Decision boundaries in signal-to-noise space for the case of two bands (g and r) and for two variants of the χ² and SED-matched detection methods. Left: in this case, the two bands are assumed to have equal levels of noise. For the χ² (chi-squared) detector, we show the original version and the ﹩{\chi }_{+}^{2}﹩ (chi-positive) version defined in Equation (40). For the SED-matched detector, we show a version that takes the union of detections from three SEDs and a Bayesian version that sums the detection probabilities between the same three SEDs. The SEDs are r-only, g = r + 1 mag (“red”), and g = r (“flat”). Pixels that lie above and to the right of these curves (or anywhere outside the χ² circle) will be detected. The thresholds have been set so that each method produces the same false-positive rate given pure Gaussian noise inputs (equivalent to a 4σ Gaussian). This illustrates how the ﹩{\chi }_{+}^{2}﹩ method improves on the χ² method by detecting more true sources, thanks to treating negative fluxes differently than positive fluxes: the blue dashed decision boundary detects more sources with positive fluxes and does not detect sources where both fluxes are negative. Notice that the SED-matched decision boundaries are shaped to have lower thresholds in directions that are consistent with the SEDs they are tuned to, as well as how the asymptotic behaviors in the r and g directions are different because we chose (for purposes of illustration) to omit a g-only filter. Right: in this case, the noise in the r filter is set to be half as large as the noise in the g filter—that is, the r image is more sensitive. The χ² methods do not take sensitivities into account, so their decision boundaries are unchanged. The SED-matched methods, in contrast, take advantage of the fact that one unit of flux will produce more signal in the r band, and sculpt the decision boundaries accordingly.