Let 
be any subset of 
,
say that generated by an aborted matching pursuit process;
has m elements.
Let us now consider the problem
of minimizing 
on the space E generated by the 
,
k spanning
By definition, E is the range of the operator:
In the case where 
is equipped with its standard scalar product,
the adjoint of S is explicitly defined by the relationship:
Indeed, for any 
,
we have from Eq. (36):
In what follows,
S is not necessarily a one-to-one map
from 
onto E :
the vectors 
lying in 
are not necessarily linearly independent.
Let 
now be a vector minimizing on 
the quantity
.
Then, the vector 
minimizes q on E .
From Eq. (2),
the vectors 
in question are such that
These vectors are therefore the solutions of
the normal equation
(the least-squares solutions of the equation
).
In most cases encountered in image reconstruction,
the conjugate-gradients method is the best suited technique for
solving Eq. (38).
The version of this method
presented below provides 
.
ALGORITHM| 1:
#&#
Step 0: &Set 
(for example) and n = 0 ;
&choose a natural starting point 
in E ;
&compute 
,
&
;
&compute 
(for all 
;
&set 
(for all 
.
10ptStep 1: &Compute
&
,
5pt&
,
5pt&
(for all 
,
5pt&
,
5pt&
,
5pt&
,
5pt&
,
5pt&
(for all 
;
6pt&if 
,
&termination.
6pt&Compute
&
,
5pt&
(for all 
;
5pt&increment n and loop to step 1. 101
Throughout this algorithm,
is the residue of the equation
for 
.
Likewise,
is the value of 
at the same iterate:
The iteration in 
results from the identity:
The sequence of vectors 
converges towards a solution of the problem with all
the remarkable properties of the 
method
(see Lannes et al. 1987b).
In practice, E is chosen so that 
is a
map.
The uniqueness of the solution
can easily be verified by modifying the starting point
of the algorithm.
The stopping criterion is based on the fact that the final residue must be
practically orthogonal to all the 
's
(Eq. (37));
the 
of the angle between
the vectors 
and 
is equal
to 
.
Here, as 
is endowed with its standard scalar product,
this algorithm cannot provide the condition number
of 
.
(The transposition of what is presented in Sect. 4.2
would give the "generalized condition number" of AS ).
We therefore recommend to use algorithm 1 only when 
is
approximately known.
REMARK| 3.
Let us consider the special case where A is the identity operator
on 
(which then coincides with 
).
The problem is then to find 
,
the projection of
onto E .
Note that 
is then equal to unity.
In this case, algorithm 1 reduces to
ALGORITHM| 2:
#&#
Step 0: &Set 
(for example) and n = 0 ;
&set 
and 
;
&compute
&
,
&
(for all 
;
&set 
(for all 
.
10ptStep 1: &Compute
&
,
5pt&
(for all 
,
5pt&
,
5pt&
,
5pt&
,
5pt&
,
5pt&
(for all 
);
6pt&if 
,
&termination.
6pt&Compute
&
,
5pt&
(for all 
) ;
5pt&increment n and loop to step 1. 101
This algorithm
converges towards the projection of 
onto E
with all the properties of the conjugate-gradients method.
In principle,
the projection operation can also be performed by using
the matching pursuit iteration (25).
In this case,
on setting 
equal to its optimal value,
this iteration reduces to
where
The residues 
are then obtained via iteration (27):
and likewise for 
(cf. Eq. (28)):
At each iteration, it is then natural to choose k so that
.
In the general case where the 
's (
)
do not form an orthogonal set,
the conjugate-gradients algorithm is of course preferable.