, with the bands beyond each zero returning phase-reversed.
Motion sits between them because it ruins one axis and leaves the perpendicular
one untouched.

**The methodological consequence is the important one.** This study originally
swept Gaussian blur alone and concluded blur was a minor effect. On the family
that actually occurs in film, the same nominal severity costs **53% error
instead of 9%** at full resolution. A threshold set from the Gaussian arm would
have been wrong by a factor of five in error rate, and the axis would probably
have been dropped as not worth its cost.

**Defocus is also the case a size gate cannot catch.** Every one of those 112 px
faces is large and confidently detected, and sails through AR-002 untouched.
That, not the Gaussian result, is what justifies a sharpness axis existing at
all.

## Result 4 — variance of Laplacian is anti-predictive at fixed degradation

Pooled across all cells, every candidate scores AUC 0.76–0.80 for predicting
correct identification, with textbook `var_laplacian` top. That number is close
to worthless: it rewards a measure for detecting *how degraded the crop is*,
which all five do. The question a per-observation discount needs is whether, at
a **fixed** degradation, the measure predicts which faces fail:

| measure | Gaussian | Defocus | Motion |
|---|---|---|---|
| `hf_energy_ratio` | **0.530** | **0.521** | **0.557** |
| `norm_var_laplacian` | 0.520 | 0.507 | 0.539 |
| `dir_min_tenengrad` | 0.524 | 0.512 | 0.506 |
| `tenengrad` | 0.433 | 0.437 | 0.457 |
| `var_laplacian` | 0.423 | 0.422 | 0.473 |

Best is 0.557 — barely above chance, and `hf_energy_ratio` wins on all three
families. `var_laplacian` is anti-predictive on all three too, so that finding
does not depend on the blur model.

**The two metrics measure different jobs, and the candidates split along that
line.** On the motion arm `dir_min_tenengrad` has the best *pooled* AUC by a
wide margin — **0.854** against 0.792 for the next — exactly as its synthetic
directional-blur ladder predicted, yet its within-cell AUC there is 0.506. It is
an excellent detector of *how badly smeared a crop is* and no guide at all to
*which face will be recognised*. Pooled AUC is the right metric for a
gross-degradation flag; within-cell AUC is the right one for a per-observation
discount; a measure can be strong at one and useless at the other.

Deciles within the 16 px Gaussian cell, where 1277 failures give the test real
power:

| `var_laplacian` decile | TPI |
|---|---|
| 0.00071–0.00192 (blurriest) | **37.1%** |
| 0.00242–0.00278 | 22.8% |
| 0.00397–0.00447 | 25.7% |
| 0.00625–0.01445 (sharpest) | **14.4%** |

The faces the measure calls sharpest are **2.6x less identifiable** than those
it calls blurriest, monotone across ten bins of 167. Within a cell every crop
received identical degradation, so the residual variance is *native contrast*,
not native detail — and hard shadows, high-contrast lighting, sharpening halos
and JPEG ringing all raise Laplacian variance while making a face harder to
match. The measure reads photographic style and encoding artifacts and calls
them sharpness.

`hf_energy_ratio` is the only candidate with a correctly-signed within-cell
trend (16.2% → 35.3% across the same deciles), being a pure ratio in which the
contrast factor cancels.

**Consequence:** a per-face quality *discount* keyed on variance of Laplacian —
the most widely used blur metric in production vision pipelines — would
systematically down-weight the *more* identifiable faces. It is worse than no
discount.

## Result 5 — as a compute gate, sharpness loses to the size filter

Skipping the embed for crops below a threshold, measured as compute saved
against true identifications lost:

| gate | skipped | true IDs lost | of skipped, doomed anyway |
|---|---|---|---|
| `hf_energy_ratio` < 0.00023 | 10.0% | 7.6% | 37.9% |
| `hf_energy_ratio` < 0.00051 | 20.0% | 15.1% | 38.7% |
| **source size < 24 px** | **16.7%** | **4.7%** | **77.3%** |

At a comparable skip rate the size filter loses **4.7% against sharpness's
15.1%** — three times less damage — and it is free, being a bbox dimension
available before alignment or embedding, where sharpness needs the warped crop
plus a colour convert, three convolutions and a 64×64 DFT.

Restricting to large faces (≥64 px) on the **defocus** arm, where the size
filter is blind, improves the gate's precision 3.5x (37% of skipped crops doomed
versus 10.7% on the Gaussian arm) but not its trade: skip 10%, lose 7.0%.

A hard ceiling explains why. **At 112 px with defocus radius 6 — visually
destroyed — 46.9% of faces still identify correctly and rank-1 is still 94.8%.**
Blur does not determine the outcome, so any gate keyed on apparent blur is
predicting a coin flip. The size filter wins not because size is better
measured, but because *smallness destroys identity more completely than blur
does*: 16 px faces succeed only 23.5% of the time, so discarding them is cheap.

## What this means for the requirements

**Do not gate on sharpness; discount on it.** Heavily defocused faces remain
~47% identifiable, so a gate destroys recoverable evidence. This is the first
hard evidence that AR-028's "**discounts the observation, never deletes the
detection**" is right on the merits rather than merely cautious. Since ranking
survives where confidence does not, the per-track accumulation (AR-025) should
recover much of what a single-frame threshold rejects — which is also the
argument for the discount living in `EvidenceDiscounter` rather than in a filter.

**`var_laplacian` and `tenengrad` are disqualified as discounts** by Result 4,
on all three blur families. They remain usable as coarse *gross-degradation*
detectors, the role in which their pooled AUC is real — the same role the size
filter plays — but they must never weight a per-observation belief.

**`hf_energy_ratio` is the only surviving discount candidate**, best on all
three families, and its within-cell signal (0.52–0.56) is weak enough that
shipping a discount on it needs justification beyond this study.

**`dir_min_tenengrad` earns a different job.** Its pooled 0.854 on the motion arm
makes it the best available detector of gross directional smear — useful as a
per-frame "this shot is unusable" flag, which is a decision about a *frame*, not
a weighting of an *observation*. If AR-029 ships two measures for two roles, this
is the second one, and it must not be confused with the first.

**Model the blur family, not just its amount.** Result 3 makes the choice of
degradation model a first-order design decision rather than a detail: the same
matched severity costs 9% or 53% error depending on the PSF. Any future study
that sweeps blur must state which family it used and why.

**Any discount curve must be flat then steep**, not linear or sigmoid over the
measure. Blur costs nothing until it costs a great deal.

## Limitations

- **Cooperative population.** Gallery mugshots are frontal and well-lit;
  within-cell failures are likely dominated by cross-view mismatch, which no
  sharpness measure can predict. Read the ~chance within-cell AUCs as "sharpness
  does not predict the dominant failure mode *here*", not as "sharpness is
  meaningless".
- **Uniform grid, not a natural distribution.** Sizes and blur levels are
  sampled evenly, so "skip 16.7%" is exactly the 16 px row. The gate comparisons
  are like-for-like on identical records, but the absolute savings are not what
  a film would show.
- **TensorRT fp16.** A different realisation of the embedder from the fp32 ONNX
  reference — VR-005 measured ~0.85 cosine agreement with separation intact.
  Gallery and probes share one session so the study is internally consistent,
  but the absolute knee belongs to the fp16 space.
- **Blur is applied in the canonical frame**, after resampling, so its width is
  independent of the cell's size. Real optics blur before sampling.
- **The top motion rung is an anchor, not an operating point.** Length 21 is a
  per-axis σ of 6.1 — 17% of the inter-ocular distance, a streak rather than a
  face — and it is swept to bound the curve, not because a frame like that is
  worth reasoning about. Its 3.4% TPI at 112 px should not be quoted as a
  headline. The same caution applies less severely to defocus radius 6 (σ = 3).
- **Per-axis σ equates spread, not perceptual damage.** It is the fairest single
  scalar for comparing PSFs, but Result 3 is precisely the finding that equal
  spread does *not* mean equal harm, so the matched-severity tables compare
  like-for-like inputs, not like-for-like severity as a face would experience it.