  We characterize the lack of compactness in the critical embedding of
functions spaces $X\subset Y$ having similar scaling properties in the
following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence
that can be expressed as a finite sum of translations and dilations of
functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as
the number of functions in the sum and $n$ tend to $+\infty$. Such a
decomposition was established by G\'erard for the embedding of the homogeneous
Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with
$0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz
potential space, using wavelet expansions. In this paper, we revisit the
wavelet-based profile decomposition, in order to treat a larger range of
examples of critical embedding in a hopefully simplified way. In particular we
identify two generic properties on the spaces $X$ and $Y$ that are of key use
in building the profile decomposition. These properties may then easily be
checked for typical choices of $X$ and $Y$ satisfying critical embedding
properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older
and BMO spaces.
