The map-sharpening problem can be written in the general form
where ρ
0 is the underlying signal that we would like to observe (actual electron density), ρ is the observed signal (model of electron density from observation),
g is a process through which blurring operates on the signal,
k is a blurring function that changes the signal (ρ
0) before observation is carried out and
n is noise. However, this formulation is too general to be practical. In order to make the problem manageable, we must make assumptions regarding the functional forms of
g and
k and assume a model for the noise
n. Therefore, for simplicity, we assume that noise is additive and the blurring function is linear,
If there were no noise then the problem would be a linear equation. This problem is ill-posed, especially when
k is near singular,
i.e. small perturbations in input parameters may cause large variations in output. For example, the effects of small noise addition, an incorrectly defined blurring function or Fourier series termination may result in an uninterpretable ‘deblurred’ electron-density map. It should be noted that in crystallography we always deal with limited noisy data and that Fourier series termination is always present. Even if there were no noise and we had knowledge of the exact blurring function
k(
x, y), solving (11)
would still not be straightforward. The numbers of equations and parameters to be estimated are equal to the number of grid points in the electron density, which can be very large.
The problem becomes manageable, whilst not completely reflecting reality, when we make the further assumption that the blurring function is independent of position. This simplification essentially means that the whole content of the asymmetric unit oscillates as a unit with no rotational component, resulting in the blurring function having the property
k(
x, y) =
k(
x −
y, 0). Using the notation
k(
x) =
k(
x, 0), (11)
becomes
Since the problem is ill-posed, we can approach its solution utilizing ideas from the field of regularization (Tikhonov & Arsenin, 1977
![[triangle]](/corehtml/pmc/pmcents/rtrif.gif)
). Under the assumption of white noise, our ill-posed problem may be replaced by the minimization problem
where ||.|| denotes the
L
2 norm,
f is a regularization function and γ is a regularization parameter to be selected. Usually, regularizers are chosen so that the resultant function obeys certain conditions. For example purposes, we shall consider two popular conditions: (i) the function should be small and (ii) the first derivatives of the function should be small (
i.e. the function should vary slowly). For the first case we have
and for the second case
which is known as a first-order Sobolev norm. Since ρ is a periodic function, we can write
where Δ is the Laplace operator [Δ =
(
![[partial differential]](/corehtml/pmc/pmcents/part.gif)
2/
xi
2)] and (.,.) denotes the scalar product in Hilbert space.
Now the problem is reduced to finding the minimum of the functional
where
L =
I (identity operator) for
L
2-type regularizers (first case) and
L = −Δ for Sobolev-type regularizers (second case).
Using Plancherel’s theorem, the convolution theorem and the fact that the Fourier transformation of the Laplacian is proportional to the negative squared length of the reciprocal-space vector, we can rewrite the problem as
where
F
hkl is the structure factor before sharpening (
e.g. 2
mF
o −
DF
c-type maps),
F
0hkl is that after sharpening and |
s| = 2sinθ/λ is the length of the reciprocal-space vector, with
t(
s) = 1, α = γ for regularization function
f
1 and
t(
s) =
s
2, α = (2π)
2γ for
f
2. This minimization problem has a very simple solution,
When
k(
x) is Gaussian then the equation has an especially simple form, since
K(
s) =
[
k(
x)] = exp(−
sTB
deblur
s/4), where
B
deblur is an anisotropic deblurring
B value.
Unfortunately, in reality neither
B values nor α are known. Whilst there are several techniques to find an ‘optimal’ value for α (Vogel, 2002
) when the blurring function is known, in our implementation such an approach did not give consistent results. Therefore, we used the following
ad hoc procedure for selection of the regularization parameter. Denoting
K
α(
s) =
K(
s)/[
K
2(
s) + α
t(|
s|)] and
A
α(
s) =
K
α(
s)
K(
s), we see that
A
α is similar to the hat function used in regression analysis (Stuart
et al., 2009
). We can define the degrees of freedom of errors (the number of observations minus the effective number of parameters) as
1
Note that when α = 0 then
n
df = 0 and when α → ∞ then
n
df is equal to the number of observations. We select α so that
n
df is equal to 10–20% of the number of observations. Since we do not know the exact values of
B and α, we also perform
ad hoc integration using an empirically derived distribution of these parameters. The necessary integral may then be written
where
B
aniso reflects the anisotropy of the data and is calculated during scaling of the calculated structure factors relative to the observed structure factors, under the conditions that it obeys crystal symmetry, and tr(
B
aniso) = 0.
The joint probability distribution of
B and α can be written
The mean value of the isotropic part is taken to be equal to the median value (
B) of the coordinates (although it may be better to use Wilson’s
B value estimated using intensity curves derived by Popov & Bourenkov (2003
). We approximate the distribution of the isotropic part of the
B values using a Gaussian distribution centred at
B
sharp with standard deviation equal to
B
sharp/10. For each
B value, we select α so that
n
df is 10–20% of the number of observations and the standard deviation of the distribution of α is taken to be α
B/10.
Note that (17)
and (18)
suggest a class of regularizers. They can be selected to use particular knowledge about the electron density in real and reciprocal space. For example, if it is desired to suppress the effect of ice rings then one can select
t(|
s|) so that the corresponding reflections are weighted down.
3.1. Implementations and an example
We have implemented anisotropic sharpening with
L
2 and Tikhonov–Sobolev regularizers with and without integration over the
ad hoc joint probability distribution of
B and α using probability distribution (24)
. We have also implemented the regularization function
t(
s) = 1 +
s
2. These are available from
REFMAC5 v.5.7. In our tests, all regularization functions gave similar results. This is not surprising, as the major problem is that the blurring function is not position-independent. Before finding accurate regularizers, the problem of modelling position-dependent blurring functions should be dealt with. All results presented here were achieved using the
L
2-type regularizer.
Map sharpening was tested for many cases using data sets from the PDB (Berman
et al., 2002
) with resolution below 3 Å. The best results were obtained for PDB entry
2r6c (Bailey
et al., 2007
). For any low-resolution data taken from the PDB, before map calculation we generally try jelly-body, local NCS (if present) and external reference structure (if applicable) restrained refinement and take the best refined results for further analysis. For
2r6c, the original
R/
R
free statistics reported in the PDB were 0.321/0.344. After refinement, these values became 0.240/0.300. Fig. 7 shows an illustration of the maps after refinement with and without unregularized and regularized map sharpening. It is apparent that in this case using regularized map-sharpening coefficients shows more features (possibly side chains) and connectivity. Whilst this example shows regularization using the
L
2-type regularizer, it should be noted that the Sobolev-type regularizer gave similar results.
This article has been 
![[set membership]](/corehtml/pmc/pmcents/x2208.gif)
R to the target function, where R is the list of external restraints and the function f
ext(
![[similar, equals]](/corehtml/pmc/pmcents/sime.gif)
N(c
1, σ1
2) and x
2 
