  This paper develops a novel spatial quantile function-on-scalar regression
model, which studies the conditional spatial distribution of a high-dimensional
functional response given scalar predictors. With the strength of both quantile
regression and copula modeling, we are able to explicitly characterize the
conditional distribution of the functional or image response on the whole
spatial domain. Our method provides a comprehensive understanding of the effect
of scalar covariates on functional responses across different quantile levels
and also gives a practical way to generate new images for given covariate
values. Theoretically, we establish the minimax rates of convergence for
estimating coefficient functions under both fixed and random designs. We
further develop an efficient primal-dual algorithm to handle high-dimensional
image data. Simulations and real data analysis are conducted to examine the
finite-sample performance.
