The generalized Helmholtz equation least squares (HELS) formulations for reconstructing transient acoustic radiation from an arbitrary object subject to an arbitrary time-dependent excitation are derived. To facilitate the derivations, the Laplace transform is employed and the vibro-acoustic quantities on the source surface are solved explicitly in terms of the acoustic pressures measured on a conformal surface around the source at close range multiplied by transfer functions in the Laplace domain first. The vibro-acoustic responses in the time domain can then be expressed as convolution integrals of the measured acoustic pressure signals over temporal kernels. Replacing the spherical Hankel functions in the transfer functions with polynomial expressions, we can recast the infinite integrals in the inverse Laplace transform as contour integrals and evaluate the temporal kernels by using residue theorem. Once the temporal kernels are determined, the vibro-acoustic quantities anywhere in the field, including those on the source surface can be reconstructed directly. Numerical examples of reconstruction of transient acoustic radiation from a baffled disk subject to impulsive and arbitrarily time-dependent excitations are demonstrated. [Work supported by NSF.]