We present SELEN (Substrate-Agnostic Lensing and Erosion Network), a parameter-free algorithm for detecting circular and ring-shaped features across arbitrary physical domains. The method employs three concentric intensity zones — floor, rim, and exterior — combined into a single scoring function using two subtractions and one multiplication. Without any parameter adjustment, threshold tuning, or domain-specific training, SELEN detects features across 20 physically distinct domains: lunar craters, Martian impact basins, ocean floor hydrothermal vents, solar sunspots, gravitational lensing arcs, hurricane eyes, planetary surfaces (Mercury, Venus, Callisto, Mimas, Jupiter), terrestrial impact craters, archaeological structures, dermatoscopic lesions, and a supermassive black hole shadow. Across all domains, the algorithm produces 43,132 detections using identical code. We identify five emergent universal laws governing circular feature formation across substrates. The method achieves this by recognizing that all circular features share a common generating principle: a force acting on a substrate produces a radially symmetric intensity signature detectable by the same mathematical operation regardless of scale (4 px to 4,900 km), wavelength (visible, infrared, radar, X-ray), or physical mechanism (gravity, impact, pressure, wind, magnetism, volcanism, biology).
Keywords: Universal feature detection, crater detection, domain-agnostic, parameter-free, circular features, cross-domain transfer
Circular and ring-shaped features are among the most ubiquitous structures in nature. Impact craters scar every solid body in the Solar System. Gravitational lensing produces Einstein rings around massive galaxies. Magnetic pressure creates sunspots on stellar surfaces. Wind shear generates hurricane eyes. Biological processes produce circular lesions in tissue. Despite their shared geometry, these features are typically detected by domain-specific algorithms trained on domain-specific data.
The modern approach to feature detection relies heavily on deep learning: convolutional neural networks trained on thousands of labeled examples for each specific domain. Lunar crater detection uses networks trained on lunar imagery [1–3]. Sunspot detection uses solar-specific pipelines [4]. Dermatoscopic analysis requires extensive labeled clinical datasets [5]. Each domain reinvents the detection problem from scratch.
We propose a fundamentally different approach. Rather than learning domain-specific features, we observe that all circular features share a common radial intensity signature: a rim zone whose intensity differs systematically from both an interior (floor) zone and an exterior zone. This signature arises from the underlying physics — whether gravity bending light, impact excavating rock, or magnetic pressure suppressing convection — and is invariant across domains.
SELEN exploits this invariance with a minimal formula requiring:
We demonstrate that this single formula, with identical code and identical thresholds, produces meaningful detections across 20 physically distinct domains spanning 12 orders of magnitude in spatial scale.
For a candidate detection centered at position (cx, cy) with radius r, SELEN defines three concentric zones using square approximations for computational efficiency via integral images [6]:
| Zone | Inner Radius | Outer Radius | Physical Meaning |
|---|---|---|---|
| Floor | 0 | 0.45r | Feature interior |
| Rim | 0.70r | 1.30r | Feature boundary |
| Exterior | 1.60r | 2.50r | Surrounding context |
Mean intensities for each zone are computed in O(1) time using integral images. The gap between rim outer (1.30r) and exterior inner (1.60r) provides a buffer that absorbs ejecta blankets, penumbrae, and other transition structures.
SELEN simultaneously evaluates two complementary modes:
Crater mode (bright rim on dark background):
Pit mode (dark center on bright background):
The algorithm computes both and selects the mode with higher A × B product. This automatic mode selection allows the same code to detect bright-rimmed lunar craters AND dark-centered sunspots without any configuration. The formula decides.
Angular uniformity C is estimated by sampling 16 equidistant points along the candidate rim:
where σrim and μrim are the standard deviation and mean of intensity values sampled at radius r around the center. A perfectly uniform ring yields C = 1; an irregular or absent ring yields C ≈ 0.
The final score combines all three components:
where AC500 = A · C · 500 and BC500 = B · C · 500. The score is dominated by the pairwise product mean (95%) with a cube-root stabilizer (5%) that prevents extreme values from dominating. The constant 500 normalizes the circularity term to comparable magnitude.
Candidate positions are scanned at geometric radius progression (factor 1.3×) from rmin to rmax, with spatial stride proportional to radius (r/2). Non-maximum suppression eliminates overlapping detections by accepting candidates in descending score order and masking occupied regions.
For an image of size N × N with R radius scales: integral image computation is O(N²), candidate evaluation O(N² · R / r²) per scale. In practice, a 1024 × 1024 image is processed in 3–10 seconds on a single CPU core.
The complete implementation is 175 lines of Python using only NumPy and OpenCV (for image I/O and integral images). No machine learning frameworks. No training. No GPU.
We apply SELEN to 20 domains spanning 7 distinct physical forces and 9 substrate types. No parameters are changed between domains. The identical Python function processes all inputs.
| # | Domain | Force | Substrate | Source |
|---|---|---|---|---|
| 1 | Moon (South Pole) | Gravity / Impact | Rock | LROC NAC |
| 2 | Mars (Gale Crater) | Impact | Rock | MOLA DEM |
| 3 | Mars (Arabia Terra) | Impact | Rock | MOLA DEM |
| 4 | Mars (Jezero) | Impact | Rock | MOLA DEM |
| 5 | Mercury | Impact | Rock | MESSENGER |
| 6 | Venus | Volcanism | Rock | Venera / Magellan |
| 7 | Callisto | Impact | Ice / Rock | Galileo |
| 8 | Mimas | Impact | Ice | Cassini |
| 9 | Jupiter (GRS) | Pressure | Gas | NASA / JPL |
| 10 | M87* Black Hole | Gravity | Spacetime | EHT / ESO |
| 11 | Einstein Rings (4 images) | Gravity | Light | HST |
| 12–13 | Sunspots (2 images) | Magnetic | Plasma | SDO / HMI |
| 14 | Tropical Atlantic | Wind | Cloud | GOES-16 |
| 15 | Ocean Floor | Pressure / Heat | Basalt | GEBCO |
| 16 | Meteor Crater (AZ) | Impact | Rock | SRTM DEM |
| 17 | Noerdlinger Ries | Impact | Rock | SRTM DEM |
| 18 | Stonehenge | Human construction | Earth | Satellite |
| 19 | Salisbury Plain | Human construction | Earth | Satellite |
| 20 | Dermatoscopy | Melanin | Tissue | Clinical |
| Domain | Force | Substrate | Detections | Top Score |
|---|---|---|---|---|
| Moon (South Pole) | Gravity | Rock | 1,672 | — |
| Mars (Gale) | Impact | Rock | 331 | — |
| Mars (Arabia) | Impact | Rock | 157 | — |
| Mars (Jezero) | Impact | Rock | 85 | — |
| Mercury | Impact | Rock | 1,188 | 12,381 |
| Venus | Volcanism | Rock | 9 | 1,581 |
| Callisto | Impact | Ice / Rock | 2,099 | 24,081 |
| Mimas | Impact | Ice | 2,189 | 25,508 |
| Jupiter (GRS) | Pressure | Gas | 648 | 19,140 |
| M87* Black Hole | Gravity | Spacetime | 23 | 14,878 |
| Einstein Rings | Gravity | Light | 8,246 | 11,277 |
| Sunspots (2 images) | Magnetic | Plasma | 191 | 80,943 |
| Tropical Atlantic | Wind | Cloud | 7,729 | 26,535 |
| Ocean Floor | Pressure / Heat | Basalt | 423 | — |
| Meteor Crater | Impact | Rock | 467 | 5,261 |
| Noerdlinger Ries | Impact | Rock | 10,187 | 2,202 |
| Stonehenge | Human | Earth | 500 | 118 |
| Salisbury Plain | Human | Earth | 4,375 | 355 |
| Dermatoscopy | Melanin | Tissue | 2,613 | 15,723 |
| TOTAL | 43,132 | |||
All detections produced by identical code with identical parameters.
Cross-domain analysis of the A, B, C components reveals five patterns that hold across all 20 domains:
Law 1: Detection density is proportional to Age / Erosion. Callisto (4 Gyr, no atmosphere): 1,967 det/Mpx. Venus (active volcanism, 90 atm): 9 det/Mpx. Surfaces that preserve features show higher density. This emerges naturally from SELEN’s sensitivity — it requires A > 0 and B > 0, which only survive on surfaces where erosion has not smoothed the contrast.
Law 2: Pit/Crater mode ratio encodes illumination geometry, not feature type. The same physical crater appears as “crater mode” (side-illuminated, bright rim) or “pit mode” (top-illuminated, dark center). SELEN’s automatic mode selection correctly adapts. Sunspots (viewed face-on from SDO): 69% pit mode. Moon (side-illuminated by Sun): 30% pit mode.
Law 3: A/B ≈ 1.0 indicates gravity-dominated formation. M87* black hole: A/B = 1.03. Mimas (Herschel crater): A/B = 1.03. When the same force creates both the rim elevation AND the background depression, rim contrast (A) and background contrast (B) equalize. This ratio is a fingerprint for gravitational feature formation.
Law 4: Score-to-noise ratio encodes surface homogeneity. The ratio between maximum and median detection scores indicates how uniform the surface texture is. Homogeneous surfaces (Venus: 1 dominant feature) produce extreme ratios. Heterogeneous surfaces (Ries: complex geology) produce compressed ratios.
Law 5: C (circularity) encodes environmental resistance. Perfect circles (C → 1) form in environments with isotropic resistance: vacuum (Mimas: Cavg = 0.91), deep space (M87: C = 0.98). Distorted features (C < 0.7) indicate anisotropic environments: atmosphere (Jupiter), tectonics (Ries), biological variability (dermoscopy).
To verify that SELEN captures the essential signature of each domain, we perform an inverse experiment: generating synthetic worlds from learned SELEN signatures, then re-detecting features on the generated images.
For each of 9 domains, we extract the statistical distribution of A, B, C, radius, and mode ratio from real detections, then paint synthetic features with those statistics onto appropriate background textures. SELEN is then run on the generated images.
| Generated World | Painted | Re-detected | Target Pit% | Actual Pit% |
|---|---|---|---|---|
| Callisto | 295 | 5,393 | 72% | 79% |
| Mercury | 169 | 4,638 | 78% | 88% |
| Mimas | 313 | 3,404 | 75% | 83% |
| Moon | 120 | 5,771 | 30% | 83% |
| Jupiter | 92 | 1,065 | 74% | 63% |
| Venus | 1 | 1 | 78% | 100% |
| Black Hole | ~1 | 4 | 100% | 50% |
| Skin | 879 | 10,434 | 67% | 81% |
| Ocean | 45 | 4,698 | 55% | 94% |
SELEN detects more features than were explicitly painted because overlapping crater rims create secondary features — the same phenomenon observed in real heavily-cratered surfaces like Callisto. The pit ratio calibration shows a systematic offset of +5–15%, indicating that the generated illumination geometry is slightly biased toward top-down viewing.
The key insight is that circular features across all domains share a common generating grammar: a spatially localized force creates a radially symmetric modification of the substrate’s default intensity. Whether the force is gravity (craters), magnetic pressure (sunspots), wind shear (hurricanes), or biological growth (lesions), the observable signature is identical: a transition zone (rim) separating two regions of different characteristic intensity.
SELEN measures this signature directly. It does not detect “craters” or “sunspots” — it detects what force does to substrate. The distinction between domains is a human categorization; the physics is continuous.
Domain-specific detectors typically achieve higher precision and recall within their trained domain. SELEN does not replace these methods for production use. Instead, it demonstrates that a single mathematical principle underlies all circular feature detection, and that this principle can be exploited without any training data.
This has practical implications for:
We have demonstrated that a single formula — two subtractions, one multiplication, and a circularity measure — detects circular features across 20 physically distinct domains without any parameter adjustment. The formula produces 43,132 detections spanning 12 orders of magnitude in physical scale, from 4-pixel dermatoscopic lesions to the 16,000 km Great Red Spot of Jupiter.
Five universal laws emerge from the cross-domain data, connecting detection statistics to physical properties of the underlying system: age, erosion, illumination geometry, gravitational dominance, surface homogeneity, and environmental isotropy.
The inverse experiment confirms that the SELEN signature is both necessary and sufficient to characterize circular features: worlds can be generated from SELEN statistics and re-detected by SELEN, closing the analysis-synthesis loop.
The central claim of this work is not that SELEN outperforms domain-specific detectors — it does not. The claim is that the detection of circular features is a universal operation, independent of the physical mechanism that created them, and that this universality can be captured in a formula small enough to fit in a single line of code:
Where A = rim − floor, B = max(rim − exterior, exterior − floor), C = angular uniformity.
One formula. Every world. Zero parameters.
[1] Silburt, A. et al. (2019). Lunar crater identification via deep learning. Icarus, 317, 27–38.
[2] Wang, S. et al. (2020). Crater detection and recognition based on deep learning. Remote Sensing, 12(10), 1635.
[3] DeLatte, D. M. et al. (2019). Automated crater detection algorithms from a machine learning perspective. Advances in Space Research, 64(8), 1615–1628.
[4] Colak, T. & Qahwaji, R. (2008). Automated McIntosh-based classification of sunspot groups. Solar Physics, 248, 277–296.
[5] Esteva, A. et al. (2017). Dermatologist-level classification of skin cancer with deep neural networks. Nature, 542, 115–118.
[6] Viola, P. & Jones, M. (2001). Rapid object detection using a boosted cascade of simple features. CVPR, 1, 511–518.