Advances in image acquisition and informatics technology have led to organism-scale spatiotemporal atlases of gene expression and protein distributions. To maximize the utility of this information for the study of developmental processes, a new generation of mathematical models is needed for discovery and hypothesis testing. Here, we develop a data-driven, geometrically accurate model of early Drosophila embryonic bone morphogenetic protein (BMP)-mediated patterning. We tested nine different mechanisms for signal transduction with feedback, eight combinations of geometry and gene expression prepatterns, and two scale-invariance mechanisms for their ability to reproduce proper BMP signaling output in wild-type and mutant embryos. We found that a model based on positive feedback of a secreted BMP-binding protein, coupled with the experimentally measured embryo geometry, provides the best agreement with population mean image data. Our results demonstrate that using bioimages to build and optimize a three-dimensional model provides significant insights into mechanisms that guide tissue patterning.