How to organize expression by symbols (like Collect), but apply different functions to each coefficient

I want to mimic the functionality of Collect[expr, {vars}, func], but with the following modification: The function f that is applied to each coefficient is different, and depends on which variable it is a coefficient of. The vars are expressions of predetermined known heads (which in the example below are _e, _f, _g, _h).

Example:

expr = -2 x e[x] + x e[y] + x^2 e[x] + y e[x] + y e[y] + x^2 f[x] + y^2 f[x] + x^2 g[x] + y^2 h[x]

1. 
   

   I need to collect expr by e[_], f[_], g[_] and h[_].

   
   


2. 
   

   Apply simpE to the coefficients of e[_], and simpF to the coefficients of f[_], and simpGen to the coefficient of everything else.

   
   


3. 
   

   This needs to work even when certain terms are absent from expr. e.g. if expr doesn't have any e[_] etc.

   
   

My idea (which doesn't work) is to do this in two steps:

1. 
   

   First collect the expression:

   
   

   Collect[expr, {_f, _e, _g, _h}]
   



2. 
   

   Then replace with the rule that makes the transformation on each coefficient.

   
   

   rule = (Plus[
      Optional[Times[exprE_., funcE_e]], 
   
      Optional[Times[exprF_., funcF_f]], 
      rest_.]) :> 
   (simpE[exprE] funcE + simpF[exprF] funcF + Collect[rest,{_g, _h}, simpRest])
   

The (not quite correct) result is:

Collect[expr, {_f, _e, _g, _h}] /. rule

e[x] simpE[-2 x + x^2 + y] + f[x] simpF[x^2 + y^2] + g[x] simpRest[x^2] + h[x] simpRest[y^2] + simpRest[(x + y) e[y]]

But this doesn't work because

1. 
   

   the rule groups e[y] with the rest of the terms and incorrectly applies simpRest to it (see last term of output), instead of simpE[x+y] e[y].

   
   


2. 
   

   If certain terms are absent, then the rule doesn't even match. Consider expr2 below which is absent of f[_]:

   
   

   expr2 = -2 x e[x] + x^2 e[x] + y e[x] + x e[y] + y e[y] + x^2 g[x] + y^2 h[x] 
   

   
   
   

   The rule doesn't match:

   
   

   Collect[expr, {_f, _e, _g, _h}] /. rule
   

   
   

   
   

   (-2 x + x^2 + y) e[x] + (x + y) e[y] + x^2 g[x] + y^2 h[x]

   
   

   
   

I need help with this particular modification of Collect. Is there a sexy way to get this done?

Answer

Maybe the following. Simplifying a left-over constant term along with the other coefficients seems hard to comprise in a single, simple function.

forms = {_f, _e, _g, _h};
funcs = {simpF, simpE, simpG, simpH};
simpConstant[c_] := simpC[c];
Times[c_simpAll, form_] ^:= (c /. simpAll -> (form /. Thread[forms -> funcs])) form;

Collect[2 + expr, {_f, _e, _g, _h}, simpAll] /. simpAll -> simpConstant
Collect[2 + expr2, {_f, _e, _g, _h}, simpAll] /. simpAll -> simpConstant
Collect[2 + x, {_f, _e, _g, _h}, simpAll] /. simpAll -> simpConstant
(*

  simpC[2] + e[y] simpE[x + y] + e[x] simpE[-2 x + x^2 + y] + 
   f[x] simpF[x^2 + y^2] + g[x] simpG[x^2] + h[x] simpH[y^2]

  simpC[2] + e[y] simpE[x + y] + e[x] simpE[-2 x + x^2 + y] + 
   g[x] simpG[x^2] + h[x] simpH[y^2]

  simpC[2 + x]
*)