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The Tikhonov regularized solution of the convolution equation (1)
is the minimizer in 
of the regularization functional (Tikhonov &Arsenin 1977, Morozov 1984, Groetsch 1984):
where 
is the regularization parameter to be chosen
depending on the
noise level. This functional can be expressed by the Plancherel theorem
in the frequency space as follows:
Hence it is
straightforward to prove, by variational calculus, that the minimizer
of 
is the solution of the equation
where 
denotes the complex conjugate of 
.
Since any image is numerically represented as a non-negative function
with finite support 
(in other words: 
outside ),
we restrict our considerations
to the rectangle 
.
The regularized solution 
, in a discrete set of equally spaced
points of , is then the inverse discrete Fourier transform
of
which can be evaluated easily by the Fast Fourier Transform (FFT) method.
Let 
be
the set of knots of a regular mesh of the space domain and 
be the B-spline of order 
relative to the 
equally spaced knots
of the extended interval
.
Then, the unknown function 
can be approximated by the 
order spline
whose coefficients, as a direct consequence of (6),
may be obtained solving in a regular mesh of points the system
where
Let 
be the set of knots of a uniform
decomposition of the frequency domain 
.
The matrix 
is an orthogonal matrix since
, where 
is the adjoint of 
and 
is the
identity matrix of order 
; then we can solve the system (8)
using the inverse discrete Fourier transform. Thus we obtain:
where
and
is the Tikhonov regularized solution
in the frequency space.
Since the approximation properties of splines are only asymptotically
satisfied, that is they hold only if 
converges to 
, we replace the above expression of 
by its limit
for 
. In this way 
converges to
and then, excep for edge effects,
.
In other words, the coefficients of B-splines converge, for each value of ,
to the values of the Tikhonov regularized solution obtained by the discrete
Fourier transform applied to 
.
This regularization technique, based on approximation by splines, can then
be seen as a method, with a very low computational complexity, for recovering
the image in all the points (except for the edges) of its domain
and not only in the knots of its mesh.
It is remarkable that these limits can also been obtained by expressing
as a linear combination of
sinc functions and then applying the Tikhonov regularization method in the
frequency space. Such an expansion is usual for band limited
functions, as a direct consequence of the sampling theorem which states that,
if 
outside the domain 
, then the values of at the sampling points
can be interpolated by a bi-infinite
series of sinc functions to reproduce .
In our numerical computations we approximate this series by the finite sum
where the coefficients 
are the values of at the sampling
points. We recall, for the reader's convenience, that the Fourier
transform of a sinc function is a square pulse.
More precisely
Therefore the Tikhonov regularized solution of equation (1) sought
in a finite-dimensional span of sinc functions, is the solution on a discrete
set of 
values of the problem
For each value of 
we have a regularized solution 
whose expansion by sinc is uniquely characterized by the coefficients
which can be easily obtained applying the FFT to
relation (12). These coefficients, as already remarked, are the
limits (for ) of the coefficients of the same regularized
solution spanned by B-splines.
This fact is not completely surprising because the interpolation in the
sampling points
by splines of order , for 
converges to the
interpolation in the same points by sinc functions (Schoenberg 1973).
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Up: Regularization and Smoothing for
Previous: Introduction