  This is the third in a series of papers that develop a new and flexible model
to predict weak-lensing (WL) peak counts, which have been shown to be a very
valuable non-Gaussian probe of cosmology. In this paper, we compare the
cosmological information extracted from WL peak counts using different
filtering techniques of the galaxy shear data, including linear filtering with
a Gaussian and two compensated filters (the starlet wavelet and the aperture
mass), and the nonlinear filtering method MRLens. We present improvements to
our model that account for realistic survey conditions, which are masks,
shear-to-convergence transformations, and non-constant noise. We create
simulated peak counts from our stochastic model, from which we obtain
constraints on the matter density $\Omega_\mathrm{m}$, the power spectrum
normalisation $\sigma_8$, and the dark-energy parameter $w_0$. We use two
methods for parameter inference, a copula likelihood, and approximate Bayesian
computation (ABC). We measure the contour width in the
$\Omega_\mathrm{m}$-$\sigma_8$ degeneracy direction and the figure of merit to
compare parameter constraints from different filtering techniques. We find that
starlet filtering outperforms the Gaussian kernel, and that including peak
counts from different smoothing scales helps to lift parameter degeneracies.
Peak counts from different smoothing scales with a compensated filter show very
little cross-correlation, and adding information from different scales can
therefore strongly enhance the available information. Measuring peak counts
separately from different scales yields tighter constraints than using a
combined peak histogram from a single map that includes multiscale information.
Our results suggest that a compensated filter function with counts included
separately from different smoothing scales yields the tightest constraints on
cosmological parameters from WL peaks.
