Beyond Colors: Probing Redshifts from Galaxy Morphology in Single-band Images with ViT-MDNz

TL;DR

ViT-MDNz provides a practical approach for redshift estimation of galaxy samples with limited photometric band coverage, contributing to improved completeness and usability of redshift catalogs for future large-scale surveys such as DESI and LSST.

Abstract

To address the challenge of estimating redshifts when only single-band images are available, this study introduces a deep learning model named ViT-MDNz. Leveraging robust statistical priors learned from large-scale data concerning the correlation between redshift and morphology, the model can directly estimate redshifts and their associated uncertainties from single-band galaxy images. It integrates a Vision Transformer (ViT) to extract deep morphological features and a Mixture Density Network (MDN) to predict the full redshift probability density function. Trained and evaluated on approximately 300,000 single-band images from the DESI Legacy Imaging Surveys (DESI-LS), the model achieves a normalized median absolute deviation $σ_{\rm NMAD} = 0.034$ and an outlier fraction $f_{\rm out} = 2.6\%$ in the $r$-band for redshifts up to $z \lesssim 1$. Evaluations using probability integral transform (PIT) and continuous ranked probability score (CRPS) confirm that the predicted probability density functions are well calibrated and closely match the true distribution. These results demonstrate that competitive redshift estimates can be obtained using morphological features alone, and that incorporating color information further enhances the accuracy and robustness of the estimation. Therefore, ViT-MDNz provides a practical approach for redshift estimation of galaxy samples with limited photometric band coverage, contributing to improved completeness and usability of redshift catalogs for future large-scale surveys such as DESI and LSST.

Figures (7)

• Figure 1: Comparison of the RNS and WNS samples: (a) number of galaxies as a function of spectroscopic redshift in linear scale, and (b) distribution of DESI $r$-band magnitudes.
• Figure 2: ViT-MDNz model architecture and data flow diagram. The input consists of single or multi-band galaxy images. These images are first processed by the ViT encoder (red dashed box): the image is split into fixed-size patches, which undergo linear projection and position encoding before being fed into the Transformer encoder layers to extract deep morphological features with global context. The extracted features are then passed to the MDN module (blue dashed box), which computes and outputs the conditional probability density function of the photometric redshift using a Gaussian Mixture Model.
• Figure 3: (Left) Predicted redshift vs. spectroscopic redshift scatter plot for the r-band. (Right) Corresponding residual distribution.
• Figure 4: (Left) PIT histogram for the r-band test set. The red dashed line indicates the ideal uniform distribution. (Right) Q-Q plot of the PIT values. Green points represent observed quantiles versus theoretical quantiles; the red dashed line is the y=x reference line.
• Figure 5: Distribution of CRPS values for the $r$-band test set. The red dashed line indicates the mean CRPS value.