Vertical Edge Profile: Models & Methods

• Vertical edge profiles are quantitative functions that describe how physical properties vary perpendicular to structural interfaces in diverse scientific contexts.
• They are extracted using methods like 1D averaging, 2D/3D model fitting, edge detection, and Abel inversion to correct for projection and instrument effects.
• These profiles inform models across galaxy structures, protoplanetary disks, plasma diagnostics, and combinatorial structures, shedding light on equilibrium and gradient formation.

A vertical edge profile is a quantitative description of how a physical (or mathematical) quantity varies along the axis perpendicular to a structural or spatial interface—typically, in context, across an image edge, through a galactic or gaseous disk, or within structured trees, as observed or modeled in edge-on geometry. Vertical edge profiles are crucial for inferring intrinsic three-dimensional structures from projected or line-integrated data, for constructing analytic dynamical or radiative models, and for algorithmic feature extraction in computational image analysis. The vertical direction is privileged in these contexts as it is orthogonal to predominant symmetries (disk planes, building walls, or tree label gradings), making the profile directly sensitive to underlying processes such as equilibrium, stratification, or gradient formation.

1. Mathematical and Physical Definitions

Within edge-on astrophysical systems, image analysis, or combinatorial structures, a vertical edge profile refers to the function—luminosity, mass density, number density, emission intensity, or count—parameterized as a function of vertical coordinate $z$ or label $k$:

• In galactic disks, the vertical profile is typically $L(z)$ or $\rho(z)$, the stellar or gas (HI, molecular, dust) density as a function of height from the mid‑plane. Common analytic forms include the exponential, $L(z) = L_0 e^{-|z|/h_z}$, and the isothermal $\sech^2$ law, $L(z) = L_0\,\sech^2(z/z_0)$, with $z_0=2h_z$ so that $\sech^2(z/2h_z)\sim \exp(-|z|/h_z)$ at large $|z|$ (Ranaivoharimina et al., 2024, Dehnen et al., 10 Jan 2025, Zheng et al., 2022).
• In disk modeling, analytic potential-density pairs are constructed with vertical dependence modeled via a smooth function $\zeta(z)$, enabling pure exponential, $\sech^2$, or cored exponential forms, which are then embedded in physically motivated gravitational potentials (Dehnen et al., 10 Jan 2025).
• In random labelled trees, the vertical edge profile (VEP) is the sequence $(X_k^+, X_k^-)$ counting edges moving upward or downward in label, which encodes Markovian properties of the underlying structure (Metz-Donnadieu, 26 Nov 2025).
• In imaging systems, the vertical edge profile may refer to the spatial variation in intensity across a detected vertical edge, crucial for Modulation Transfer Function (MTF) analysis (Dasgupta, 2015).

Vertical profiles are also central in laboratory plasma diagnostics (e.g., vertical emissivity profiles in tokamaks), where measured quantities such as $I(z)$ or the derived local emissivity $\epsilon(r)$ (via Abel inversion) characterize the edge plasma (Hu et al., 26 Feb 2025).

2. Methodologies for Extraction and Measurement

Astrophysical Disks

Empirical vertical edge profiles are typically obtained from resolved images of edge-on galaxies or disks. Several approaches are combined:

• 1D Averaging: Image slices parallel to the major axis are averaged to form $I(z)$, often after dust correction and sky subtraction. The profile is fitted with analytic laws (exponential, $\sech^2$) convolved with the instrumental point-spread function (e.g., IRAC FWHM ≈ 2.1″ in S${}^4$G) (Ranaivoharimina et al., 2024).
• 2D/3D Model Fitting: Full images are fitted with axisymmetric disk models (e.g., EdgeOnDisk in GALFIT or ExponentialDisk3D in IMFIT) that encode both the radial and vertical structure, with mid-plane parameters recovered via Poisson-likelihood or least-squares methods (Ranaivoharimina et al., 2024).
• Correction for Projection and Instrumental Effects: For HI profiles, moment-0 strips in highly inclined galaxies are fitted with Gaussians to obtain a 'photometric width' $h_{\rm phot}$. Systematic broadening from beam smearing, planar and edge-on projection, and disk flaring are quantified and corrected via mock-cube calibrations with explicit correction formulae (Zheng et al., 2022).

Image Processing

• Edge Detection: Vertical edge profiles are extracted from pixelated images by calculating gradients (e.g., using Sobel or other vertical-favoring convolution kernels), thresholding for candidate edges, and linking pixels into subpixel-accurate chain profiles. The local Edge Spread Function (ESF) is constructed by accumulating pixel intensities in an over-sampled binning along the edge normal (Dasgupta, 2015).
• Algorithmic Automation: GPU-accelerated workflows parallelize the full chain: gradient computation, candidate pixel selection, curve tracing, ESF construction, and subsequent MTF computation (Dasgupta, 2015).

Spectroscopy and Remote Sensing

• Line-Integrated Measurements: In plasma physics, an optical fiber bundle samples the vertical extent of the plasma, yielding $I(z)$ at line emission wavelengths. Local emissivity profiles $\epsilon(r)$ are obtained using Abel inversion under axisymmetry assumptions (Hu et al., 26 Feb 2025).

3. Analytic Models and Functional Forms

Several canonical vertical profile laws are utilized, each mapping to a physical or dynamical context:

Model Name	Profile Equation	Features
Exponential	$L(z)=L_0 e^{-|z|/h_z}$	Sharp central cusp, steep decay
Isothermal ($\sech^2$)	$L(z)=L_0\,\sech^2(z/z_0)$ ($z_0=2h_z$)	Smooth core, slower near-plane falloff
Gaussian	$n(z)=n_0\exp\left[-(z/H(r))^2\right]$	Used for dust/gas in disks (Foucher et al., 6 Oct 2025)
Cored exponential	See text (combination of $e^{-|z|/h}$ and a central core parameter $w$)	Tunable inner core width

Pure exponentials yield the strongest mid-plane gradient and are supported by stellar number count data. The $\sech^2$ law arises naturally as the equilibrium solution of an isothermal sheet. Gaussian profiles are empirical fits to the vertical distribution of gas and dust in disks, as directly constrained by ALMA observations (Foucher et al., 6 Oct 2025). The cored exponential interpolates between these limits and provides additional flexibility for modeling (Dehnen et al., 10 Jan 2025).

4. Applications Across Domains

Galaxy Structure

Vertical edge profiles in stellar and HI disks are indispensable for quantifying disk thickness ($h_z$), identifying thick and thin disk components, and diagnosing disk flaring—defined by an increase in $h_z$ with radius. In S${}^4$G edge-on spirals, the mean thin-disk scale height is $0.14\pm0.07$ kpc and mean thick-disk height is $0.33\pm0.16$ kpc, with an average ratio $2.65\pm0.56$ (Ranaivoharimina et al., 2024). Empirical laws $h_z\propto h_R^{0.9}$ and $h_z\propto R_{25}^{0.6}$ establish scaling relations linking vertical and radial structure.

For neutral hydrogen, vertical HI scale heights are measured photometrically and then corrected for beam, projection, and flaring, achieving $\sim 10\%$ accuracy using the method outlined by Zheng et al. (Zheng et al., 2022). The recovered $h_{\rm true}$ values display expected scaling with $R_{\rm HI}$ and strong anticorrelation with mass surface density, consistent with vertical hydrostatic equilibrium.

Protoplanetary Disks

By leveraging nearly edge-on orientation, tomographically reconstructed profiles reveal direct stratification: mid-plane regions with $T\sim 7$–$11$ K, molecular layers (15–20 K), and a hotter CO atmosphere (31 K at 100 au). The dust and gas density follow Gaussian vertical profiles with tracer-dependent scale heights (24–31 au at $r=100$ au), and mid-plane H$_2$ densities exceeding $10^8$ cm$^{-3}$ (Foucher et al., 6 Oct 2025).

Imaging and Instrument Characterization

Vertical edge profiles of intensity in imaging systems enable precise evaluation of spatial resolution (via MTF). The chain ESF $\to$ LSF $\to$ MTF protocol, as implemented with GPU acceleration, delivers subpixel edge localization and full automation for gigapixel-scale images, critical for satellite and surveillance data (Dasgupta, 2015).

Random Trees and Markov Processes

In discrete mathematics, the vertical edge profile (VEP) in random labelled trees $(X_k^+, X_k^-)$ encodes the Markovian evolution of edge traversals across label levels. For a broad class of Galton–Watson labelled trees, the VEP forms a time-homogeneous Markov chain, with explicit transition kernels available for incomplete binary trees (Metz-Donnadieu, 26 Nov 2025). The process links naturally to super-Brownian motion and ISE in scaling limits, providing a bridge between combinatorics and continuum stochastic processes.

Spectroscopy in Plasmas

Full vertical profile measurements of impurity line emission with endoscopic fiber bundles yield $I(z)$ profiles resolved over 1.7 m (60 channels, Δz ≈ 2.9 cm), with absolute calibration uncertainties ≲10% and spatial resolution set by fiber geometry. Local emissivity profiles are recovered via Abel inversion (Hu et al., 26 Feb 2025).

5. Correction, Uncertainties, and Model Limitations

Extracting intrinsic vertical profiles from observed data invariably requires careful correction:

• Beam Smearing: Instrumental PSF broadens the apparent vertical profile. The correction is quadratic in the Gaussian regime (Zheng et al., 2022).
• Projection Effects: Planar and edge-on projection act to broaden and skew measured profiles. Polynomial corrections in $\cos i$ and empirical mass–radius offsets are used for HI (Zheng et al., 2022).
• Vertical Model Choice: Exponential versus $\sech^2$ versus Gaussian fits can yield differences near the mid-plane; for stellar disks, two-component $\sech^2$ fits (thin/thick) are justified by observed profile shapes (Ranaivoharimina et al., 2024).
• Edge Detection: In image analysis, the granularity of subpixel localization and parameter tuning (e.g., oversampling pitch, bin size) determine the fidelity of the reconstructed profile (Dasgupta, 2015).

Propagation of errors is treated via mock cubes (astrophysics) or Monte Carlo techniques, yielding uncertainties of a few percent (instrumental) to $\sim$10% (fitting) depending on domain and data quality (Zheng et al., 2022).

6. Physical and Algorithmic Interpretation

Physical interpretations of vertical edge profiles depend strongly on domain:

• In disk galaxies, vertical scalings and thick/thin dichotomies provide insight into disk heating (external accretion, internal secular processes), flaring mechanisms, and overall dynamical equilibrium (Ranaivoharimina et al., 2024).
• In protoplanetary disks, resolved vertical profiles directly expose temperature and density gradients underpinning planet formation environments (Foucher et al., 6 Oct 2025).
• In random trees, analysis of the VEP reveals underlying Markov structures, path-count combinatorics, and encodes universality classes linked to super-Brownian motion (Metz-Donnadieu, 26 Nov 2025).
• In imaging, the edge profile quantifies sharpness and maximal frequency transfer, forming the basis of quantitative system benchmarking via the MTF (Dasgupta, 2015).

A more general significance lies in the fact that vertical edge profiles act as cross-sectional fingerprints of multidimensional structure, with direct implications for modeling, inversion, and simulation across the physical sciences and computational domains.