ElbowSig: Multidomain Signal Analysis

• ElbowSig is a multifaceted concept representing biomechanical elbow-angle signals, anthropomorphic signature features, and statistical clustering indicators.
• It employs CNN-based pose estimation and inverse kinematics to derive real-time elbow angles with RMS errors of ~3° (sagittal) and ~1° (coronal), supporting diverse applications.
• In clustering, ElbowSig formalizes the elbow heuristic using normalized curvature statistics to test for significant multiscale structure, enabling error-controlled discovery.

Searching arXiv for the cited ElbowSig-related works to ground the article in current records. ElbowSig is an overloaded term that appears in at least three distinct research contexts: a continuous elbow-angle time series produced from monocular RGB video for upper-limb motion capture, elbow-centered anthropomorphic time signals derived from a virtual skeletal arm for on-line signature verification, and a statistical framework for assessing multiscale clustering significance by formalizing the elbow heuristic (Yahya et al., 2018, Diaz et al., 15 Jan 2025, Perez-Reche, 3 Mar 2026). Across these usages, the common thread is not a single invariant algorithm, but the extraction or interpretation of diagnostically informative “elbow” signals: biomechanical in the first two cases, inferential in the third.

1. Terminological scope and disambiguation

In the motion-capture setting, ElbowSig denotes a continuous stream of elbow flexion/extension angles in degrees, produced in real time from RGB frames by estimating shoulder, elbow, and wrist keypoints and computing the elbow angle at each frame (Yahya et al., 2018). The underlying paper is "Real Time Elbow Angle Estimation Using Single RGB Camera" (Yahya et al., 2018), which presents a markerless and cost-effective solution using a single RGB camera and part affinity field inference.

In the on-line signature literature, ElbowSig refers to elbow-based anthropomorphic features extracted from pen trajectories through a Virtual Skeletal Arm (VSA) model. These signals include the elbow flexion angle $\theta_{\mathrm{elbow}}(t)$, its derivatives, and the 3D elbow trajectory $p_{\mathrm{elbow}}(t)$, all intended to characterize signer neuromotor style (Diaz et al., 15 Jan 2025). The 2024 paper on neural modelling of kinematic and dynamic signature features is adjacent rather than terminologically identical: it estimates angular positions, angular velocities, and joint torques from 2D trajectories with a small neural network, but it does not define ElbowSig by name (Diaz et al., 2024).

In clustering, ElbowSig is the explicit name of a framework that converts the heuristic elbow method into a formal hypothesis-testing procedure using a normalized discrete curvature statistic computed from the heterogeneity sequence $H_k$ (Perez-Reche, 3 Mar 2026). Here, the “signal” is not biomechanical; it is the scale-wise evidence that reductions in clustering heterogeneity depart from a null baseline.

This polysemy is consequential. A plausible implication is that any technical discussion of ElbowSig should identify the operative domain explicitly, since the same label can denote a time-domain biomechanical series, a set of anthropomorphic signature features, or a multiscale inferential statistic.

2. Monocular RGB ElbowSig for real-time elbow-angle estimation

The RGB-camera formulation begins with a monocular camera capturing frames at approximately $15$ fps while a participant performs a simple upper-limb task such as cup-to-mouth movement (Yahya et al., 2018). Each input image is processed by a convolutional neural network following the bottom-up part-affinity-field design of Cao et al. (2016): the network predicts confidence maps $P_1,\dots,P_G$ for body keypoints and part affinity fields $Q_1,\dots,Q_H$ for limb associations. Peaks in the confidence maps identify candidate joints, and line integrals over the corresponding affinity fields score limb connectivity, allowing shoulder $\rightarrow$ elbow $\rightarrow$ wrist to be assembled into a coherent articulated skeleton (Yahya et al., 2018).

From the right-arm skeleton, the pixel coordinates $X_s=(x_s,y_s)$, $X_e=(x_e,y_e)$, and $p_{\mathrm{elbow}}(t)$0 are extracted for shoulder, elbow, and wrist, respectively. Two vectors are formed at the elbow, $p_{\mathrm{elbow}}(t)$1 and $p_{\mathrm{elbow}}(t)$2, and the elbow angle is computed by

$p_{\mathrm{elbow}}(t)$3

The resulting $p_{\mathrm{elbow}}(t)$4 values are emitted frame by frame, yielding a continuous elbow-angle signal in real time (Yahya et al., 2018).

The CNN architecture uses a shared feature extractor $p_{\mathrm{elbow}}(t)$5 comprising the first $p_{\mathrm{elbow}}(t)$6 convolutional layers, followed by two parallel branches—one for confidence maps and one for affinity fields—unfolded over $p_{\mathrm{elbow}}(t)$7 stages, typically $p_{\mathrm{elbow}}(t)$8 (Yahya et al., 2018). At stage $p_{\mathrm{elbow}}(t)$9, the model refines both outputs according to

$H_k$0

where $H_k$1 and $H_k$2 are convolution-only subnets. Training uses stage-wise $H_k$3 losses for both map and field predictions, masked by $H_k$4 to exclude missing annotations, with total loss

$H_k$5

The implementation is explicitly markerless, single-view, and requires no pre-calibration (Yahya et al., 2018). Kinect is used only as a comparison device, operating at $H_k$6 fps with a depth-based skeleton API via the C# Coding4Fun toolkit; RGB frames are timestamp-matched to subsampled Kinect frames with millisecond accuracy (Yahya et al., 2018). Within this formulation, ElbowSig is therefore a derived kinematic observable obtained from 2D pose estimation rather than a direct sensor measurement.

3. Experimental behavior and limitations of the RGB signal

The evaluation in (Yahya et al., 2018) recruited five healthy volunteers with height $H_k$7 cm and mass $H_k$8 kg, each performing three cup-to-mouth trials in both sagittal and coronal planes. The metric was RMS error between the RGB-camera elbow angle and the Kinect elbow angle after time synchronization. Median RMS error over all subjects was approximately $H_k$9 in the sagittal plane and approximately $15$0 in the coronal plane (Yahya et al., 2018).

Per-subject RMSE values reported in Table 1 are as follows:

Subject	Sagittal RMSE	Coronal RMSE
1	1.5°	1.0°
2	4.0°	2.0°
3	2.0°	0.5°
4	2.5°	3.5°
5	0.5°	1.0°

The paper states that markerless and cost-effective RGB camera has a median RMS errors of $15$1 and $15$2 in sagittal and coronal plane respectively as compared to Microsoft Kinect (Yahya et al., 2018). It further characterizes the significance as under $15$3 error in coronal tasks and $15$4–$15$5 in sagittal, and describes this as sufficient accuracy for many rehabilitation applications (Yahya et al., 2018).

The stated limitations are specific. Lower accuracy in the sagittal plane is largely attributed to wrist-keypoint localization errors caused by limb overlaps and foreshortening (Yahya et al., 2018). The proof of concept handles a single, relatively unobstructed subject; multi-person and cluttered scenes remain challenging, and frame rate is limited by single-view CNN inference at roughly $15$6 fps (Yahya et al., 2018). Proposed extensions include improving wrist detection via temporal filtering or finer PAF stages, extending to multi-person scenarios and non-rigid backgrounds, fusing a second camera or lightweight IMUs for depth and occlusion robustness, and incorporating domain-specific temporal smoothing to reduce jitter in ElbowSig (Yahya et al., 2018). This suggests that, in this usage, ElbowSig is best understood as a real-time but perception-limited kinematic signal whose quality depends heavily on distal keypoint localization.

4. Anthropomorphic ElbowSig in on-line signature verification

In the on-line signature setting, ElbowSig is grounded in a Virtual Skeletal Arm model that represents signing as the motion of a serial chain with six revolute joints plus a pen-holding link (Diaz et al., 15 Jan 2025). Joint $15$7 is the elbow flexion-extension degree of freedom and joint $15$8 is elbow pronation/supination. For joints $15$9, the Denavit–Hartenberg parameters reported are: for $P_1,\dots,P_G$0, $P_1,\dots,P_G$1, $P_1,\dots,P_G$2, $P_1,\dots,P_G$3, $P_1,\dots,P_G$4 shoulder roll; for $P_1,\dots,P_G$5, $P_1,\dots,P_G$6, $P_1,\dots,P_G$7, $P_1,\dots,P_G$8, $P_1,\dots,P_G$9 elbow flexion; and for $Q_1,\dots,Q_H$0, $Q_1,\dots,Q_H$1, $Q_1,\dots,Q_H$2, $Q_1,\dots,Q_H$3, $Q_1,\dots,Q_H$4 pronation axis (Diaz et al., 15 Jan 2025). The frame transform is given by the standard homogeneous matrix

$Q_1,\dots,Q_H$5

and the chain to the elbow-related frame is

$Q_1,\dots,Q_H$6

Pen pose is lifted into the VSA end-effector frame via

$Q_1,\dots,Q_H$7

where $Q_1,\dots,Q_H$8 is built from recorded $Q_1,\dots,Q_H$9, lift $\rightarrow$0, azimuth, and inclination (Diaz et al., 15 Jan 2025). In practice, inverse kinematics are solved from $\rightarrow$1 in order to recover $\rightarrow$2, after which elbow position is computed by forward kinematics.

The elbow flexion angle is then obtained by decoupling the proximal chain from the forearm and wrist. Let

$\rightarrow$3

then

$\rightarrow$4

$\rightarrow$5

$\rightarrow$6

and by the law of cosines

$\rightarrow$7

The closed-form elbow angle reported is

$\rightarrow$8

Once $\rightarrow$9 and $\rightarrow$0 are available, the extracted elbow-based time signals are: $\rightarrow$1, $\rightarrow$2, $\rightarrow$3, $\rightarrow$4, $\rightarrow$5, and $\rightarrow$6 (Diaz et al., 15 Jan 2025). The source text interprets these physically: $\rightarrow$7 captures how much the signer bends the elbow, $\rightarrow$8 and $\rightarrow$9 measure smoothness and neuromotor control, and $X_s=(x_s,y_s)$0 encodes global arm posture and reach (Diaz et al., 15 Jan 2025).

5. Verification performance and relation to neural kinematic modelling

The VSA-based elbow features were evaluated on MCYT-100, MCYT-330, BiosecurID-SONOF, SUSIG, mobile SG-NOTE, and OnOffSig Bengali/Devanagari datasets (Diaz et al., 15 Jan 2025). The protocol used five genuine signatures for training per signer, with remaining genuine signatures and random and skilled forgeries for testing; Equal-Error-Rate was reported, and FRR/FAR at operating points were also measured (Diaz et al., 15 Jan 2025). On MCYT-100 with a DTW-based verifier, the example results reported are: position-only (3D joints) with $X_s=(x_s,y_s)$1 and $X_s=(x_s,y_s)$2; elbow-angle-only $X_s=(x_s,y_s)$3 with $X_s=(x_s,y_s)$4 and $X_s=(x_s,y_s)$5; and score-level fusion with $X_s=(x_s,y_s)$6 and $X_s=(x_s,y_s)$7, described as an approximately $X_s=(x_s,y_s)$8 relative skilled-forgery gain (Diaz et al., 15 Jan 2025). Across all six corpora and two verifier back-ends, adding elbow-based dynamic features consistently reduced EER, especially for skilled forgeries, by $X_s=(x_s,y_s)$9–$X_e=(x_e,y_e)$0 relative over a strong position-only baseline (Diaz et al., 15 Jan 2025).

The paper "Neural network modelling of kinematic and dynamic features for signature verification" (Diaz et al., 2024) is relevant because it addresses the same broader objective—recovering latent arm kinematics and dynamics from 2D signature traces—but with a different methodology. Instead of a VSA inverse-kinematics pipeline, it frames the task as multi-target regression from sliding windows of trajectory points to $X_e=(x_e,y_e)$1 outputs: $X_e=(x_e,y_e)$2 (Diaz et al., 2024). The MLP uses $X_e=(x_e,y_e)$3 input features, one hidden layer with $X_e=(x_e,y_e)$4 ReLU units, Dropout$X_e=(x_e,y_e)$5, and three output heads with $X_e=(x_e,y_e)$6 sigmoid units each (Diaz et al., 2024). Training on MCYT300 uses

$X_e=(x_e,y_e)$7

with Adam at learning rate $X_e=(x_e,y_e)$8 and early stopping on a $X_e=(x_e,y_e)$9 validation subset (Diaz et al., 2024).

The reported estimation accuracy on DS1 shows that the MLP is markedly stronger than bidirectional RNN, LSTM, and GRU for angular velocities and torques, while sequence models are slightly better on angular positions (Diaz et al., 2024). For example, MLP MAE/MSE are $p_{\mathrm{elbow}}(t)$00 for $p_{\mathrm{elbow}}(t)$01, $p_{\mathrm{elbow}}(t)$02 for $p_{\mathrm{elbow}}(t)$03, and $p_{\mathrm{elbow}}(t)$04 for $p_{\mathrm{elbow}}(t)$05, whereas torque errors for the sequence baselines are much larger (Diaz et al., 2024). In verification on DS1, MLP-estimated features outperform UR5e-based features for both random and skilled forgery EERs in several cases; for instance, for random forgeries, $p_{\mathrm{elbow}}(t)$06 yields $p_{\mathrm{elbow}}(t)$07 for UR5e-based features versus $p_{\mathrm{elbow}}(t)$08 for MLP-estimated features (Diaz et al., 2024). Cross-dataset tests further indicate that angular velocities and torques generalize better than angular positions (Diaz et al., 2024). A plausible implication is that elbow-centered signature signals can be embedded within a broader family of latent kinematic descriptors that need not be computed exclusively through explicit biomechanical inversion.

6. ElbowSig as multiscale clustering significance

In "The elbow statistic: Multiscale clustering statistical significance" (Perez-Reche, 3 Mar 2026), ElbowSig is not a biomechanical time signal but a framework for testing whether apparent elbows in the clustering heterogeneity curve correspond to statistically meaningful structure. Let $p_{\mathrm{elbow}}(t)$09 and let a clustering algorithm with $p_{\mathrm{elbow}}(t)$10 clusters yield parameters $p_{\mathrm{elbow}}(t)$11 and a non-negative point-wise heterogeneity function $p_{\mathrm{elbow}}(t)$12. The total heterogeneity is defined as

$p_{\mathrm{elbow}}(t)$13

with $p_{\mathrm{elbow}}(t)$14 (Perez-Reche, 3 Mar 2026). The framework is algorithm-agnostic and accommodates hard, fuzzy, and model-based clustering through different choices of $p_{\mathrm{elbow}}(t)$15.

The first and second discrete differences are

$p_{\mathrm{elbow}}(t)$16

for $p_{\mathrm{elbow}}(t)$17, and the normalized discrete curvature, termed the elbow statistic, is

$p_{\mathrm{elbow}}(t)$18

Local peaks in $p_{\mathrm{elbow}}(t)$19 indicate values of $p_{\mathrm{elbow}}(t)$20 at which the reduction in heterogeneity slows abruptly, which the paper interprets as statistically meaningful cluster resolutions (Perez-Reche, 3 Mar 2026).

The null theory is developed in two asymptotic regimes. In the large-sample limit with fixed $p_{\mathrm{elbow}}(t)$21 and $p_{\mathrm{elbow}}(t)$22, one has

$p_{\mathrm{elbow}}(t)$23

hence

$p_{\mathrm{elbow}}(t)$24

For uniform data on bounded support in the optimal quantization regime, $p_{\mathrm{elbow}}(t)$25, giving

$p_{\mathrm{elbow}}(t)$26

for large $p_{\mathrm{elbow}}(t)$27 (Perez-Reche, 3 Mar 2026). In the high-dimensional limit with fixed $p_{\mathrm{elbow}}(t)$28 and $p_{\mathrm{elbow}}(t)$29 under isotropic sub-Gaussian assumptions,

$p_{\mathrm{elbow}}(t)$30

so

$p_{\mathrm{elbow}}(t)$31

Special cases are also given: for hard-clustering inertia, $p_{\mathrm{elbow}}(t)$32 implies $p_{\mathrm{elbow}}(t)$33 and $p_{\mathrm{elbow}}(t)$34; for GMM, asymptotically

$p_{\mathrm{elbow}}(t)$35

(Perez-Reche, 3 Mar 2026). The paper’s central claim is that the null elbow statistic concentrates around a known smooth baseline with vanishing variance, and that departures above this baseline at specific $p_{\mathrm{elbow}}(t)$36 indicate genuine clustering structure.

The algorithm takes as input the data, clustering method, $p_{\mathrm{elbow}}(t)$37, number of references $p_{\mathrm{elbow}}(t)$38, per-scale error $p_{\mathrm{elbow}}(t)$39, and FDR level $p_{\mathrm{elbow}}(t)$40 (Perez-Reche, 3 Mar 2026). Reference data are generated either by a bounding-box or PCA-aligned uniform scheme; empirical $p_{\mathrm{elbow}}(t)$41-values are computed as

$p_{\mathrm{elbow}}(t)$42

followed by a subsampling-based per-scale significance threshold and Benjamini–Hochberg FDR adjustment (Perez-Reche, 3 Mar 2026). Under the null, Appendix A calibration ensures exact per-scale Type I rate $p_{\mathrm{elbow}}(t)$43, while FDR control bounds the expected false-discovery proportion by $p_{\mathrm{elbow}}(t)$44 (Perez-Reche, 3 Mar 2026).

Empirically, unstructured experiments show per-scale detections at $p_{\mathrm{elbow}}(t)$45 at an approximately $p_{\mathrm{elbow}}(t)$46 nominal rate, and global FDR reduces false positives to below $p_{\mathrm{elbow}}(t)$47 in high-dimensional settings (Perez-Reche, 3 Mar 2026). In Gaussian-mixture power simulations, ElbowSig identifies the true number of components in more than $p_{\mathrm{elbow}}(t)$48 of replicates under moderate cluster separation, while also flagging coarser resolutions when components overlap (Perez-Reche, 3 Mar 2026). On real datasets, it detects multiscale structure in Iris, breast cancer, Campylobacter host, human populations, and insulin resistance profiles, with retained significant $p_{\mathrm{elbow}}(t)$49 depending on reference generator and multiplicity correction (Perez-Reche, 3 Mar 2026). This suggests that the clustering sense of ElbowSig is best viewed as an inferential generalization of the traditional elbow heuristic rather than a model-selection rule that returns a single $p_{\mathrm{elbow}}(t)$50.

7. Common themes, misconceptions, and research directions

A common misconception would be to treat ElbowSig as a single standardized method. The evidence does not support that interpretation. In (Yahya et al., 2018), ElbowSig is a real-time elbow-angle stream from 2D keypoints; in (Diaz et al., 15 Jan 2025), it is a family of anthropomorphic elbow time signals reconstructed from pen pose; in (Perez-Reche, 3 Mar 2026), it is a statistical significance framework for clustering. These usages are conceptually connected by the notion of an “elbow” as a salient locus of information, but they are methodologically independent.

Another possible misconception is that all ElbowSig variants are direct measurements. The RGB and VSA formulations are both derived signals: one from image-based pose estimation, the other from kinematic reconstruction and inverse kinematics (Yahya et al., 2018, Diaz et al., 15 Jan 2025). Likewise, the clustering formulation is not a descriptive curvature plot alone; it explicitly constructs empirical null distributions, computes $p_{\mathrm{elbow}}(t)$51-values, and applies per-scale or FDR-controlled significance rules (Perez-Reche, 3 Mar 2026).

Future directions stated in the sources remain domain-specific. For RGB elbow-angle estimation, the proposed extensions are temporal filtering, finer PAF stages, multi-person handling, non-rigid backgrounds, second-camera or IMU fusion, and temporal smoothing for jitter reduction (Yahya et al., 2018). For anthropomorphic signature features, the stated optimizations include fixing or denoising pen orientation, calibrating link lengths to individual anthropometrics, accelerating inverse kinematics through lookup tables or vectorization, and enriching the joint model with shoulder or compliance effects (Diaz et al., 15 Jan 2025). For learned kinematic signature modelling, the proposed improvements include anthropomorphic robot redesign, attention mechanisms, more interpretable architectures, per-signer calibration of D–H parameters or torque coefficients, and deeper or graph-based skeletal models (Diaz et al., 2024). For clustering ElbowSig, the significance is its compatibility with hard, fuzzy, and model-based methods and its explicit multiscale perspective, which contrasts with traditional criteria such as Gap, CH, DB, and Silhouette that the paper describes as lacking either error control or multiscale awareness (Perez-Reche, 3 Mar 2026).

Taken together, the literature shows that ElbowSig functions less as a singular technical artifact than as a domain-dependent construct for encoding informative elbow-related structure—whether biomechanical or statistical—into analyzable signals (Yahya et al., 2018, Diaz et al., 15 Jan 2025, Perez-Reche, 3 Mar 2026).