Empty-Signal Detection (ESD)

• Empty-Signal Detection (ESD) is a decision problem that determines whether an observation is empty or non-empty based on domain-specific criteria and operational definitions.
• It spans applications such as medical image segmentation, quantum communication with auxiliary measurements, sensor network fusion, and ambient backscatter detection with NP thresholds.
• Explicitly handling empty signals instead of combining them with standard inference improves metric reliability, error control, and adapts to unique operational challenges.

Empty-Signal Detection (ESD) denotes the problem of deciding whether an observation is empty or non-empty, with the meaning of “empty” determined by the application domain. In the literature considered here, emptiness refers to an empty reference mask in medical image segmentation (Ostmeier et al., 2022), a vacuum input at a quantum communication receiver (Shu, 19 Sep 2025), signal absence at all sensors in a network (Fellouris et al., 2013), or the absence of a Zero-Energy-Device beacon in ambient backscatter reception (Yang et al., 14 Apr 2025). The practical role of ESD therefore varies: it may replace overlap scoring by classification-style evaluation, suppress dark-count-induced quantum bit errors before message measurement, drive a sequential hypothesis test, or set a Neyman–Pearson decision threshold. This suggests a family of domain-specific procedures unified by one principle: empty observations must be handled explicitly rather than absorbed into ordinary downstream inference.

1. Domain-specific meanings of emptiness

In medical image segmentation, an empty reference annotation means that the ground-truth mask contains no positive voxels. The USE-Evaluator framework treats such cases as qualitatively different from ordinary segmentation examples because overlap and surface metrics may be undefined, unstable, or clinically uninformative when the reference is empty, very small, or uncertain (Ostmeier et al., 2022). The same source also distinguishes low-signal pathology, where weak imaging contrast or poorly defined boundaries produce uncertain annotations and small lesion masks.

In quantum communication, empty signals are pulses that arrive at the receiver with no photon in the message-encoding degree of freedom. The quantum ESD proposal separates photon-existence information from the message-encoding degree of freedom by placing the existence flag on an auxiliary degree of freedom, so that vacuum events can be identified and excluded before message processing (Shu, 19 Sep 2025).

In unstructured sequential testing for sensor networks, emptiness is the null hypothesis that signal is absent everywhere. The alternative is composite: signal is present somewhere, but the subset of affected sensors is unknown and may be any non-empty subset of the network (Fellouris et al., 2013).

In ambient backscatter detection for Zero-Energy-Devices, emptiness is the hypothesis that no beacon is present, or that the device is transparent so that the backscatter term is zero. The detector must then distinguish that case from the presence of a coded backscatter beacon using non-coherent processing and an NP threshold selected for a target false alarm probability (Yang et al., 14 Apr 2025).

These usages are not interchangeable. In one case ESD is an evaluation protocol, in another a receiver-side physical-layer filter, in another a sequential fusion rule, and in another a classical detection test. A common misconception is therefore to treat ESD as a single algorithmic template. The cited work instead supports a broader interpretation in which ESD is a recurring decision problem instantiated differently across sensing and communication systems.

2. Decision structures and operational consequences

Across the four settings, ESD is implemented by explicit branching rules rather than by leaving empty cases inside a generic score.

Domain	Empty condition	Operational consequence
Medical image segmentation	Ground-truth mask empty	Evaluate with image-classification metrics; report segmentation metrics as NA for truly empty cases
Quantum communication	Incoming signal vacuum in the message DOF	Discard events classified as empty before message measurement
Sensor networks	All sensors observe noise under $H_0$	Stop sequentially for the null when evidence for signal absence is sufficient
Ambient backscatter	ZED absent or transparent	Threshold a detection statistic at a target false alarm probability

In the segmentation setting, the recommended branch is between empty and non-empty cases. Empty cases are assessed by specificity, false positive rate, negative predictive value, positive predictive value, and balanced accuracy, while overlap or distance metrics are not reported for truly empty ground truth (Ostmeier et al., 2022). In the quantum setting, the branch is implemented through a $k$-of-$n$ auxiliary-measurement rule: a received event is accepted as non-empty only if at least $k$ out of $n$ auxiliary measurements report $|1\rangle$ (Shu, 19 Sep 2025). In the sensor-network setting, the branch is sequential, with one stopping rule favoring $H_0$ and another favoring $H_1$ (Fellouris et al., 2013). In ambient backscatter, the branch is classical NP detection based on a Gaussian test statistic and a threshold fixed by the desired $P_{FA}$ (Yang et al., 14 Apr 2025).

This shared structure clarifies why ordinary downstream metrics often fail in empty regimes. If emptiness is not handled separately, one obtains undefined overlap scores, dark-count-dominated QBER, intractable subset enumeration, or false-alarm behavior uncontrolled by design. The individual literatures differ in implementation, but each makes emptiness a first-class state of the system.

3. Quantum communication: auxiliary-DOF verification of non-vacuum events

The quantum ESD scheme is receiver-side and uses an auxiliary degree of freedom $L$ that is independent of the message degree of freedom $k$0. The message qubit is denoted $k$1, while the auxiliary degree of freedom is treated as a qutrit with basis $k$2, where $k$3 denotes vacuum in $k$4 (Shu, 19 Sep 2025). For a non-empty incoming signal, the receiver prepares

$k$5

and then applies $k$6 cascaded controlled gates $k$7 with $k$8 as control and $k$9 as targets.

The core decision rule is based on projective measurements of the auxiliary targets in the $n$0 basis. Only $n$1 outcomes are registered, the control $n$2 is discarded, and the message degree of freedom $n$3 proceeds intact. An event is accepted as non-empty if at least $n$4 of the $n$5 auxiliary measurements return $n$6; otherwise it is rejected as empty. Because the gates act only on $n$7 and $n$8, the message state remains $n$9 after the auxiliary operations and measurements (Shu, 19 Sep 2025).

The paper defines single-auxiliary acceptance probabilities for non-empty and vacuum inputs,

$k$0

$k$1

and corresponding binomial-tail acceptance probabilities

$k$2

False acceptance and false rejection are then

$k$3

The central feasibility condition is

$k$4

which is equivalent to $k$5 and to $k$6. Under this condition, increasing $k$7 and choosing $k$8 appropriately can make $k$9 arbitrarily small, so that the post-filter non-empty signal rate

$n$0

approaches $n$1 even when the channel transmission rate $n$2 is arbitrarily small (Shu, 19 Sep 2025).

The operational motivation is suppression of QBER growth caused by dark counts when empty signals dominate at long distance. With ESD, the QBER is expressed as

$n$3

so vacuum-induced errors are attenuated as $n$4. Simulations in the paper use $n$5, $n$6, $n$7, $n$8, $n$9, and $|1\rangle$0. Under those settings, with $|1\rangle$1, $|1\rangle$2, and $|1\rangle$3, NESR increases by $|1\rangle$4 orders of magnitude and QBER remains below $|1\rangle$5 at $|1\rangle$6; with $|1\rangle$7, NESR increases by $|1\rangle$8 orders of magnitude and QBER remains below $|1\rangle$9 at $H_0$0 (Shu, 19 Sep 2025).

The phrase “arbitrarily long-distance” is technically qualified in this work. The paper states that ESD does not change the repeaterless secret-key capacity, does not overcome the PLOB bound, and may leave the secret key rate extremely low because both acceptance and channel transmittance can vanish. The claim is instead that QBER can be kept within the secure regime for arbitrary distance in principle, provided a valid ESD block exists and the message-channel error satisfies the stated corollary bound on $H_0$1 (Shu, 19 Sep 2025).

4. Sensor networks: unstructured sequential testing of empty versus non-empty activation

In the sensor-network formulation, ESD is cast as a sequential hypothesis test with a simple null and a composite alternative. There are $H_0$2 sensors, observations are independent across sensors, and the local log-likelihood ratio process is

$H_0$3

The empty-signal hypothesis is

$H_0$4

while under the non-empty hypothesis the signal is present in an unknown subset $H_0$5 with $H_0$6 (Fellouris et al., 2013).

The proposed global procedure uses two one-sided stopping rules. The downward statistic for $H_0$7 is

$H_0$8

The upward statistic for $H_0$9 is

$H_1$0

where the local statistics $H_1$1 satisfy

$H_1$2

and

$H_1$3

The fusion center stops at

$H_1$4

with decision $H_1$5 if $H_1$6 and $H_1$7 otherwise (Fellouris et al., 2013).

Thresholds are selected as

$H_1$8

where

$H_1$9

is the survival function of an Erlang random variable with rate $P_{FA}$0 and shape $P_{FA}$1. With these thresholds, the paper proves that the type-I error is bounded by $P_{FA}$2 and the worst-case type-II error by $P_{FA}$3, so $P_{FA}$4 (Fellouris et al., 2013).

The principal theoretical result is asymptotic optimality. Under $P_{FA}$5,

$P_{FA}$6

while under $P_{FA}$7, for any affected subset $P_{FA}$8,

$P_{FA}$9

As $L$0 with $L$1, the proposed test minimizes expected sample size asymptotically both under the null and uniformly over all non-empty subsets (Fellouris et al., 2013).

A distinctive feature of this ESD formulation is communication efficiency. The downward side is one-shot: each sensor transmits at most once when its local LLR crosses below $L$2. The upward side can be implemented by level-triggered sampling,

$L$3

with either transmitted LLR increments or a one-bit scheme

$L$4

The paper shows that the one-bit scheme preserves asymptotic optimality under finite second moments, while computation remains $L$5 per time step and communication can be made sparse by increasing $L$6 slowly relative to $L$7 (Fellouris et al., 2013).

This formulation makes explicit that, in a network setting, ESD is not merely detection of a weak signal. It is detection of non-emptiness under uncertainty about where the signal is located, combined with stopping-time optimality and communication constraints.

5. Ambient backscatter: Neyman–Pearson ESD for Zero-Energy-Device beacons

For ambient backscatter Zero-Energy-Devices, ESD is the binary test of “no beacon present” versus “beacon present,” implemented on LTE/4G reference-signal observations at a smartphone receiver (Yang et al., 14 Apr 2025). The received complex baseband sample on subcarrier $L$8 and RS index $L$9 is

$k$00

where $k$01 denotes reflective versus transparent ZED state, and $k$02.

Under $k$03, the ZED is absent or transparent so the backscatter term is zero. Under non-coherent magnitude processing,

$k$04

with $k$05. Under $k$06,

$k$07

where

$k$08

The assumptions used are $k$09 and $k$10 (Yang et al., 14 Apr 2025).

The detector exploits two orthogonal chip sequences $k$11 and $k$12 and corresponding pseudo-filters $k$13 and $k$14. Over a bit window, normalized non-coherent correlator outputs are formed as

$k$15

If the active bit is $k$16 and $k$17, then

$k$18

with real-valued Gaussian noise terms whose variances depend on $k$19 and $k$20 (Yang et al., 14 Apr 2025).

A per-bit contrast is then defined as

$k$21

where

$k$22

This is stated to be an unbiased ML estimator of the backscatter path power $k$23. Correlation with the sync bit sequence $k$24 yields

$k$25

For large $k$26, $k$27 is approximately Gaussian at the aligned beacon start $k$28, with mean $k$29 under $k$30 and mean $k$31 under $k$32 (Yang et al., 14 Apr 2025).

The NP-optimal multi-subcarrier statistic is

$k$33

with

$k$34

where

$k$35

For a target false alarm probability,

$k$36

and the detection probability is

$k$37

The experimental setup uses FIT/CorteXlab in a two-node configuration, with bandwidth $k$38 MHz, FFT length $k$39, cyclic prefix length $k$40 subcarriers, OFDM symbol duration including CP $k$41, FSK modulation with $k$42 Hz and $k$43 Hz, a $k$44th-order Butterworth low-pass filter with cutoff $k$45 Hz, and $k$46 observed subcarriers. A Barker $k$47 code $k$48 is used in the sync sequence (Yang et al., 14 Apr 2025). The resulting combined statistic exhibits clear peaks, and the paper reports an experimental peak-to-lobe ratio of approximately $k$49 dB.

A recurrent issue in this domain is the trade-off between stringent false-alarm control and detection probability. The paper makes this relation explicit through the NP threshold formula: lowering $k$50 raises $k$51 and therefore reduces $k$52 unless $k$53 is increased by larger $k$54, larger $k$55, longer $k$56, or lower $k$57 (Yang et al., 14 Apr 2025).

6. Medical image segmentation: ESD as explicit handling of empty reference annotations

In the USE-Evaluator framework, ESD is not a new segmentation model but a recommendation for evaluation and optimization when reference annotations are uncertain, small, or empty. The motivating problem is a mismatch between public datasets and clinical practice: BraTS 2019 is characterized as a high-signal benchmark with certain, larger, and no empty reference annotations, whereas clinical datasets may include low-signal pathology with uncertain, small, or empty ground truth (Ostmeier et al., 2022). Under those conditions, common overlap metrics can fail to represent clinical value.

The central operational claim is that “very small and empty reference annotations are better evaluated with image-classification metrics” (Ostmeier et al., 2022). For empty ground-truth masks, the synthesis recommends identifying emptiness explicitly and assessing it as an ESD classification problem. If $k$58 is empty and the prediction $k$59 is also empty, overlap formulations such as Dice and IoU are mathematically undefined in set form because numerator and denominator are both zero; these cases should instead count as correct absence in the classification subtask, and segmentation metrics should be marked as not applicable. If $k$60 is empty and $k$61 is non-empty, the result is a false detection contributing to false positive rate rather than a clinically meaningful segmentation error (Ostmeier et al., 2022).

For non-empty but very small masks, the same source emphasizes the instability of overlap metrics. Dice,

$k$62

and IoU,

$k$63

become highly sensitive to small voxel-level perturbations when $k$64 is tiny. Hausdorff distance and surface-based metrics are also undefined if either boundary set is empty. The proposed remedy is stratified evaluation: use classification-style ESD metrics for empty and very small annotations, and evaluate segmentation quality separately on non-empty cases (Ostmeier et al., 2022).

For non-empty cases, the preferred metric is Surface Dice at Tolerance,

$k$65

which is described as facilitating clinically meaningful evaluation better than classical overlap metrics when annotations are uncertain, small, or empty (Ostmeier et al., 2022). The rationale given is reduced sensitivity to small boundary shifts and greater independence from reference volume size.

The framework also introduces a U-score to quantify uncertainty in reference annotations. The available description does not specify the exact U-score formula, but it explicitly validates the USE-Evaluator for models trained on uncertain annotations and motivates uncertainty-aware losses or weighted Soft Dice when confidence maps are available (Ostmeier et al., 2022). This leaves the exact operationalization of uncertainty as a domain-dependent component rather than a fixed universal definition.

A key misconception addressed by this work is that a single pooled Dice score remains adequate when empty or tiny lesions are common. The paper argues the opposite: pooled overlap scores can be misleading because they collapse detection and segmentation into one number, while empty cases require classification-style reporting and non-empty cases require boundary-aware or uncertainty-aware segmentation evaluation (Ostmeier et al., 2022).