  The radiosotope $^{44}$Ti is produced through $\alpha$-rich freezeout and
explosive helium burning in type Ia supernovae (SNe Ia). In this paper, we
discuss how the detection of $^{44}$Ti, either through late-time light curves
of SNe Ia, or directly via gamma rays, can uniquely constrain the origin of SNe
Ia. In particular, building upon recent advances in the hydrodynamical
simulation of helium-ignited double white dwarf binaries, we demonstrate that
the detection of $^{44}$Ti in a nearby SN Ia or in a young galactic supernova
remnant (SNR) can discriminate between the double-detonation and
double-degenerate channels of sub-Chandrasekhar (sub-$M_{\rm Ch}$) and
near-Chandrasekhar (near-$M_{\rm Ch}$) SNe Ia. In addition, we predict that the
late-time light curves of calcium-rich transients are entirely dominated by
$^{44}$Ti.
