Scale-Vector Alignment: A Scale-Aware Framework for Spatially Resolved Morphological Similarity in Astronomical Images
Astronomical maps made with different tracers are not expected to have identical morphology. Excitation, optical depth, chemistry, radiation, and ISM phase alter the response of a tracer, and the resulting differences can depend on both position and spatial scale. We propose scale-vector alignment, a scale-aware method based on Constrained Diffusion Decomposition (CDD). CDD decomposes an image into localized scale components; at each position, their amplitudes define a scale vector that describes how the measured intensity is distributed over spatial scale. We define the pixel-wise similarity $\Spix(x,y)$ as the normalized alignment of two local scale vectors. The normalization removes the overall amplitude, so $\Spix$ compares relative scale composition rather than absolute flux. We also define the scale-wise similarity $\Sscale(l)$ by comparing the two CDD component maps at each spatial scale. Spatial shifts are used to construct an empirical shifted reference distribution for $\Spix$. In Orion~A, the tracer with the highest similarity to the dust-derived column-density map changes from $^{12}$CO to $^{13}$CO to C$^{18}$O toward higher column density. In NGC~6334I(N), the line--continuum similarity decreases locally around the brightest compact structures, where radiative-transfer effects can alter the observed line morphology. In NGC~3627, CO is most similar to 21~$μ$m emission, and $\Sscale$ reaches its maximum at an intermediate sub-kpc scale. The method measures where two tracers have similar multiscale structure and at which scales their spatial distributions agree. The implementation is publicly available at https://github.com/meng-ke/Scale-Vector-Alignment.