A Generalized Iterative Polar Transform for Readable Time-Series Imaging

1. Introduction

Time series classification is a central problem in data mining [1,2,3], with applications in medicine, industrial monitoring, finance, and the physical sciences. One productive strategy converts each series into an image and then applies the mature toolbox of convolutional neural networks [4,5,6]. The conversion step, which we call an encoder, is where most of the design effort goes, and a range of encoders is now standard: Gramian Angular Fields and Markov Transition Fields [7], recurrence plots [8,9], spectrograms from the short-time Fourier transform [10], scalograms from the continuous wavelet transform, and polar or spiral embeddings that lay the trajectory out in polar coordinates [11].

These encoders are almost always evaluated by a single criterion: the accuracy of a classifier trained on the resulting images. There is, however, a second property that matters whenever a person must audit, trust, debug, or directly act on the representation. We call it legibility: can a human look at the encoded image and tell which class it belongs to? For the encoders listed above the answer is usually no. The class information is present, since a network recovers it, but it survives only as a distributed, low-contrast perturbation that the human visual system, tuned to salient shape, contrast, and topology, cannot pick up. The image is accurate fuel for a network and visual noise for a person.

This paper studies a specific polar-spiral encoder that produces good accuracy numbers and illegible images, and asks a direct question: can we change the encoder so that its images become classifiable by the unaided eye, without giving up classifier accuracy? Our answer is the Generalized Iterative Polar Transform (GIPT), a family of polar encodings arrived at through a sequence of designed and iterated variants, each one rendered and inspected, and each one measured both for classifier accuracy and for legibility. Two members of the family carry the results of this paper: GIPT-H, the single-channel human-facing silhouette, and GIPT-MC, its aligned multi-channel extension for convolutional networks.

Contributions. 

• A geometric diagnosis (Section 3.3) proving why the existing spiral encoder is opaque: the time term dominates the radius and drives the plotted angle toward zero, so the trajectory collapses into a thin wedge and the turns parameter is inert. This explains the diminishing returns the original work observed when stacking more spirals.
• GIPT, a principled readable encoding (Section 3.4): swap the polar axes so time drives the angle and value drives the radius, fill the silhouette so the network sees area rather than a one-pixel curve, shade the interior by local value, and choose the normalization deliberately. We prove a lobe-counting property making a frequency difference countable by eye.
• An analysis of the readability-accuracy trade-off (Section 6.0.0.1), which we show is controlled entirely by normalization.
• A one-shot classifier (Section 3.5): because GIPT enforces cross-sample consistency, a training-free nearest-neighbor classifier on the encoded images matches or beats the convolutional network at a fraction of the cost. This is a significant practical result that turns the auditability of the representation into a cheap, instant classifier.
• A paired evaluation protocol and a thorough empirical study (Section 4 and Section 5), coupling a stabilized multi-seed accuracy benchmark, a comparison against DTW nearest-neighbor and ROCKET with statistical significance testing, an ablation and a noise-robustness analysis, a vision zero-shot legibility test, and a reproducible human test-set protocol.

Summary of findings. 

GIPT’s structure does not cost accuracy: GIPT-MC is competitive with the opaque baseline, and the residual tension lives entirely in the normalization, resolved by rendering the human-facing and network-facing images separately. The same cross-sample consistency that makes the images readable makes them nearest-neighbor friendly: on twenty-eight datasets the one-shot classifier averages $83.1\%$ against the network’s $76.0\%$ at roughly $1,200$ times lower cost, statistically on par with the classical DTW nearest-neighbor standard (Section 5.3).

2. Related Work

Time series classification. 

The field offers nearest-neighbor rules under elastic distances [12,13], symbolic and dictionary methods [14,15,16], shapelets [17,18], heterogeneous ensembles [19,20], random convolution kernels [21,22], and deep networks [23,24,25,26,27], with large comparative studies on the UCR and UEA archives [2,3,28,29]. None of these measures whether the intermediate representation a method builds is readable by a person, which is the axis this paper adds.

Imaging time series. 

Standard encoders include Gramian Angular and Markov Transition Fields [7], recurrence plots [8,9,30], spectrograms and scalograms [10], and stacked combinations classified with convolutional backbones inherited from vision [4,5,6,11]. Polar and spiral embeddings, the family studied here, rasterize the trajectory directly in polar coordinates. In every case the encoder is tuned and reported on classifier accuracy alone; the visual legibility of the produced image is not a stated objective.

Interpretability. 

Interpretability work in this setting is largely post hoc: saliency maps [31,32] or local surrogates [33,34] explain the classifier rather than the representation, and can themselves be unreliable [35]. A parallel line argues for inherently interpretable models [36] and for precise, testable definitions of interpretability [37,38]. We ask instead for intrinsic legibility, that the representation itself be readable by a person before any network is involved, a property not previously quantified for time-series imaging encoders.

Distance-based, one-shot, and model-judged evaluation. 

1-NN with an elastic distance [12,39] is a famously strong time-series baseline [2,13,28], and one-shot learning in vision classically learns embeddings or metric networks [40,41,42]. Our readable image space needs no learned embedding at all, because cross-sample consistency, the property a nearest-neighbor rule exploits, is designed in. Finally, multimodal models classify images zero-shot [43,44] and model-as-judge protocols increasingly stand in for human raters [45]; we repurpose this capability as a legibility instrument and pair it with a reproducible human test-set protocol.

3. Methodology

This section develops the technical core of the paper in five steps: we formalize the two objectives (Section 3.1), summarize the existing encoder (Section 3.2), prove why its images are unreadable (Section 3.3), derive the readable encoding (Section 3.4), and introduce the training-free one-shot classifier that the readable encoding enables (Section 3.5).

3.1. Problem Formulation

A univariate time series of length n is $\mathbf{s}=(s_0,\dots ,s_{n-1})\in {\mathbb{R}}^n$. An encoder is a map $\mathsf{\Phi }:{\mathbb{R}}^n\to {\mathbb{R}}^{C\times H\times W}$, producing a C-channel image classified by a convolutional network $f_{\theta }:{\mathbb{R}}^{C\times H\times W}\to {\mathbb{R}}^K$ over K classes. We write the time fraction ${\varphi }_i=i/(n-1)\in [0,1]$. The image has center $(\frac{H}{2},\frac{W}{2})$, and a polar pair $(r,\vartheta )$ maps to a pixel by

$$\mathrm{row}(r,\vartheta )=\frac{H}{2}-rsin\vartheta ,\mathrm{col}(r,\vartheta )=\frac{W}{2}+rcos\vartheta .$$

(1)

We make explicit the two objectives the paper balances. (G1) Accuracy: the classifier applied to $\mathsf{\Phi }\left(\mathbf{s}\right)$ should reach high test accuracy. (G2) Legibility: a human shown a few labeled examples per class should be able to classify a new $\mathsf{\Phi }\left(\mathbf{s}\right)$ by eye. The original encoder targets only G1; this paper adds G2. Because the analysis necessarily juggles the coordinates of two different encoders, Table 1 collects every recurring symbol in one place; the reader can treat it as a legend for Section 3.2 to Section 3.5.

3.2. The Existing Polar-Spiral Encoder

We summarize the encoder under study because the diagnosis depends on its exact form; we refer to this original, opaque encoder as OLD in the text and in every table. Let $\tau$ be a turns parameter. The encoder normalizes the series with a robust median / median-absolute-deviation rule,

$${\tilde{s}}_i={\displaystyle \frac{s_i-med\left(\mathbf{s}\right)}{3MAD\left(\mathbf{s}\right)}},MAD\left(\mathbf{s}\right)={med}_i\left(|s_i-med\left(\mathbf{s}\right)|\right),$$

(2)

defines $t_i=2\pi \tau {\varphi }_i\in [0,2\pi \tau ]$, and sets

$$r_i=\sqrt{t_i^2+{\left(2{\tilde{s}}_i\right)}^2},{\theta }_i=atan2({\tilde{s}}_i,t_i).$$

(3)

The radius is rescaled per series by min and max, points are placed inward or outward, and mass is accumulated into a blurred 2D histogram. Multi-spiral variants repeat this for several $\tau$ and fuse the results into three channels. Figure 1 shows the result: a small colored glyph in a largely empty field. The next section explains why.

3.3. Why the Images Are Not Readable

Four failure modes by inspection.

(1) The signal occupies a small fraction of the canvas, so there is almost nothing to fixate on. (2) Samples within a class do not resemble each other, so the eye cannot form a template. (3) The classes share the same visual vocabulary and differ only in subtle proportions. (4) Three unrelated features overlap spatially in the color channels, so the colors mix into mud.

The geometric cause.

Failure modes (1) to (3) follow from one fact about Equation (3). The time variable $t_i$ ranges up to $2\pi \tau$; with $\tau \in [1,10]$ this is roughly 30 to 60, while the value term $2{\tilde{s}}_i$ is of order one. Hence the time term dominates the radius:

$$r_i=t_i\sqrt{1+{\left(\frac{2{\tilde{s}}_i}{t_i}\right)}^2}\approx t_i.$$

(4)

After per-series rescaling ${\widehat{r}}_i\approx {\varphi }_i$, a deterministic ramp in time. The plotted angle satisfies

$${\theta }_i=atan2({\tilde{s}}_i,t_i)\approx {\displaystyle \frac{{\tilde{s}}_i}{2\pi \tau {\varphi }_i}}\underset{i\mathrm{large}}{\to }0.$$

(5)

So points pile into a thin wedge near the positive horizontal axis.

Proposition 1  

(Canvas collapse and inert turns). Under Equation (3) with $t_i=2\pi \tau {\varphi }_i$ and bounded values $2\left|{\tilde{s}}_i\right|\le M$: (i) for any fixed ${\varphi }_0\in (0,1]$, every point with ${\varphi }_i\ge {\varphi }_0$ has plotted angle $\left|{\theta }_i\right|\le M/\left(2\pi \tau {\varphi }_0\right)$, so the signal-bearing trajectory is confined to an angular wedge whose width is $O(M/\tau )$ and tends to zero as τ grows; (ii) the turns parameter τ does not increase the number of visible turns: it only rescales $t_i$, which is removed by the per-series min and max in ${\widehat{r}}_i$, while making the angle ${\theta }_i$ smaller.

Point (ii) explains the diminishing returns when the original work stacks five, seven, and nine spirals: raising $\tau$ produces near-duplicate, even more collapsed channels. The discriminative content survives only as a tiny distributed perturbation, which a network integrates statistically while the eye sees near-identical smears. The technical appendix (supplementary material) gives the full proof and a quantitative occupancy bound: the signal-bearing region covers at most an $O\left({\tau }^{-2/3}\right)$ fraction of the canvas, about one eighth at $\tau =3$.

3.4. GIPT: A Readable Polar Encoding

We now derive the proposed Generalized Iterative Polar Transform (GIPT): it generalizes the polar embedding by decoupling which signal feature drives which polar coordinate, and it was reached iteratively through rendered and measured candidates (Section 5.1). The diagnosis suggests four design principles: (P1) fill the canvas; (P2) cross-sample consistency, so two samples of one class look alike; (P3) categorical inter-class difference, mapping the class property to a large-scale shape difference; (P4) clean contrast and color.

Swap the axes.

The central change inverts the two polar coordinates: time drives the angle, so the trajectory winds around and fills the plane (P1), and value drives the radius, so the waveform becomes the visible outline (P3). With $v_i=\mathcal{N}\left(s_i\right)\in [0,1]$,

$${\vartheta }_i=2\pi \tau {\varphi }_i,{\rho }_i=\left(r_{min}+(r_{max}-r_{min})v_i\right)\frac{H}{2},$$

(6)

with $r_{min}=0.10$, $r_{max}=0.97$. For one turn this is a closed polar curve whose radius traces the waveform.

Proposition 2  

(Lobe counting). Let $\tau =1$ and $v\left(\vartheta \right)=\frac{1}{2}(1+sin(\omega \vartheta +\phi ))$ for integer frequency ω and $\vartheta \in [0,2\pi )$. Then $\rho \left(\vartheta \right)$ in Equation (6) attains exactly ω local maxima, so the closed curve shows exactly ω lobes.

Proof. 

$\rho$ is an increasing affine function of v, so its maxima coincide with those of $sin(\omega \vartheta +\phi )$, which completes $\omega$ periods on $[0,2\pi )$ and has exactly $\omega$ maxima.    □

Thus a 3 Hz versus 5 Hz difference appears as a three- versus five-petal shape a person can count (rendered in the technical appendix). The technical appendix extends the statement to general band-limited profiles and shows the count is stable under noise.

Filled silhouette operator.

A one-pixel curve activates only $O\left(H\right)$ of $H^2$ pixels, so a convolutional filter mostly sees zeros. We therefore fill the shape. For a feature profile $u:\{0,\dots ,n-1\}\to [0,1]$ we resample u in angle to get $\overline{u}\left(\alpha \right)$ and fill each spoke from the center out to the profile,

$${\mathcal{S}}_1\left[u\right](r,\alpha )=\left\{\begin{array}{cc}1.0,\hfill & r=\frac{H}{2}\overline{u}\left(\alpha \right)\left(\mathrm{outline}\right),\hfill \\ g\left(\alpha \right),\hfill & 0\le r<\frac{H}{2}\overline{u}\left(\alpha \right)\left(\mathrm{interior}\right),\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(7)

with shaded interior $g\left(\alpha \right)=0.25+0.6\overline{u}\left(\alpha \right)$. The technical appendix gives the vectorized procedure.

Normalization.

The map $\mathcal{N}$ to the unit interval is the crux of the whole problem:

$$\begin{array}{cc}\hfill \mathrm{logistic}\text{:}& \mathcal{N}\left(s_i\right)=\sigma \left(\frac{s_i-med\left(\mathbf{s}\right)}{4MAD\left(\mathbf{s}\right)}\right);\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \min \text{-}\max \text{:}& \mathcal{N}\left(s_i\right)=\frac{s_i-{min}_j{s}_j}{{max}_j{s}_j-{min}_j{s}_j};\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \mathrm{global}\text{:}& \mathcal{N}\left(s_i\right)=\mathrm{clip}\left(\frac{s_i-\ell }{h-\ell },0,1\right),(\ell ,h)=(P_1,P_{99}),\hfill \end{array}$$

(10)

where the global percentiles $(\ell ,h)$ are fit once on the pooled training values. The logistic map preserves relative amplitude but compresses contrast; min-max uses the full range of each image but differs sample to sample; the global map uses a wide range and the same mapping for every sample. The technical appendix makes this precise: per-image min-max is invariant to affine changes of the input and therefore annihilates amplitude information, while the global map is a single Lipschitz function of the data, which is exactly the consistency property P2 requires.

Multi-channel reconciliation.

A single filled value silhouette is the most legible image but not the most accurate. A person reads one channel; the network can consume several. We stack three aligned silhouettes sharing the angular frame, so the composite stays one coherent shape. With value field v, gradient field $d=\mathcal{N}\left(\right|\nabla \mathbf{s}\left|\right)$, and a lightly smoothed field m,

$$\mathsf{\Phi }\left(\mathbf{s}\right)=\left[{\mathcal{S}}_1\left[v\right],{\mathcal{S}}_1\left[d\right],{\mathcal{S}}_1\left[m\right]\right].$$

(11)

Channel one is the image a human classifies; channel two places the derivative magnitude (edges, overshoots) on the same silhouette; channel three carries a low-frequency trend. We call the single-channel human-facing encoder GIPT-H ($\mathsf{\Phi }={\mathcal{S}}_1\left[v\right]$, shaded) and this aligned multi-channel extension GIPT-MC; these two names are used in every table.

3.5. One-Shot Nearest-Neighbor Classification

A reviewer-style question motivates this section: is a full convolutional network necessary, or can a cheap classifier exploit the readable structure? We evaluate a parameter-free one-shot classifier: 1-nearest-neighbor [39] under Euclidean distance on the flattened encoded images, with no training whatsoever. Unlike one-shot learning methods that meta-train an embedding [41,42], no component of this classifier is fit to data. The prediction for a test image is the label of the closest training image,

$$\widehat{y}\left(\mathbf{x}\right)=y_{arg{min}_j{\parallel \mathsf{\Phi }\left(\mathbf{x}\right)-\mathsf{\Phi }\left({\mathbf{x}}_j\right)\parallel }_2}.$$

(12)

This is the natural classifier for our design, because principle P2 (cross-sample consistency) is exactly the property a nearest-neighbor rule needs: if two samples of a class produce visually similar silhouettes, they are close in pixel space.

The result, reported in Table 3, is significant: on the encoded silhouettes the one-shot classifier matches and frequently exceeds the convolutional network, while requiring no gradient steps and running about three orders of magnitude faster in wall-clock time. The readable global-normalized encoding, which is the most consistent across samples, is especially nearest-neighbor friendly. In other words, the same property we introduced for human legibility yields a fast, training-free, and competitive classifier as a free by-product. The technical appendix explains this effect at a deeper level: Euclidean distance between filled silhouettes is metrically equivalent to a robust $L_1$ distance between the underlying normalized series, so the encoder is not merely a visualization but an embedding of a good series metric into pixel space. The technical appendix gives the cost accounting behind the wall-clock ratio.

4. Experimental Setup

Datasets. 

We evaluate on a broad suite of univariate datasets from the UCR archive [2,28]; the encoder as defined takes a single-channel series, and we do not evaluate on the multivariate UEA archive (see the limitations in Section 6 for the multivariate extension). The suite spans binary and multi-class problems, short and long series, and morphological, spectroscopic, and shape-based class differences. The core five-seed comparison uses twelve datasets (table in the technical appendix); the expanded study in Table 2 covers the larger suite. The technical appendix gives per-dataset notes.

Classifier and training. 

The classifier is a three-block convolutional network: $3\times 3$ convolutions with $32,64,128$ channels, each followed by GroupNorm [46] (eight groups) and ReLU, with $2\times 2$ max-pooling after the first two blocks, then global average pooling and a linear head. We optimize cross-entropy with Adam [47] at ${10}^{-3}$ for a fixed budget of fifty epochs on the full training set, and report mean and standard deviation over multiple seeds.

A methodological note on stability. 

Early stopping on a held-out split is counterproductive at this data scale: with twenty-three training series a validation split is five examples, and training halts at a majority-class predictor (even the baseline fell to chance on several datasets). A fixed epoch budget on the full training set, with GroupNorm replacing batch normalization [48] to remove batch dependence, restored sensible numbers; we report this because it is an easily made error on small UCR sets.

Vision zero-shot legibility protocol. 

To turn legibility into a number we render a reference sheet of a few labeled examples per class and a query sheet of unlabeled, numbered test images in shuffled order (rendered in the technical appendix). A grader studies the reference and then names each query; accuracy against the hidden truth measures legibility on the same footing as classifier accuracy, with all predictions committed before the truth is revealed.

5. Results

5.1. From Iterations To The Final Encoder

An exploratory single-seed study on four datasets (reported in full in the technical appendix) shaped the design: the bare axis-swapped curve is legible but fragile, filling the silhouette recovers accuracy, and spiral bands help only with a dataset-dependent number of turns; these iterations fixed the filled one-turn silhouette and motivated multi-seed evaluation. On the twelve core datasets with five seeds, GIPT-MC (min-max) then matches and slightly exceeds the OLD baseline on average ($75.3$ vs $73.8$), winning six of twelve datasets including a perfect score on Trace (full table in the technical appendix). Structurally, a readable polar silhouette does not cost classifier accuracy.

5.2. Expanded Benchmark and One-Shot Classifier

Table 2 reports the expanded study on the larger univariate UCR suite, comparing the convolutional network against the one-shot nearest-neighbor classifier of Section 3.5 on the same encoded images. Two findings stand out. First, the one-shot classifier is competitive with, and on many datasets superior to, the trained network, confirming that the readable encoding places same-class samples close together in pixel space. Second, it does so at a tiny fraction of the cost: the network requires fifty epochs of gradient descent per seed, whereas the one-shot classifier requires no training at all. Table 3 summarizes the head-to-head and the speedup.

A particularly telling pattern concerns the human-readable global-normalized channel GIPT-H (column H). It is the weakest representation under the convolutional network, because the single global channel offers the network little texture, yet it is among the strongest under the one-shot classifier: on Coffee its one-shot accuracy is $96.4\%$ against a network accuracy of $53.6\%$, and on ItalyPowerDemand $96.2\%$ against $50.1\%$. The reason is precisely principle P2. The global normalization makes same-class silhouettes nearly congruent, which is exactly what a nearest-neighbor rule rewards and what a convolutional network, deprived of discriminative texture, cannot exploit. The representation we designed for human legibility is therefore best classified not by a large trained network but by the simplest possible distance rule, a satisfying alignment of the three goals of legibility, accuracy, and cheapness.

Table 2. Expanded UCR benchmark, twenty-eight univariate datasets (% test accuracy). Left block: the trained CNN (mean over three seeds) on OLD (the opaque baseline), MC (GIPT-MC, per-image min-max), and H (the GIPT-H channel, global normalization). Right block: the training-free one-shot 1-NN classifier (Eq. 12) on the same encoded images, and on the z-normalized raw series as a reference.

	trained CNN			training-free 1-NN
dataset	OLD	MC	H	MC	H	raw
GunPoint	86.7	78.4	56.2	89.3	91.3	91.3
Coffee	75.0	83.3	53.6	100.0	96.4	100.0
ECG200	73.0	75.3	64.0	86.0	90.0	88.0
ItalyPower.	85.3	63.8	54.2	96.0	96.2	95.5
TwoLeadECG	68.1	79.3	50.0	78.2	71.3	74.7
ECGFiveDays	74.6	69.5	49.7	84.0	80.4	79.7
MoteStrain	79.2	80.5	62.2	83.0	83.4	87.9
BeetleFly	68.3	51.7	58.3	70.0	70.0	75.0
BirdChicken	75.0	85.0	73.3	65.0	65.0	55.0
Plane	89.8	79.7	49.8	99.0	98.1	96.2
Trace	76.3	100.0	38.7	100.0	79.0	76.0
Wine	50.0	50.0	50.0	61.1	63.0	61.1
SonyAIBO1	42.9	81.6	42.9	70.7	72.0	69.6
SonyAIBO2	71.4	76.5	61.7	84.6	88.4	85.9
Chinatown	75.9	68.1	76.0	92.1	92.1	95.3
PowerCons	84.6	82.0	93.1	91.7	93.9	93.3
GunPtAgeSpan	89.5	85.9	58.0	94.6	98.4	89.9
GunPtMvF	95.6	99.4	86.8	98.4	99.1	97.5
ArrowHead	49.5	48.6	36.6	78.3	76.6	80.0
Ham	63.5	59.7	52.7	62.9	48.6	60.0
Herring	59.4	59.4	59.4	56.2	50.0	51.6
ToeSeg1	67.8	83.2	49.1	72.4	71.1	68.0
ToeSeg2	58.2	77.9	60.5	76.9	83.1	80.8
DistPhalanxOC	68.7	70.2	69.9	73.9	68.5	71.7
ProxPhalanxOC	73.8	73.9	68.3	79.7	80.1	80.8
Strawberry	86.8	90.2	87.7	94.1	95.4	94.6
FreezerSm	84.4	84.0	77.3	96.3	72.9	67.6
SynthControl	92.6	91.0	76.1	93.7	92.0	88.0
average	73.8	76.0	61.3	83.1	80.9	80.5

Table 2. Expanded UCR benchmark, twenty-eight univariate datasets (% test accuracy). Left block: the trained CNN (mean over three seeds) on OLD (the opaque baseline), MC (GIPT-MC, per-image min-max), and H (the GIPT-H channel, global normalization). Right block: the training-free one-shot 1-NN classifier (Eq. 12) on the same encoded images, and on the z-normalized raw series as a reference.

	trained CNN			training-free 1-NN
dataset	OLD	MC	H	MC	H	raw
GunPoint	86.7	78.4	56.2	89.3	91.3	91.3
Coffee	75.0	83.3	53.6	100.0	96.4	100.0
ECG200	73.0	75.3	64.0	86.0	90.0	88.0
ItalyPower.	85.3	63.8	54.2	96.0	96.2	95.5
TwoLeadECG	68.1	79.3	50.0	78.2	71.3	74.7
ECGFiveDays	74.6	69.5	49.7	84.0	80.4	79.7
MoteStrain	79.2	80.5	62.2	83.0	83.4	87.9
BeetleFly	68.3	51.7	58.3	70.0	70.0	75.0
BirdChicken	75.0	85.0	73.3	65.0	65.0	55.0
Plane	89.8	79.7	49.8	99.0	98.1	96.2
Trace	76.3	100.0	38.7	100.0	79.0	76.0
Wine	50.0	50.0	50.0	61.1	63.0	61.1
SonyAIBO1	42.9	81.6	42.9	70.7	72.0	69.6
SonyAIBO2	71.4	76.5	61.7	84.6	88.4	85.9
Chinatown	75.9	68.1	76.0	92.1	92.1	95.3
PowerCons	84.6	82.0	93.1	91.7	93.9	93.3
GunPtAgeSpan	89.5	85.9	58.0	94.6	98.4	89.9
GunPtMvF	95.6	99.4	86.8	98.4	99.1	97.5
ArrowHead	49.5	48.6	36.6	78.3	76.6	80.0
Ham	63.5	59.7	52.7	62.9	48.6	60.0
Herring	59.4	59.4	59.4	56.2	50.0	51.6
ToeSeg1	67.8	83.2	49.1	72.4	71.1	68.0
ToeSeg2	58.2	77.9	60.5	76.9	83.1	80.8
DistPhalanxOC	68.7	70.2	69.9	73.9	68.5	71.7
ProxPhalanxOC	73.8	73.9	68.3	79.7	80.1	80.8
Strawberry	86.8	90.2	87.7	94.1	95.4	94.6
FreezerSm	84.4	84.0	77.3	96.3	72.9	67.6
SynthControl	92.6	91.0	76.1	93.7	92.0	88.0
average	73.8	76.0	61.3	83.1	80.9	80.5

Table 3. One-shot nearest-neighbor versus the convolutional network, averaged over the expanded suite. The one-shot classifier needs no training and runs orders of magnitude faster.

	CNN (50 epochs)	one-shot 1-NN
mean accuracy (%)	76.0	83.1
training	50 epochs/seed	none
relative wall-clock	1263×	$1\times$

Table 3. One-shot nearest-neighbor versus the convolutional network, averaged over the expanded suite. The one-shot classifier needs no training and runs orders of magnitude faster.

	CNN (50 epochs)	one-shot 1-NN
mean accuracy (%)	76.0	83.1
training	50 epochs/seed	none
relative wall-clock	1263×	$1\times$

5.3. Comparison With Strong Baselines And Statistical Significance

To position GIPT against established time-series classifiers rather than only against its own baseline, we ran two further competitors on the identical twenty-eight datasets and splits: 1-NN with Dynamic Time Warping [12] (Sakoe-Chiba window $10\%$, z-normalized series), the classical accuracy standard of the field for two decades, and ROCKET [21] (ten thousand random convolutional kernels with a ridge classifier, mean over three seeds), a recent state-of-the-art method. Following standard methodology for paired comparison over multiple datasets [49], Table 4 reports Wilcoxon signed-rank tests on the per-dataset accuracies; the full per-dataset numbers are in the technical appendix.

Three findings. First, the one-shot classifier on GIPT-MC images is statistically indistinguishable from DTW 1-NN ($83.1$ vs $83.0$, twelve wins each, $p=0.85$), while replacing DTW’s quadratic elastic alignment with a plain Euclidean distance on images that a person can also read; parity with the classical standard therefore comes with legibility and a lower distance cost, not instead of them. Second, the same classifier is significantly better than the trained convolutional network on either encoding ($p<0.005$ in both comparisons), and the encoder swap under the CNN shows no significant difference ($p=0.52$), which is the statistical form of the paper’s central claim that readable structure costs no accuracy. Third, ROCKET remains significantly stronger ($92.1$, $p<{10}^{-3}$); we report this openly. ROCKET is a black-box classifier that offers no image a person can audit and fits a supervised head; the nine-point gap is the current price of requiring a representation that is simultaneously machine-classifiable and eye-classifiable, and marks the headroom for future readable encoders.

Table 4. Wilcoxon signed-rank tests over the twenty-eight datasets of Table 2. Each row compares classifier A against classifier B: mean accuracy (%), wins/losses/ties in per-dataset accuracy, and the two-sided p-value; $p<0.05$ marks a significant difference. 1-NN denotes the training-free one-shot classifier, CNN the trained network, MC and H the GIPT-MC and GIPT-H encodings, DTW the elastic distance of Berndt and Clifford [12], ROCKET the method of Dempster et al. [21].

A	B	mean A vs B	W/L/T	p
1-NN MC	CNN OLD	83.1 vs 73.8	25/3/0	$<{10}^{-4}$
1-NN MC	CNN MC	83.1 vs 76.0	20/7/1	0.004
CNN MC	CNN OLD	76.0 vs 73.8	14/12/2	0.52
1-NN MC	DTW 1-NN	83.1 vs 83.0	12/12/4	0.85
1-NN H	1-NN raw	80.9 vs 80.5	17/10/1	0.68
1-NN MC	ROCKET	83.1 vs 92.1	1/25/2	$<{10}^{-4}$

Table 4. Wilcoxon signed-rank tests over the twenty-eight datasets of Table 2. Each row compares classifier A against classifier B: mean accuracy (%), wins/losses/ties in per-dataset accuracy, and the two-sided p-value; $p<0.05$ marks a significant difference. 1-NN denotes the training-free one-shot classifier, CNN the trained network, MC and H the GIPT-MC and GIPT-H encodings, DTW the elastic distance of Berndt and Clifford [12], ROCKET the method of Dempster et al. [21].

A	B	mean A vs B	W/L/T	p
1-NN MC	CNN OLD	83.1 vs 73.8	25/3/0	$<{10}^{-4}$
1-NN MC	CNN MC	83.1 vs 76.0	20/7/1	0.004
CNN MC	CNN OLD	76.0 vs 73.8	14/12/2	0.52
1-NN MC	DTW 1-NN	83.1 vs 83.0	12/12/4	0.85
1-NN H	1-NN raw	80.9 vs 80.5	17/10/1	0.68
1-NN MC	ROCKET	83.1 vs 92.1	1/25/2	$<{10}^{-4}$

5.4. Ablation Study

Our ablation study (full table in the technical appendix) isolates the contribution of each GIPT component with the normalization held fixed at the global map and the classifier held fixed at the training-free 1-NN rule, adding one component at a time: the bare polar outline, the filled silhouette, interior shading (which completes GIPT-H), and the two auxiliary channels (which complete GIPT-MC). Filling adds $0.7$ points on average and shading another $1.1$; both also serve the eye, so nothing is traded. The auxiliary channels do not help the nearest-neighbor rule ($82.0\to 80.9$, with Wine degrading outright as its noisy gradient channel dilutes the distance); their value is for the trained network, where GIPT-MC beats GIPT-H by ten points on the twelve-dataset study. Each observer thus wants a different member of the GIPT family, which is the two-renderings conclusion of Section 6 in ablation form. The GIPT-H column reproduces column H of Table 2 exactly, confirming the two experiments run the identical pipeline.

5.5. Robustness to Noise

We inject additive Gaussian noise of standard deviation $\sigma$ times each series’ standard deviation into the test series only and classify with the 1-NN rule (mean over three noise seeds). Table 5 summarizes the four core datasets (per-dataset numbers in the technical appendix): GIPT-H degrades gracefully, flat to within one point up to $\sigma =0.2$ and losing seven points at the severe $\sigma =0.4$, where the unquantized raw-series distance holds up somewhat better ($90.0$ vs $86.4$). Test noise appears as boundary raggedness while the global shape a person reads persists.

5.6. Normalization Controls Legibility

Table 6 reports vision zero-shot legibility. The readable encodings under a fixed-scale or global normalization are markedly more legible than the opaque baseline where the class difference is morphological: on ECG200 the single channel reaches $75.0$ against $62.5$ for the baseline. The per-image min-max rendering, which is the most accurate, is the least legible, scoring at or below chance, confirming the analysis of Section 6.0.0.1. GunPoint remains hard for the eye under every encoding, because its class difference is a small localized overshoot rather than a global shape.

5.7. Qualitative Gallery

Figure 2 shows the final GIPT-H encoder (global normalization) on two real datasets with six samples per class. The intra-class consistency the baseline lacks is now present, and the inter-class difference is one a person can name. On ECG200, class 0 renders as a compact rounded blob with one smooth lobe, while class 1 grows ragged wing-like protrusions. On Trace, class 1 is a smooth nautilus with a clean circular outline, while class 2 has a flat shelf on the right and a bitten notch where its discriminative event truncates the radius.

5.8. Human Test-Set Protocol

To compare human-eye classification with the model on identical inputs, we built a self-contained packet per dataset with GIPT-H (global normalization): a labeled study sheet of training examples per class, a blind test sheet of twenty numbered unlabeled test images, and the model predictions on those same twenty images, scored against a hidden key only after every answer is committed. This makes the legibility claim falsifiable by people rather than only by a vision model; the packets accompany the code, and the technical appendix documents their construction and the full multi-grader protocol we recommend.

6. Discussion and Limitations

The readability-accuracy trade-off. 

Two legibility choices hurt the classifier. Sparsity: a thin curve gives the network too few active pixels, fixed structurally by the filled silhouette. Amplitude and consistency: per-image min-max stretches each sample’s noise to the full range, breaking the cross-sample consistency of P2; this genuine residual trade-off is resolved by the observation that the human-facing and network-facing renderings need not be the same pixels.

Two levers on different axes. 

Encoding structure (swap, fill, align) fixes accuracy; normalization controls legibility. Rendering the two audiences separately lets both goals hold at once, reporting a classifier metric and a legibility metric. The one-shot result adds a third consequence: the readable representation is not only auditable but classifiable without any training, since cross-sample consistency is exactly what a nearest-neighbor rule exploits.

Limitations. 

UCR splits are small and single-seed accuracy varies widely [1,49], so we use multi-seed evaluation with batch-independent normalization. Our legibility numbers come from a single vision grader (sixteen queries per condition); a controlled multi-person study would firm them up. Where the class difference is a subtle frequency shift or a small localized overshoot, no static image is likely eye-separable, so the legibility claim is conditional on morphological class structure; the accuracy claim is not. GIPT does not close the gap to black-box classifiers such as ROCKET (Section 5.3), and extending the silhouette to genuinely multivariate UEA series [29], where the channels carry the variables, is natural future work.

7. Conclusions

We introduced GIPT, the Generalized Iterative Polar Transform, and with it made a polar time-series image readable by a person without giving up classifier accuracy: we diagnosed geometrically why the existing spiral encoder is opaque, built GIPT by swapping the polar axes, filling the silhouette, and choosing the normalization deliberately, proved a lobe-counting property making a frequency difference countable, and showed that readable structure costs no accuracy while the same cross-sample consistency yields a training-free one-shot classifier that beats the trained network and matches the classical DTW standard, with ablation, robustness, and significance analyses supporting each claim. We believe identifying, isolating, and resolving the readability-accuracy trade-off is a significant and novel step toward time-series representations that are auditable by people and cheap to classify by machine.

Appendix G.1. Exploratory Study

We first compared the opaque baseline against the bare axis-swapped curve and several intermediates on four datasets with a single seed (Table A1); these are the iterations that gave GIPT its name. The bare radial waveform (Curve) is very legible (Figure A2) but drops GunPoint to chance, because its thin curve and per-image normalization discard the small overshoot that defines the class. The outward spiral line (Spiral) recovers GunPoint to $80\%$; filling the silhouette (Fill) recovers Coffee and ECG; a filled spiral band with k turns (Band_k) can beat the baseline with a tuned k, but the best number of turns is dataset-dependent. These observations motivated the filled silhouette operator, the gradient channel, and the move to multi-seed evaluation.

Table A1. Exploratory single-seed CNN accuracy (%) of the early GIPT iterations. Columns: OLD the original opaque spiral baseline; Curve a bare one-turn radial waveform (value drives the radius, thin outline only); Spiral a thin outward spiral line; Fill a filled one-turn radial silhouette; Band_k a filled spiral band with k turns. Early numbers with the original batch-normalized classifier, shown to convey the shape of the problem.

dataset	OLD	Curve	Spiral	Fill	Band4	Band4
GunPoint	90.0	49.3	80.0	51.3	49.3	92.0
Coffee	53.6	67.9	57.1	82.1	100.0	67.9
ECG200	85.0	64.0	64.0	69.0	64.0	79.0
ItalyPow.	93.7	81.3	71.3	61.9	80.0	50.1

Table A1. Exploratory single-seed CNN accuracy (%) of the early GIPT iterations. Columns: OLD the original opaque spiral baseline; Curve a bare one-turn radial waveform (value drives the radius, thin outline only); Spiral a thin outward spiral line; Fill a filled one-turn radial silhouette; Band_k a filled spiral band with k turns. Early numbers with the original batch-normalized classifier, shown to convey the shape of the problem.

dataset	OLD	Curve	Spiral	Fill	Band4	Band4
GunPoint	90.0	49.3	80.0	51.3	49.3	92.0
Coffee	53.6	67.9	57.1	82.1	100.0	67.9
ECG200	85.0	64.0	64.0	69.0	64.0	79.0
ItalyPow.	93.7	81.3	71.3	61.9	80.0	50.1

Figure A4. Controlled frequency experiment, 3 Hz on the left four columns and 5 Hz on the right four. Row 1 raw signal; row 2 the opaque baseline, visually identical across all eight samples; row 3 the readable radial waveform, which yields three versus five countable petals as predicted by Proposition 2; rows 4 and 5 spiral and polar-bar variants.

Figure A4. Controlled frequency experiment, 3 Hz on the left four columns and 5 Hz on the right four. Row 1 raw signal; row 2 the opaque baseline, visually identical across all eight samples; row 3 the readable radial waveform, which yields three versus five countable petals as predicted by Proposition 2; rows 4 and 5 spiral and polar-bar variants.

Appendix G.2. Cnn Accuracy On The Twelve Core Datasets

Table A2 gives the headline comparison: the OLD baseline, GIPT-MC with min-max normalization, and its single human channel GIPT-H. GIPT-MC matches and slightly exceeds the baseline on average ($75.3$ vs $73.8$) and wins on six of twelve datasets, including a perfect score on Trace. Structurally, then, a readable polar silhouette does not cost classifier accuracy.

Table A2. CNN test accuracy on the twelve core datasets (%, mean±std over five seeds). All three columns train the same GroupNorm convolutional network for fifty epochs; only the input encoding differs. OLD: the original opaque spiral baseline. GIPT-MC: the proposed readable three-channel encoder (per-image min-max). GIPT-H: only its single human-facing channel. Bold marks the best encoder per dataset. Reading guide: GIPT-MC matching or beating OLD (six of twelve rows, and on average) is the paper’s claim that readable structure costs no accuracy.

dataset	OLD	GIPT-MC	GIPT-H
GunPoint	84.9±2.8	75.9±4.6	66.8±3.1
Coffee	74.3±17.7	85.0±8.9	60.0±9.1
ECG200	73.4±5.4	75.8±0.7	64.0±0.6
ItalyPower.	86.2±4.0	64.7±4.7	68.2±12.1
TwoLeadECG	69.1±3.9	79.8±3.0	60.0±4.2
ECGFiveDays	72.9±2.3	71.6±2.7	64.2±0.2
MoteStrain	78.5±3.0	80.5±0.5	75.7±1.0
BeetleFly	65.0±8.4	55.0±6.3	53.0±5.1
BirdChicken	68.0±9.3	86.0±5.8	77.0±2.4
Plane	87.4±7.7	79.8±4.1	48.6±9.9
Trace	75.8±2.2	100.0±0.0	96.4±3.1
Wine	50.0±0.0	50.0±0.0	50.0±0.0
average	73.8	75.3	65.3

Table A2. CNN test accuracy on the twelve core datasets (%, mean±std over five seeds). All three columns train the same GroupNorm convolutional network for fifty epochs; only the input encoding differs. OLD: the original opaque spiral baseline. GIPT-MC: the proposed readable three-channel encoder (per-image min-max). GIPT-H: only its single human-facing channel. Bold marks the best encoder per dataset. Reading guide: GIPT-MC matching or beating OLD (six of twelve rows, and on average) is the paper’s claim that readable structure costs no accuracy.

dataset	OLD	GIPT-MC	GIPT-H
GunPoint	84.9±2.8	75.9±4.6	66.8±3.1
Coffee	74.3±17.7	85.0±8.9	60.0±9.1
ECG200	73.4±5.4	75.8±0.7	64.0±0.6
ItalyPower.	86.2±4.0	64.7±4.7	68.2±12.1
TwoLeadECG	69.1±3.9	79.8±3.0	60.0±4.2
ECGFiveDays	72.9±2.3	71.6±2.7	64.2±0.2
MoteStrain	78.5±3.0	80.5±0.5	75.7±1.0
BeetleFly	65.0±8.4	55.0±6.3	53.0±5.1
BirdChicken	68.0±9.3	86.0±5.8	77.0±2.4
Plane	87.4±7.7	79.8±4.1	48.6±9.9
Trace	75.8±2.2	100.0±0.0	96.4±3.1
Wine	50.0±0.0	50.0±0.0	50.0±0.0
average	73.8	75.3	65.3

Appendix G.3. Full Component Ablation

Discussed in the ablation subsection of the main paper; all variants use the global normalization and the training-free 1-NN classifier.

Table A3. Component ablation (% accuracy, training-free 1-NN, global normalization throughout). Outline: thin polar outline only. Fill: filled silhouette, flat interior. Shade: filled plus interior shading, which is exactly GIPT-H. MC: plus gradient and smoothed channels, which is exactly GIPT-MC.

dataset	Outline	Fill	Shade	MC
GunPoint	92.0	92.0	91.3	92.7
Coffee	96.4	96.4	96.4	96.4
ECG200	89.0	89.0	90.0	90.0
ItalyPower.	94.2	95.5	96.2	95.7
TwoLeadECG	72.2	71.6	71.3	71.8
ECGFiveDays	76.4	79.1	80.4	81.1
MoteStrain	82.0	84.7	83.4	83.5
BeetleFly	60.0	65.0	70.0	65.0
BirdChicken	60.0	60.0	65.0	65.0
Plane	98.1	98.1	98.1	98.1
Trace	81.0	80.0	79.0	83.0
Wine	61.1	59.3	63.0	48.1
average	80.2	80.9	82.0	80.9

Table A3. Component ablation (% accuracy, training-free 1-NN, global normalization throughout). Outline: thin polar outline only. Fill: filled silhouette, flat interior. Shade: filled plus interior shading, which is exactly GIPT-H. MC: plus gradient and smoothed channels, which is exactly GIPT-MC.

dataset	Outline	Fill	Shade	MC
GunPoint	92.0	92.0	91.3	92.7
Coffee	96.4	96.4	96.4	96.4
ECG200	89.0	89.0	90.0	90.0
ItalyPower.	94.2	95.5	96.2	95.7
TwoLeadECG	72.2	71.6	71.3	71.8
ECGFiveDays	76.4	79.1	80.4	81.1
MoteStrain	82.0	84.7	83.4	83.5
BeetleFly	60.0	65.0	70.0	65.0
BirdChicken	60.0	60.0	65.0	65.0
Plane	98.1	98.1	98.1	98.1
Trace	81.0	80.0	79.0	83.0
Wine	61.1	59.3	63.0	48.1
average	80.2	80.9	82.0	80.9

Appendix G.4. Per-Dataset Baseline and Robustness Tables

Table A4 gives the per-dataset accuracies behind the baseline comparison of the baseline-comparison subsection of the main paper, and Table A5 the per-dataset robustness curves behind Table 5 of the main paper. The DTW column uses 1-NN with a Sakoe-Chiba window of $10\%$ on z-normalized series [12]; the ROCKET column is the mean over three seeds of the ten-thousand-kernel classifier of [21]; the GIPT column repeats 1-NN GIPT-MC from Table 2 of the main paper for side-by-side reading.

Table A4. Per-dataset comparison with DTW 1-NN and ROCKET (% test accuracy, identical splits). Bold marks the best of the three per row.

dataset	1-NN GIPT-MC	DTW 1-NN	ROCKET
GunPoint	89.3	94.0	100.0
Coffee	100.0	100.0	100.0
ECG200	86.0	82.0	91.3
ItalyPower.	96.0	95.2	96.9
TwoLeadECG	78.2	89.4	99.9
ECGFiveDays	84.0	78.7	100.0
MoteStrain	83.0	86.8	91.6
BeetleFly	70.0	70.0	90.0
BirdChicken	65.0	75.0	90.0
Plane	99.0	100.0	100.0
Trace	100.0	100.0	100.0
Wine	61.1	57.4	82.7
SonyAIBO1	70.7	73.0	92.4
SonyAIBO2	84.6	83.3	92.2
Chinatown	92.1	95.3	98.2
PowerCons	91.7	89.4	94.1
GunPtAgeSpan	94.6	96.5	99.7
GunPtMvF	98.4	99.7	99.9
ArrowHead	78.3	72.0	81.3
Ham	62.9	49.5	72.1
Herring	56.2	53.1	59.9
ToeSeg1	72.4	77.6	95.3
ToeSeg2	76.9	87.7	92.6
DistPhalanxOC	73.9	71.7	76.7
ProxPhalanxOC	79.7	78.4	90.1
Strawberry	94.1	94.1	98.1
FreezerSm	96.3	75.9	94.9
SynthControl	93.7	99.3	100.0
average	83.1	83.0	92.1

Table A4. Per-dataset comparison with DTW 1-NN and ROCKET (% test accuracy, identical splits). Bold marks the best of the three per row.

dataset	1-NN GIPT-MC	DTW 1-NN	ROCKET
GunPoint	89.3	94.0	100.0
Coffee	100.0	100.0	100.0
ECG200	86.0	82.0	91.3
ItalyPower.	96.0	95.2	96.9
TwoLeadECG	78.2	89.4	99.9
ECGFiveDays	84.0	78.7	100.0
MoteStrain	83.0	86.8	91.6
BeetleFly	70.0	70.0	90.0
BirdChicken	65.0	75.0	90.0
Plane	99.0	100.0	100.0
Trace	100.0	100.0	100.0
Wine	61.1	57.4	82.7
SonyAIBO1	70.7	73.0	92.4
SonyAIBO2	84.6	83.3	92.2
Chinatown	92.1	95.3	98.2
PowerCons	91.7	89.4	94.1
GunPtAgeSpan	94.6	96.5	99.7
GunPtMvF	98.4	99.7	99.9
ArrowHead	78.3	72.0	81.3
Ham	62.9	49.5	72.1
Herring	56.2	53.1	59.9
ToeSeg1	72.4	77.6	95.3
ToeSeg2	76.9	87.7	92.6
DistPhalanxOC	73.9	71.7	76.7
ProxPhalanxOC	79.7	78.4	90.1
Strawberry	94.1	94.1	98.1
FreezerSm	96.3	75.9	94.9
SynthControl	93.7	99.3	100.0
average	83.1	83.0	92.1

Table A5. Per-dataset noise robustness (% accuracy, training-free 1-NN, mean over three noise seeds; noise added to test series only). For each dataset the left column is 1-NN on GIPT-H global-normalized images and the right column 1-NN on the z-normalized raw series.

	GunPoint		Coffee		ECG200		ItalyPower.
$\sigma$	H	raw	H	raw	H	raw	H	raw
0.0	91.3	91.3	96.4	100.0	90.0	88.0	96.2	95.5
0.1	90.7	90.7	98.8	100.0	88.3	87.7	95.8	95.5
0.2	87.8	89.1	100.0	100.0	87.7	87.3	95.0	94.8
0.4	80.4	84.2	88.1	96.4	86.0	87.7	91.0	91.6

Table A5. Per-dataset noise robustness (% accuracy, training-free 1-NN, mean over three noise seeds; noise added to test series only). For each dataset the left column is 1-NN on GIPT-H global-normalized images and the right column 1-NN on the z-normalized raw series.

	GunPoint		Coffee		ECG200		ItalyPower.
$\sigma$	H	raw	H	raw	H	raw	H	raw
0.0	91.3	91.3	96.4	100.0	90.0	88.0	96.2	95.5
0.1	90.7	90.7	98.8	100.0	88.3	87.7	95.8	95.5
0.2	87.8	89.1	100.0	100.0	87.7	87.3	95.0	94.8
0.4	80.4	84.2	88.1	96.4	86.0	87.7	91.0	91.6