LensExplorer Visualization Framework

• LensExplorer is a suite of interactive visualization tools and algorithmic frameworks designed to explore gravitational lensing effects in astrophysics and data analysis.
• It integrates methodologies for cluster lens modeling, UMAP-based manifold learning, and real-time strong-field ray tracing to simulate lensing phenomena.
• The system supports practical applications from Hubble Frontier Fields observation planning to mobile simulations of black hole lensing, advancing reproducible research.

LensExplorer refers to a class of interactive visualization tools and algorithmic frameworks for exploring gravitational lensing phenomena, traditionally in astrophysical contexts but also in high-dimensional data analysis. Over the past decade, the term has become associated with versatile applications enabling detailed investigation of deflection fields, magnification maps, and lensing-induced connectivity structures, leveraging both domain-specific GUI platforms and algorithmic augmentation of dimensionality reduction methods. Below, key implementations and principles are delineated, drawing on major published instances in cluster lens modeling, UMAP-based manifold learning, and real-time strong-field ray tracing (Diego, 2014, Bot et al., 2024, Berens et al., 6 Mar 2026).

1. Conceptual Overview and Motivation

LensExplorer denotes both concrete software (IDL tool for Hubble Frontier Fields cluster lensing) and generalized paradigms for visualization and analysis of lensing effects. Its primary objectives are:

• Visualization and exploration of gravitational lensing fields and theoretical constructs (e.g., deflection, magnification, multiple imaging).
• Interactive, research-grade GUI for astrophysical observation planning and model validation.
• Frameworks for “lensed” projections in manifold learning: sculpting high-dimensional graphs via user-supplied lens functions to reveal hidden structure.
• Generalization to dynamic, real-time environments (mobile applications simulating strong-field black hole lensing).

The archetype is the IDL-based code for exploring galaxy-cluster gravitational lensing models using precomputed surface mass density and deflection fields, but the term also encompasses contemporary approaches in algorithmic data visualization and general ray-tracing architectures (Diego, 2014, Bot et al., 2024, Berens et al., 6 Mar 2026).

2. Astrophysical LensExplorer: GUI and Theoretical Basis

The original LensExplorer (Diego, 2014) was designed for gravitational lens modeling in the Hubble Frontier Fields (HFF), supporting clusters like Abell 2744 and MACSJ0416.1–2403. Its architecture and salient features:

• Thin-lens formalism: Implements

$$\boldsymbol{\beta} = \boldsymbol{\theta} - \boldsymbol{\alpha}(\boldsymbol{\theta})$$

where $\boldsymbol{\beta}$ is the source position, $\boldsymbol{\theta}$ the observed position, and $\boldsymbol{\alpha}$ the deflection angle, with associated lensing potential $\psi(\boldsymbol{\theta})$ and Jacobian matrix $\mathbf{A}$ for magnification $\mu(\boldsymbol{\theta}) = 1/|\det\mathbf{A}|$.

• GUI Structure: Six-window layout coordinating explore mode, HST field visualization, zoomed views, source-plane analysis, system constraint review, and relensed model-data comparison.
• Dynamic exploration: Clickable prediction of counterimage positions, on-the-fly magnification and critical-curve computation for arbitrary source redshift $z_s$.
• Delensing/Relensing: Users select arcs for delensing to the source plane, relensing through the model, and overplotting all predicted images for model verification.
• Model inputs: Leverages WSLAP+ free-form reconstructions for $\psi(\theta)$ and $\kappa(\theta)$ fields; no non-standard dependencies beyond IDL.

The software was intended for both research (e.g., observing strategy, candidate identification) and education (visualizing caustics, magnification effects), with model and code availability for reproducibility and extension.

3. Algorithmic LensExplorer: Lens Functions for Data Manifold Projections

Building on topological data analysis, a newer instantiation of the LensExplorer concept applies to high-dimensional data via lens functions incorporated into UMAP embeddings (Bot et al., 2024). Core innovations include:

• Lens Function ($\boldsymbol{\beta}$0): Any transformation $\boldsymbol{\beta}$1 acting on the data graph $\boldsymbol{\beta}$2 after UMAP’s fuzzy simplicial set construction, producing a subgraph $\boldsymbol{\beta}$3. Such filters can leverage features, meta-data, or custom metrics.
• Three Principal Lenses:

1. Global Lens: Discretize a scalar feature; retain only edges between points assigned to the same or adjacent bins:

   $\boldsymbol{\beta}$4

2. Global Mask: Intersect the original kNN graph with a kNN graph in lens space.
3. Local Mask: For each i, restrict edges to k nearest neighbors in lens-feature space; symmetrize.

• Integration: Edge-pruned graph $\boldsymbol{\beta}$5 is re-embedded (SGD rerun) using the previous UMAP layout as initialization, leading to “lensed” projections making feature-conditioned substructures explicit.
• Applications: Enhanced interpretability for gene expression data (clarifying relapse/non-relapse branches) and environmental datasets (uncovering air-quality macroperiods and pollutant-specific slices).
• Computational complexity: Edge filtering costs remain subdominant to embedding, global mask lens has highest pre-processing demands; see the following summary.

Lens Type	Preprocessing Cost	Edge Filter Cost
Global Lens	$\boldsymbol{\beta}$6 (balanced bins) or $\boldsymbol{\beta}$7	$\boldsymbol{\beta}$8
Global Mask	$\boldsymbol{\beta}$9	$\boldsymbol{\theta}$0
Local Mask	—	$\boldsymbol{\theta}$1

This framework is available as the Python package lensed_umap.

4. Real-Time Strong-Field Visualization: Ray-Tracing Black Hole Lensing

Recent developments extend LensExplorer to solve full strong-field geodesic equations for Schwarzschild and Kerr space-times in real time, facilitating interactive visualization of black hole lensing effects via mobile applications (Berens et al., 6 Mar 2026):

• Formulation: Light propagation follows geodesics in Schwarzschild or Kerr metric, with lensing equations

$\boldsymbol{\theta}$2

for Schwarzschild, and generalized Mino-time separated equations for Kerr involving conserved quantities $\boldsymbol{\theta}$3.

• Algorithmic Structure:

1. Screen pixel $\boldsymbol{\theta}$4 impact parameter $\boldsymbol{\theta}$5
2. Setup initial 4-momentum at the observer
3. Numerically integrate geodesic backwards to the source surface
4. Map source-sphere coordinates to on-device camera feeds, accounting for hemisphere mapping
5. Output per-pixel lensed frame

• Extensibility: Supports analytic weak-field point mass lens, singular isothermal sphere, NFW profiles, and multi-lens-plane scenarios. Allows arbitrary surface mass densities via Poisson equation solution for $\boldsymbol{\theta}$6. Batch operations and GPU acceleration are considered essential for interactivity.
• Characteristic Signatures: Einstein rings (Schwarzschild), D-shaped rings and photon sub-rings (Kerr), field-of-view parameter dependencies, and explicit control of observer/source locations.

A plausible implication is that this architectural generalization enables empirical classroom/observational demonstrations and simulation-based theory testing for a wide spectrum of lensing scenarios.

5. Practical Workflows and Use Cases

HFF Cluster Lensing (IDL GUI)

• Compile and launch the code in IDL.
• Select cluster, enter “Explore” mode, and probe field: visual feedback of predicted image locations and local magnification at arbitrary redshifts.
• DeRe-lens workflow: select arc, delens to source, relens, inspect model-data agreement.

Lensed UMAP Projections (Python)

• Fit a standard UMAP embedding, extract the neighborhood graph.
• Apply lens (global, global mask, local mask) via lensed_umap, specifying the feature and type.
• Visualize and analyze the revealed connectivity or branching structure, with color overlays for domain relevance.

Black Hole Vision (Mobile)

• For each frame, for each screen pixel, map to a ray and compute lensing deflection via geodesic integration, lookup color on live camera sphere, assemble lensed view.
• Real-time adjustment of mass, spin, observer inclination, and system geometry.

These distinct use cases exemplify the flexibility and technical depth achievable by instantiating the LensExplorer concept.

6. Future Extensions and Limitations

• Astrophysics: Expansion of cluster coverage (new HFF models, CLASH, A1689) and automated batch and snapshot modalities in the IDL GUI (Diego, 2014).
• Algorithmic Lenses: Implementation of alternative masking and reweighting strategies, extension to other unsupervised graph embeddings, and deeper integration with statistical significance testing (Bot et al., 2024).
• Strong-Field Ray Tracing: Abstraction to modular “LensModel” interfaces, GPU/compute-shader based Poisson solvers for user-supplied mass maps, and unified support for both analytic and numerically reconstructed lenses (e.g., cluster-mass FITS images) (Berens et al., 6 Mar 2026).

A plausible implication is that continued co-development across these strands will foster cross-pollination of methods for gravitation, data science, and real-time visualization.

• "LensExplorer, a tool for the visualization of the gravitational lensing effect in the Hubble Frontiers Fields clusters" (Diego, 2014)
• "Lens functions for exploring UMAP Projections with Domain Knowledge" (Bot et al., 2024)
• "Black Hole Vision: An Interactive iOS Application for Visualizing Black Holes" (Berens et al., 6 Mar 2026)