  In order to constrain and possibly detect unusual physics during inflation,
we allow the power spectrum of primordial matter density fluctuations,
P_{in}(k), to be an arbitrary function in the estimation of cosmological
parameters from data. The multi-resolution and good localization properties of
orthogonal wavelets make them suitable for detecting features in P_{in}(k). We
expand P_{in}(k) directly in wavelet basis functions. The likelihood of the
data is thus a function of the wavelet coefficients of P_{in}(k), as well as
the H_0, \Omega_b h^2, $\Omega_c h^2 and the \tau_{ri}, in a flat $\Lambda$CDM
cosmology. We derive constraints on these parameters from CMB anisotropy data
(WMAP, CBI, and ACBAR) and large scale structure (LSS) data (2dFGRS and PSCZ)
using the Markov Chain Monte Carlo (MCMC) technique. The direct wavelet
expansion method is different and complimentary to the wavelet band power
method of Mukherjee & Wang (2003a,b), and results from the two methods are
consistent. In addition, as we demonstrate, the direct wavelet expansion method
has the advantage that once the wavelet coefficients have been constrained, the
reconstruction of P_{in}(k) can be effectively denoised, i.e., P_{in}(k) can be
reconstructed using only the coefficients that, say, deviate from zero at
greater than 1\sigma. In doing so the essential properties of P_{in}(k) are
retained. The reconstruction also suffers much less from the correlated errors
of binning methods. The shape of the primordial power spectrum, as
reconstructed in detail here, reveals an interesting new feature at 0.001 \la
k/{Mpc}^{-1} \la 0.005. It will be interesting to see if this feature is
confirmed by future data. The reconstructed and denoised P_{in}(k) is favored
over the scale-invariant and power-law forms at \ga 1\sigma.
