The Paired Thermal Channel Is Two Orthogonal Axes: A Controlled Attribution Study of Calibration Cost and Detection Gain in Infrared Person Detection
Authors: Tian Zhang, Yang Chen, Bing Bai, Weikai He, Xiang Gao, Zhirong Su
Organizations: School of Information Engineering, Shandong Management University, Jinan, China. · School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. · School of Aeronautics, Shandong Jiaotong University, Jinan, China. · School of Business, Shandong Management University, Jinan, China.
The calibrated thermal channel in infrared person detection comprises two structurally orthogonal axes. True calibrated ΔT outperforms scale-matched grayscale pseudo-ΔT by +0.0221 mAP (paired Wilcoxon p=0.0098, Cohen's d=+1.08), confirming that the calibrated channel carries information the grayscale channel lacks. The calibration axis is driven by the joint pairing-scale condition. Preserving both paired correspondence and absolute calibrated scale raises expected calibration error (ECE) from ≈0.05 to 0.142. The 2×2 interaction (p=0.0059) localizes the rise to this condition. Breaking either factor alone returns ECE to the broken-factor level (true-vs-const p=0.002; true-vs-shuffled p=0.002). Under post-hoc temperature scaling, the pairing cost survives recalibration (p=0.002), while the scale cost is largely absorbed (p=0.16). As regards the detection axis, the gain stems from structural de-structuring. Phase randomization, which destroys spatial structure while preserving per-sample variance, yields +0.0185 mAP over true ΔT on FLIR ADAS (p=0.014, d=+0.96). The FLAME3 gain is +0.0064 mAP (p=0.078, n=8). On the calibration axis, the same intervention is insignificant on FLIR ADAS (p=0.47) but significant on FLAME3 (p=0.0078). These findings rest on a five-arm controlled attribution with temperature-evidential learning (TEL) as the testbed. TEL obtains a single evidence-derived routing scalar together with a complementary diagnostic entropy, and establishes a finite-evidence degeneracy theorem proving that the acquisition product degenerates to this scalar with a closed-form bound. A physical 0.5∘C threshold admits cross-dataset physical readings, whereas a gray threshold of 20 inverts without physical interpretation.
Figures & tables
Figure 1: Temperature-evidential learning (TEL) as one evidence chain. (a) Dual-channel decoding. Channel A renders G , while Channel B retains the calibrated temperature matrix T . The annular contrast geometry is sampled at dilation factors r∈{0.1,0.2,0.4} , using T values from Channel B and box positions from the detector backbone. (b) Temperature-evidential modulation. The multi-scale contrast ΔT(s) (Eq. ( 3 )) passes through the sigmoid gate m(ΔT) (Eq. ( 4 )). The temperature-aware prior gk,πk produces eaug,k (Eq. ( 9 )), and the modulation factor 1+γm produces emod,k (Eq. ( 10 )), from which a single evidence-derived routing scalar u is read; a diagnostic entropy Ψ=E[H(p)] is computed alongside on a dashed branch and does not participate in routing. (c) Two-ended routing. The same u drives annotation, pseudo-labeling, and discard (Eq. ( 22 )); the partition relies on separation of the two ends of the distribution, not on monotone ranking in between. (d) Replay and distillation weighting (Eqs. ( 35 ) and ( 37 )) appear as dashed design extensions. Solid arrows denote forward data flow and core computation. Dashed arrows denote auxiliary inputs, control signals, and the closed-loop feedback that returns the updated detector to the next round. All downstream decisions in (b)-(d) use the temperature matrix retained in Channel B.
Figure 2: Physical evidence of the temperature matrix. (a) Two nighttime patches in which the grayscale contrast is low ( ∣ΔG∣≈2.2 ) while the calibrated ΔT remains measurable ( ≈0.28∘ C). The inset text reports the exact number of interior and annular background pixels. (b) Conditional distribution P(ΔT∣y=person) as a function of the illumination drift coordinate s , stratified into quintiles. P(y∣s) is stationary, while P(ΔT∣y,s) drifts with s . Data points mark median ± IQR. (c) Empirical validation of the KL separability sandwich in Eq. ( 19 ). The red line is the measured median Λ(b) with interquartile range (IQR) error bars, obtained by using the thermal camera’s native 8-bit grayscale rendering (strictly pixel-aligned with the 14-bit TIFF) to compute the true local grayscale ratio x . The black dashed line is the theoretical upper bound Λ(b) . The empirical median grayscale-to-temperature noise ratio is x≈1.10 . The empirical median strictly falls within the theoretical bounds, confirming that the calibrated temperature channel ΔT provides a reliable source of local separability evidence. The empirical median temperature noise is σmed=0.39∘ C (IQR: [0.22,0.72]∘ C).
Signal
ε
ρ
err∣acc
ub
Coverage
Mean ± std across the 45 seed pairs
u / margin / conf
0.086±0.021
0.712±0.072
0.0364±0.014
0.0660±0.016
45/45
H
0.116±0.021
0.520±0.084
0.0281±0.010
0.0736±0.014
45/45
95% bootstrap CI (2000 resamples, pair as unit)
u / margin / conf
[0.080,0.092]
[0.690,0.734]
[0.032,0.041]
[0.061,0.071]
—
H
[0.110,0.122]
[0.495,0.545]
[0.025,0.031]
[0.070,0.078]
—
Table 1: Empirical validation of Eq. ( 27 ) over all 45 seed pairs. Margin and conf are identical to each other; u is numerically indistinguishable from them because K=2 . Coverage is the fraction of seed pairs for which the bound exceeds the accepted error.
Pair
Pearson ρ
95% CI
Spearman ρ
95% CI
Signal-to-signal correlations
u , H
−0.0765
[−0.0782,−0.0748]
−0.2662
[−0.2687,−0.2639]
u , margin
−0.9255
[−0.9263,−0.9247]
−0.3079
[−0.3109,−0.3047]
u , conf
−0.9255
[−0.9263,−0.9247]
−0.3079
[−0.3110,−0.3050]
H , margin
−0.2590
[−0.2608,−0.2573]
−0.6202
[−0.6224,−0.6180]
H , conf
−0.2590
[−0.2608,−0.2573]
−0.6202
[−0.6225,−0.6180]
Table 2: Signal correlations on the FLIR ADAS prediction protocol, pooled over ten seeds. Values are Pearson ρ and Spearman ρ with 95% bootstrap confidence intervals on the pooled RoIs.
τlow
measured (mean ± std)
[min,max]
bound
0.40
0.0802±0.0704
[0.0126,0.2231]
0.1667
0.45
0.0308±0.0158
[0.0129,0.0592]
0.1837
0.50
0.0237±0.0058
[0.0162,0.0378]
0.2000
0.55
0.0332±0.0051
[0.0254,0.0454]
0.2157
0.60
0.0470±0.0027
[0.0423,0.0524]
0.2308
Table 3: Empirical validation of Corollary 3 on an oracle-based image-level protocol, averaged over ten seeds. Thresholds below 0.40 are omitted because the discard branch contains too few samples for a reliable estimate.
a
b
consistency κroute
(qlo,qhi)
1.0
0.0
1.0000
(0.4520,0.9883)
0.5
0.0
1.0000
(0.4520,0.9883)
1.0
10.0
1.0000
(0.4520,0.9883)
2.0
−5.0
1.0000
(0.4520,0.9883)
Table 4: Empirical validation of Corollary 4 . Four strictly increasing affine reparametrizations of the temperature channel, T′=aT+b , are applied; the routing partition is unchanged in all four cases.
mean
std
var
u
0.3673
0.2175
0.047301
Ψ
0.4982
0.0204
0.000416
Δ
0.1827
0.1086
0.011792
Table 5: Empirical statistics of the routing scalar, the diagnostic entropy, and the acquisition product, pooled over n=5.4×105 RoIs and ten seeds on FLIR ADAS.
regime
u range
frac(IoU>0.5)
mean IoU
router branch
background
bins 0–6
≈0
≈0
discard
transition
bin 7
0.0948
0.0861
pseudo
detection
bin 8
0.9274
0.7652
pseudo
saturated
bin 9
0.3184
0.2895
annotate
Table 6: Three-state structure of the routing scalar u on FLIR ADAS, pooled over ten seeds. The umed values of the four regimes are 0.263 – 0.271 (bins 0–6), 0.2718 (bin 7), 0.5620 (bin 8), and 0.9943 (bin 9). Regime labels describe the u -stratified structure. Router branches follow Eq. ( 23 ) with pooled ten-seed thresholds Q0.3=0.267 and Q0.9=0.714 . Bins 0–6 are grouped as a single background regime. Their median u spans Q0.3 , so some samples within this regime route to pseudo. The saturated regime is the only regime where mean IoU decreases relative to the preceding bin; this drop is what makes u a two-ended separator and not a monotone ranking.
Dataset
Modality
Calibrated T ?
Role
FLIR ADAS
RGB + 14-bit TIFF
✓
Primary
FLAME 3
Radiometric Long-Wave Infrared (LWIR)
✓
Auxiliary
Table 7: Datasets used in this work.
Contrast
Metric
Δ
t -test p
Wilcoxon p
Cohen’s d
Reading
true − pseudo
mAP
+0.0221
0.0077
0.0098 ( W=3 )
+1.08
primary, Wilcoxon
true − shuffled
ECE
+0.0882
0.039
0.002 ( W=0 )
+0.76
primary, true higher
true − pseudo
ECE
−0.0097
0.77
0.064
−0.095
secondary, n.s.
true − pseudo
Recall
+0.0008
0.66
n.s.
+0.14
secondary, n.s.
true − pseudo
LCR
+0.0044
0.60
n.s.
+0.17
secondary, n.s.
true − const- ΔT
ECE
+0.0916
0.019
0.002 ( W=0 )
+0.90
secondary, Holm-pass
Table 8: Paired contrasts on FLIR ADAS over ten seeds under the two-stage protocol. All Cohen’s d values are paired-effect sizes, computed from the per-seed paired differences. The primary family is true-vs-pseudo on mAP and true-vs-shuffled on ECE ( m=2 , Holm threshold 0.025 ); the secondary family is true-vs-pseudo ECE/Recall/LCR and true-vs-const- ΔT ECE ( m=4 , Holm thresholds 0.0125 , 0.0167 , 0.025 ). The detection-axis attribution (var-phase-vs-true mAP and ECE) is pre-specified and reported as a dedicated contrast pair. Rows below the third rule are exploratory. Positive Δ means the first arm is larger.
Probe
emod mean
Δemod
Top score
zero modulation
0.002453
reference
0.8647
true ΔT
0.003395
+38.42% ( p=4.2×10−25 )
0.8862 ( +2.48% , p=8.7×10−43 )
random modulation
0.002668
+8.78% ( p=2.97×10−2 )
0.8651 ( +0.05% , p=9.5×10−3 )
Table 9: Attribution-consumption probe on FLIR ADAS ( 200 images). The modulated evidence magnitude emod and the top detection score are compared across zero modulation, true calibrated ΔT , and random modulation.
Table 10: Overall comparison on FLIR ADAS (10-seed prediction, all bins pooled). The primary acceptance test is TEL(true ΔT ) versus Grayscale pseudo- ΔT .
Table 11: Auxiliary validation on FLAME 3 (radiometric wildfire detection, ∣ΔT∣≈30 – 60∘ C). Mean ± std over eight seeds. All p values are paired Wilcoxon signed-rank tests over the eight paired seeds, where the smallest attainable two-sided p is 0.0078 .
Variant
ECE (raw)
T (median)
ECE TS
Δ ECE
Baseline (no modulation)
0.4585
2.645
0.4588
+0.0003
Random modulation
0.0373
0.807
0.0166
−0.0206
Grayscale pseudo- ΔT
0.1512
0.661
0.0860
−0.0652
Shuffled true ΔT
0.0534
0.706
0.0172
−0.0361
const- ΔT
0.0500
0.872
0.0342
−0.0156
TEL (true ΔT )
0.1415
0.763
0.0837
−0.0578
Table 12: Post-hoc temperature scaling on FLIR ADAS (10 seeds). T is fitted per arm on the validation set with NLL as the objective; ECETS uses M=15 . T is the median across seeds; the true arm’s mean is affected by seed s2 ( T=3.89 ; Section 4.5 ). Raw ECE is reproduced from Table 10 ; Δ ECE is computed from per-seed means before rounding.
Variant
paired
absolute
mean
p50
p90
none
–
–
0.493
0.500
0.500
random
×
×
0.083
0.032
0.062
true
✓
✓
0.222
0.110
0.637
shuffled
×
✓
0.115
0.062
0.164
const- ΔT
✓
×
0.158
0.059
0.652
Table 13: Confidence distribution over 10 seeds (paired protocol).
Quantity
Value
95% CI
p
Interaction contrast cˉ
+0.0756
[+0.011,+0.140]
t -test 0.0475 ; Wilcoxon 0.0059
Cohen’s d
+0.73
—
—
Leave-one-out cˉ (sans s2)
+0.0437
—
<0.005
Component ECEtrue−ECEconst
+0.0916
—
—
Component ECEshuffled−ECErandom
+0.0161
—
—
Table 14: Joint-condition interaction test on ECE over ten paired seeds. The 2 × 2 contrast c=(ECE1,1−ECE1,0)−(ECE0,1−ECE0,0) is positive and significant under the Wilcoxon signed-rank test.
Figure 3: Δ ECE (true − random) versus per-box separability ratio ρ bins, with GT-derived quintile boundaries q20/q40/q60/q80=0.703/1.271/1.843/2.588 .
Figure 4: Cross-dataset reversal of the two axes. The same phase-randomized intervention (var-phase − true) produces a detection-axis effect of the same sign and comparable magnitude on FLIR ADAS and FLAME 3, and a calibration-axis effect that is absent on FLIR ADAS ( p=0.47 ) and significant on FLAME 3 ( p=0.0078 , the smallest attainable at eight seeds). Each axis is significant on the dataset where its structural counterpart is most pronounced, and insignificant on the other.
true vs. pseudo redundancy; random superiority check
Table 15: Data conditions and their discriminating power. Each dataset supplies a distinct validation axis.
Figure 5: Cross-dataset physical contrast.
Threshold τ
FLIR person pass
FLAME3 fire pass
Physical reading
0.5∘ C
10.5%
99.95%
FLIR: cold-background persons; FLAME 3: all fires
5∘ C
0%
99.9%
FLIR: no person reaches 5∘ C; FLAME 3: most fires
40∘ C
0%
53.5%
FLIR: no person reaches 40∘ C; FLAME 3: strong fires only
τG=20 (8-bit gray)
73.6%
36.0%
no physical reading
Table 16: Cross-dataset pass rates of physical and gray thresholds.
Figure 6: [email protected] versus ΔT bucket midpoint on FLIR ADAS for five variants: none, pseudo, random, TEL (true), and var-phase. All five curves rise with ΔT up to B4, the high-transition bucket, and then decline at the redundant end (B5). The var-phase curve lies above true at B1, B2, and B4, below it at B3 and B5, with the largest positive gap at B4, the bucket where the injected-variance and true-variance scales are best matched. This is the qualitative counterpart of the paired +0.0185 mAP effect reported in Table 8 . Buckets are test-set GT ΔT median quintiles with boundaries q20/q40/q60/q80=0.2424/0.2721/0.3392/0.5015∘ C.
Appendix figures & tables7 assets
Supplementary material from the paper’s appendix.
Appendix
Signal
Pair
err∣acc
ub
Gap
u
(7,8)
0.0556
0.0575
+0.0019
u
(2,9)
0.0273
0.0352
+0.0078
u
(1,8)
0.0616
0.0698
+0.0082
u
(3,9)
0.0360
0.0480
+0.0120
u
(2,3)
0.0267
0.0396
+0.0129
H
(7,8)
0.0348
0.0551
+0.0203
Appendix
Table 17: Tightest seed pairs for the pseudo-label error bound in Eq. ( 27 ). Only u and H are shown; margin and conf are omitted because they track u under K=2 .
Seed
τ=0.40
τ=0.45
τ=0.50
τ=0.55
τ=0.60
0
0.0140
0.0144
0.0227
0.0355
0.0486
1
0.0619
0.0234
0.0162
0.0340
0.0484
2
0.1667
0.0541
0.0285
0.0311
0.0463
3
0.0436
0.0306
0.0245
0.0311
0.0423
4
0.0179
0.0179
0.0192
0.0295
0.0464
5
0.2231
0.0592
0.0378
0.0454
0.0524
Appendix
Table 18: Per-seed discard error Pr(y^=y∣u<τlow) under the oracle-based image-level protocol.
Pair
s0
s1
s2
s3
s4
s5
s6
s7
s8
s9
Pearson ρ
u , H
−0.092
−0.145
−0.008
−0.075
−0.009
−0.060
−0.013
−0.147
−0.062
−0.189
u , margin
−0.922
−0.930
−0.943
−0.922
−0.921
−0.929
−0.932
−0.926
−0.915
−0.921
H , margin
−0.259
−0.191
−0.275
−0.267
−0.330
−0.268
−0.298
−0.193
−0.296
−0.169
margin, conf
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
+1.000
Δ , H
−0.027
−0.091
+0.047
−0.008
+0.057
−0.005
+0.049
−0.089
+0.007
−0.130
Appendix
Table 19: Per-seed signal correlations on the FLIR ADAS prediction protocol. Redundant pairs ( u –conf, H –conf, Δ –conf) are omitted; they are identical to the corresponding margin pairs because K=2 .
Set
n
Signed Pearson
Signed Spearman
Unsigned Pearson
Unsigned Spearman
FLIR train
48,649
+0.7126
+0.7183
+0.5068
+0.5709
FLIR test
12,267
+0.8136
+0.8847
+0.8073
+0.8812
FLAME 3
2,126
+0.7570
+0.8450
+0.7569
+0.8450
FLIR subset correlations
FLIR subset
Spearman
n
LCR ( ∣ΔT∣< p25)
+0.9338
3,067
Appendix
Table 20: ΔG – ΔT correlation.
Dataset
Quantity
p25
p50
p75
Other
FLIR ADAS
∣ΔT∣ ( ∘ C)
0.208
0.271
0.357
LOW_DT 0.207
FLIR ADAS
eff. SNR
5.75
8.96
13.98
SNR >1 97.4%, SNR >3 91.0%
FLAME 3
∣ΔT∣ ( ∘ C)
29.89
42.04
59.58
—
FLAME 3
∣ΔG∣
11.81
16.56
23.56
—
Appendix
Table 21: physical-quantity measurements.
α
true
random
0.00
0.5005±0.0433
0.5223±0.0029
0.25
0.4879±0.0290
0.5199±0.0007
0.50
0.4827±0.0260
0.5157±0.0039
0.75
0.4776±0.0231
0.5148±0.0046
1.00
0.4720±0.0207
0.5133±0.0062
1.25
0.4665±0.0187
0.5116±0.0076
Appendix
Table 22: [email protected] as a function of modulation amplitude α over seeds s0, s1, and s2. All points except α=1.0 are out-of-distribution extrapolations.
Router
Set
νTK / 152
per-image fraction (mean ± std)
max
fixed ( τhigh=0.7 , τlow=0.3 )
train
132
0.031±0.072
0.382
quantile ( Q0.9 , Q0.3 )
train
147
0.092±0.118
0.428
fixed
test
138
0.069±0.110
0.421
quantile
test
148
0.152±0.131
0.467
Appendix
Table 23: Per-image switch fraction under the fixed-threshold and quantile routers, on the training and test evaluation sets. νTK is the aggregate count out of 152 checkpoint pairs. The per-image fraction appears alongside.
China University of Petroleum-Beijing at Karamay, No. 355, Anding Road, Karamay, 834000, Xinjiang, China. · Guangdong University of Technology, Guangzhou, Guangdong, China. · Tianjin University, Tianjin, China. +1