Revision 69bcba365f630b1e0e8f61f8ea3414f251988705 authored by Philippe Canal on 25 April 2016, 17:15:39 UTC, committed by Philippe Canal on 21 June 2016, 20:28:10 UTC
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Spectrum.md
# BACKGROUND ELIMINATION #
##1-DIMENSIONAL SPECTRA ##
This function calculates background spectrum from source spectrum. The
result is placed in the vector pointed by spectrum pointer. On
successful completion it returns 0. On error it returns pointer to the
string describing error.
``` {.cpp}
char *Background1(float *spectrum,
int size,
int number_of_iterations);
```
Function parameters:
- **`spectrum`** pointer to the vector of source spectrum
- **`size`** length of spectrum
- **`number_of_iterations`** or width of the clipping window
The function allows to separate useless spectrum information (continuous
background) from peaks, based on Sensitive Nonlinear Iterative Peak
Clipping Algorithm. In fact it represents second order difference filter
(-1,2,-1). The basic algorithm is described in detail in [1], [2].
$$ v_p(i)= min\left\{v_{p-1} , \frac{[v_{p-1}(i+p)+v_{p-1}(i-p)]}{2} \right\} $$
where p can be changed
a. from 1 up to a given parameter value w by incrementing it in each
iteration step by 1-INCREASING CLIPPING WINDOW
b. from a given value w by decrementing it in each iteration step by
1- DECREASING CLIPPING WINDOW
An example of the original spectrum and estimated background (INCREASING
CLIPPING WINDOW) is given in the Figure 1.1 .
One can notice that on the edges of the peaks the estimated background
goes under the peaks. An alternative approach is to decrease the
clipping window from a given value to the value of one (DECREASING
CLIPPING WINDOW). Then the result obtained is given in the Figure 1.2.
The estimated background is smoother. The method does not deform the
shape of peaks.
However sometimes the shape of the background is very complicated. The
second order filter is insufficient. Let us illustrate such a case in
the Figure 1.3. The forth order background estimation filter gives better
estimate of complicated background (the clipping window w=10).
4-th order algorithm ignores linear as well as cubic component of the
background. In this case the filter is (1,-4,6,-4,1). In general the
allowed values for the order of the filter are 2, 4, 6, 8. An example of
the same spectrum estimated with the clipping window w=40 and with
filters of the orders 2, 4, 6, 8 is given in the Figure 1.4.
Sometimes it is necessary to include also the Compton edges
into the estimate of the background. In Figure 1.5 we present the example of
the synthetic spectrum with Compton edges. The background was estimated
using the 8-th order filter with the estimation of the Compton edges and
decreasing clipping window. In the lower part of the Figure we present
the background, which was added to the synthetic spectrum. One can
observe good coincidence with the estimated background. The method of
the estimation of Compton edge is described in detail in [3].
The generalized form of the algorithm is implemented in the function.
``` {.cpp}
char *Background1General(float *spectrum,
int size,
int number_of_iterations,
int direction,
int filter_order,
bool compton);
```
The meaning of the parameters is as follows:
- **`spectrum`** pointer to the vector of source spectrum
- **`size`** length of spectrum vector
- **`number_of_iterations`** maximal width of clipping window,
- **`direction`** direction of change of clipping window. Possible values:
- BACK1_INCREASING_WINDOW
- BACK1_DECREASING_WINDOW
- **`filter_order`** order of clipping filter. Possible values:
- BACK1_ORDER2
- BACK1_ORDER4
- BACK1_ORDER6
- BACK1_ORDER8
- **`compton`** logical variable whether the estimation of Compton edge will be included. Possible values:
- BACK1_EXCLUDE_COMPTON
- BACK1_INCLUDE_COMPTON
##2-DIMENSIONAL SPECTRA##
This basic background estimation function allows to separate useless
spectrum information (2D-continuous background and coincidences of peaks
with background in both dimensions) from peaks. It calculates background
spectrum from source spectrum. The result is placed in the array
pointed by spectrum pointer. On successful completion it returns 0. On
error it returns pointer to the string describing
error.
``` {.cpp}
char *Background2(float **spectrum,
int sizex,
int sizey,
int number_of_iterations);
```
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations`** width of the clipping window
In Figure 1.6 we present an example of 2-dimensional spectrum before background elimination.
Estimated background is shown in Figure 1.7. After subtraction we get pure
2-dimensional peaks.
Analogously to 1-dimensional case we have generalized also the function
for 2-dimensional background estimation. Sometimes the width of peaks in
both dimensions are different. As an example we can introduce n-gamma
2-dimensional spectra. Then it is necessary to set different widths of
clipping window in both dimensions. In Figure 1.8 we give an example of such
a spectrum.
Spectrum after background estimation (clipping window 10 for
x-direction, 20 for y- direction) and subtraction is given in Figure 1.9.
Background estimation can be carried out using the algorithm of
successive comparisons [1] or .based on one-step filtering algorithm
given by formula:
$$
v_{p}(i) = min\left\{v_{p-1}(i) ,
\frac{
\left[\begin{array}{c}
-v_{p-1}(i+p,j+p)+2*v_{p-1}(i+p,j)-v_{p-1}(i+p,j-p) \\
+2*v_{p-1}(i,j+p)+2*v_{p-1}(i,j-p) \\
-v_{p-1}(i-p,j+p)+2*v_{p-1}(i-p,j)-v_{p-1}(i-p,j-p)
\end{array}\right]
}{4}
\right\}
$$
Illustrating example is given in
the following 3 Figures. In Figure 1.10 we present original (synthetic)
2-dimensional spectrum. In the Figure 1.11. we have spectrum after background
elimination using successive comparisons algorithm and in Figure 1.12 after
elimination using one-step filtering algorithm.
One can notice artificial ridges in the spectrum in Figure 1.11. In the
estimation using filtration algorithm this effect disappears. The
general function for estimation of 2-dimensional background with
rectangular ridges has the form
``` {.cpp}
char *Background2RectangularRidges(float **spectrum,
int sizex,
int sizey,
int number_of_iterations_x,
int number_of_iterations_y,
int direction,
int filter_order,
int filter_type);
```
This function calculates background spectrum from source spectrum. The
result is placed to the array pointed by spectrum pointer.
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations_x-maximal`** x width of clipping window
- **`number_of_iterations_y-maximal`** y width of clipping window
- **`direction`** direction of change of clipping window. Possible values:
- BACK2_INCREASING_WINDOW
- BACK2_DECREASING_WINDOW
- **`filter_order-order`** of clipping filter. Possible values:
- BACK2_ORDER2 BACK2_ORDER4
- BACK2_ORDER6 BACK2_ORDER8
- **`filter_type`** determines the algorithm of the filtering. Possible values:
- BACK2_SUCCESSIVE_FILTERING
- BACK2_ONE_STEP_FILTERING
In what follows we describe a function to estimate continuous
2-dimensional background together with rectangular and skew ridges. In
Figure 1.13 we present a spectrum of this type.
The goal is to remove rectangular as well as skew ridges from the
spectrum and to leave only 2-dimensional coincidence peaks. After
applying background elimination function and subtraction we get the two
dimensional peaks presented in Figure 1.14
In Figures 1.15 and 1.16 we present experimental spectrum with skew ridges and
estimated background, respectively.
The function for the estimation of background together with skew ridges
has the form
``` {.cpp}
char *Background2SkewRidges(float **spectrum,
int sizex,
int sizey,
int number_of_iterations_x,
int number_of_iterations_y,
int direction,
int filter_order);
```
The result is placed to the array pointed by spectrum pointer.
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations_x`** maximal x width of clipping window
- **`number_of_iterations_y`** maximal y width of clipping window
- **`direction`** direction of change of clipping window. Possible values:
- BACK2\_INCREASING\_WINDOW
- BACK2\_DECREASING\_WINDOW
- **`filter_order`** order of clipping filter. Possible values:
- BACK2\_ORDER2
- BACK2\_ORDER4
- BACK2\_ORDER6
- BACK2\_ORDER8
Next we present the function that estimates the continuous
background together with rectangular, and nonlinear ridges. To
illustrate the data of such a form we present synthetic data shown in
Figure 1.17. The estimated background is given in Figure 1.18. Pure Gaussian
after subtracting the background from the original spectrum is shown in
Figure 1.19
The function to estimate also the nonlinear ridges has the form
``` {.cpp}
char *Background2NonlinearRidges(float **spectrum,
int sizex,
int sizey,
int number-of-iterations-x,
int number-of-iterations-y,
int direction,
int filter_order);
```
The result is placed to the array pointed by spectrum pointer..
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations_x`** maximal x width of clipping window
- **`number_of_iterations_y`** maximal y width of clipping window
- **`direction`** direction of change of clipping window. Possible values:
- BACK2\_INCREASING\_WINDOW
_ BACK2\_DECREASING\_WINDOW
- **`filter_order`** order of clipping filter. Possible values:
- BACK2\_ORDER2
- BACK2\_ORDER4
- BACK2\_ORDER6
- BACK2\_ORDER8
The information contained in the skew ridges and non-linear
ridges can be interesting and one may wish to separate it from the
rectangular ridges. Therefore we have implemented also two functions
allowing to estimate ridges only in the direction rectangular to x- axis
and y-axis. Let us have both rectangular and skew ridges from spectrum
given in Figure 1.13 estimated using above described function
**Background2SkewRidges**(Figure 1.20).
Now let us estimate ridges rectangular to x-axis and y-axis (Figures 1.21, 1.22)
After subtraction these data from spectrum in Figure 1.20, we get separated
skew ridge given in Figure 1.23.
The functions for estimation of 1-dimensional ridges in 2-dimensional
spectra have the forms
``` {.cpp}
char *Background2RectangularRidgesX(float **spectrum,
int sizex,
int sizey,
int number-of-iterations,
int direction,
int filter_order);
```
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations`** maximal x width of clipping window
- **`direction`** direction of change of clipping window. Possible values:
- BACK2\_INCREASING\_WINDOW
- BACK2\_DECREASING\_WINDOW
- **`filter_order`** order of clipping filter. Possible values:
- BACK2\_ORDER2
- BACK2\_ORDER4
- BACK2\_ORDER6
- BACK2\_ORDER8
``` {.cpp}
char *Background2RectangularRidgesY(float **spectrum,
int sizex,
int sizey,
int number_of_iterations,
int direction,
int filter_order);
```
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`number_of_iterations`** maximal width of clipping window
- **`direction`** direction of change of clipping window. Possible values:
- BACK2\_INCREASING\_WINDOW
- BACK2\_DECREASING\_WINDOW
- **`filter_order`** order of clipping filter. Possible values
- BACK2\_ORDER2
- BACK2\_ORDER4
- BACK2\_ORDER6
- BACK2\_ORDER8
# SMOOTHING #
**-1-DIMENSIONAL SPECTRA**
The operation of the smoothing is based on the convolution of the
original data with the filter of the type
(1,2,1)/4 -three points smoothing
(-3,12,17,12,-3)/35 -five points smoothing
(-2,3,6,7,6,3,-2)/21 -seven points smoothing
(-21,14,39,54,59,54,39,14,-21)/231 -nine points smoothing
(-36,9,44,69,84,89,84,69,44,9,-36)/429 -11 points smoothing
(-11,0,9,16,21,24,25,24,21,16,9,0,-11)/143 -13 points smoothing
(-78,-13,42,87,122,147,162,167,162,147,122,87,42,-13,-78)/1105 -15
points smoothing. The function for one-dimensional smoothing has the
form
``` {.cpp}
char *Smooth1(float *spectrum,
int size,
int points);
```
This function calculates smoothed spectrum from source spectrum. The
result is placed in the vector pointed by spectrum pointer.
Function parameters:
- **`spectrum`** pointer to the vector of source spectrum
- **`size`** length of spectrum
- **`points`** width of smoothing window. Allowed values
- SMOOTH1\_3POINTS
- SMOOTH1\_5POINTS
- SMOOTH1\_7POINTS
- SMOOTH1\_9POINTS
- SMOOTH1\_11POINTS
- SMOOTH1\_13POINTS
- SMOOTH1\_15POINTS
An example of 1-dimensional spectra smoothing and the influence of the
filter width on the data is presented in Figure 2.1
## 2-DIMENSIONAL SPECTRA ##
The smoothing of the two dimensional data is analogous to
the one dimensional case. The width of filter can be chosen
independently for each dimension. The form of the 2-D smoothing function
is as follows
``` {.cpp}
char *Smooth2(float **spectrum,
int sizex,
int sizey,
int pointsx,
int pointsy);
```
This function calculates smoothed spectrum from source spectrum. The
result is placed in the array pointed by spectrum pointer.
Function parameters:
- **`spectrum`** pointer to the array of source spectrum
- **`sizex`** x length of spectrum
- **`sizey`** y length of spectrum
- **`pointsx,pointsy`** width of smoothing window. Allowed values:
- SMOOTH2\_3POINTS
- SMOOTH2\_5POINTS
- SMOOTH2\_7POINTS
- SMOOTH2\_9POINTS
- SMOOTH2\_11POINTS
- SMOOTH2\_13POINTS
- SMOOTH2\_15POINTS
An example of 2-D original data and data after smoothing is given in Figures 2.2, 2.3.
# PEAK SEARCHING #
## 1-DIMENSIONAL SPECTRA##
The basic function of the 1-dimensional peak searching is in
detail described in [4], [5]. It allows to identify automatically the
peaks in a spectrum with the presence of the continuous background and
statistical fluctuations - noise. The algorithm is based on smoothed
second differences that are compared to its standard deviations.
Therefore it is necessary to pass a parameter of sigma to the peak
searching function. The algorithm is selective to the peaks with the
given sigma. The form of the basic peak searching function is
```{.cpp}
Int-t Search1(const float *spectrum,
int size,
double sigma);
```
This function searches for peaks in source spectrum. The number of found
peaks and their positions are written into structure pointed by
one\_dim\_peak structure pointer.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`p`** pointer to the one\_dim\_peak structure pointer
- **`size`** length of source spectrum
- **`sigma`** sigma of searched peaks
The structure one\_dim\_peak has the form:
```{.cpp}
struct one_dim_peak{
int number_of_peaks;
double position[MAX_NUMBER_OF_PEAKS1];
};
```
An example of simple one-dimensional spectrum with identified peaks is
given in Figure 3.1.
An example of 1-dimensional experimental spectrum with many identified
peaks is given in Figure 3.2.
However when we have noisy data the number of peaks can be enormous. One
such an example is given in Figure 3.3. Therefore it can be useful to have
possibility to set a threshold value and to consider only the peaks
higher than this threshold (see Figure 3.4, only three peaks were
identified, threshold=50.) The value in the center of the peak value[i]
minus the average value in two symmetrically positioned channels
(channels i-3\*sigma, i+3\*sigma) must be greater than threshold.
Otherwise the peak is ignored.
An alternative approach was proposed in [6].. The algorithm generates
new invariant spectrum based on discrete Markov chains. In this spectrum
the noise is suppressed, the spectrum is smoother than the original one.
On the other hand it emphasizes peaks (depending on the averaging
window). The example of the part of original noisy spectrum and Markov
spectrum for window=3 is given in Figure 3.5 Then the peaks can found in
Markov spectrum using standard above presented algorithm.
The form of the generalized peak searching function is as follows.
```{.cpp}
Int-t Search1General(float *spectrum,
int size,
float sigma,
int threshold,
bool markov,
int aver-window);
```
This function searches for peaks in source spectrum. The number of found
peaks and their positions are written into structure pointed by
one\_dim\_peak structure pointer.
Function parameters:
- **`spectrum`** pointer to the vector of source spectrum
source spectrum is replaced by new spectrum calculated
using Markov chains method.
- **`size`** length of source spectrum
- **`sigma`** sigma of searched peaks
- **`threshold`** threshold value for selecting of peaks
- **`markov`** logical variable, if it is true, first the source spectrum
is replaced by new spectrum calculated using Markov chains method.
- **`aver_window`** averaging window used in calculation of Markov spectrum,
applies only for markov variable is true
The methods of peak searching are sensitive to the sigma. Usually the
sigma value is known beforehand. It also changes only slightly with the
energy. We have investigated as well the robustness of the proposed
algorithms to the spectrum with the peaks with sigma changing from 1 to
10 (see Figure 3.6).
We applied peak searching algorithm based on Markov approach. We changed
sigma in the interval from 1 to 10. The spectra for averaging windows 3,
5, 10 are shown in Figure 3.7.
When we applied peak searching function to the Markov spectrum averaged
with the window=10, we obtained correct estimate of all 10 peak
positions for sigma=2,3,4,5,6,7,8. It was not the case when we made the
same experiment with the original spectrum. For all sigmas some peaks
were not discovered.
## 2-DIMENSIONAL SPECTRA##
The basic function of the 2-dimensional peak searching is in
detail described in [4].. It identifies automatically the peaks in a
spectrum with the presence of the continuous background, statistical
fluctuations as well as coincidences of background in one dimension and
peak in the other one-ridges. The form of the basic function of
2-dimensional peak searching is
```{.cpp}
Int-t Search2(const float **source,
int sizex,
int sizey,
double sigma);
```
This function searches for peaks in source spectrum .The number of found
peaks and their positions are written into structure pointed by
two\_dim\_peak structure pointer.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`sizex`** x length of source spectrum
- **`sizey`** y length of source spectrum
- **`sigma`** sigma of searched peaks
An example of the two-dimensional spectrum with the identified peaks is
shown in Figure 3.8.
We have also generalized the peak searching function analogously to one
dimensional data. The generalized peak searching function for two
dimensional spectra has the form
```{.cpp}
Int-t Search2General(float **source,
int sizex,
int sizey,
double sigma,
int threshold,
bool markov,
int aver-window);
```
This function searches for peaks in source spectrum. The number of found
peaks and their positions are written into structure pointed by
two\_dim\_peak structure pointer.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`sizex`** x length of source spectrum
- **`sizey`** y length of source spectrum
- **`sigma`** sigma of searched peaks
- **`threshold`** threshold value for selection of peaks
- **`markov`** logical variable, if it is true, first the source spectrum
is replaced by new spectrum calculated using Markov chains method.
- **`aver_window`** averaging window of searched peaks (applies only for Markov method)
An example of experimental 2-dimensional spectrum is given in Figure 3.9.
The number of peaks identified by the function now is 295.
The function works even for very noisy data. In Figure 3.10 we present
synthetic 2-dimensional spectrum with 5 peaks. The method should
recognize what is real 2-dimensional peak and what is the crossing of
two 1-dimensional ridges The Markov spectrum with averaging window=3 is
given in Figure 3.11. One can observe that this spectrum is smoother than
the original one. After applying the general peak searching function to
the Markov spectrum with sigma=2, and threshold=600, we get correctly
identified peaks.
# DECONVOLUTION - UNFOLDING #
## 1-DIMENSIONAL SPECTRA##
Mathematical formulation of the convolution system is:
$$ y(i) = \sum_{k=0}^{N-1}h(i-k)x(k), i=0,1,2,...,N-1 $$
where h(i) is the impulse response function, x, y are input and output
vectors,
respectively, N is the length of x and h vectors. In matrix form we have:
$$
\left[\begin{array}{c}
y(0)\\
y(1)\\
.\\
.\\
.\\
.\\
.\\
y(2N-2)\\
y(2N-1)
\end{array}\right] =
\left[\begin{array}{cccccc}
h(0) & 0 & 0 & 0 & ... & 0\\
h(1) & h(0) & 0 & 0 & ... & 0\\
. & h(1) & h(0) & 0 & ... & 0\\
h(N-1) & . & h(1) & h(0) & ... & 0\\
0 & h(N-1) & . & h(1) & ... & 0\\
0 & 0 & h(N-1) & . & ... & h(0)\\
0 & 0 & 0 & h(N-1) & ... & h(1)\\
. & . & . & . & ... & .\\
0 & 0 & 0 & 0 & ... & h(N-1)
\end{array}\right]
\left[\begin{array}{c}
x(0)\\
x(1)\\
x(2)\\
.\\
.\\
y(N-1)
\end{array}\right]
$$
Let us assume that we know the response and the output vector (spectrum)
of the above given system. The deconvolution represents solution of the
overdetermined system of linear equations, i.e., the calculation of the
vector **x**.
The goal of the deconvolution methods is to improve the
resolution in the spectrum and to decompose multiplets. From
mathematical point of view the operation of deconvolution is extremely
critical as well as time consuming operation. From all the methods
studied the Gold deconvolution (decomposition) proved to work as the
best one. It is suitable to process positive definite data (e.g.
histograms). In detail the method is described in [7], [8].
## Gold deconvolution algorithm ##
- proved to work as the best, other methods (Fourier, VanCittert etc) oscillate
$$ y = Hx $$
$$ H^T=H^THx $$
$$ y^{'} = H^{'}x $$
$$x_{i}^{(k+1)}=\frac{y_{i}^{'}}{\sum_{m=0}^{N-1}H_{im}^{'}x_{m}^{(k)}}x_{i}^{(k)}, i=0,1,...,N-1, $$
where:
$$ k=1,2,3,...,I $$
$$ x^{(0)} = [1,1,...,1]^T $$
The basic function has the form
```{.cpp}
char *Deconvolution1(float *source,
const float *resp,
int size,
int number-of-iterations);
```
This function calculates deconvolution from source spectrum according to response spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`resp`** pointer to the vector of response spectrum
- **`size`** length of source and response spectra
- **`number_of_iterations`** for details see [8]
As an illustration of the method let us introduce small example. In
Figure 4.1 we present original 1-dimensional spectrum. It contains
multiplets that cannot be directly analyzed. The response function (one
peak) is given in Figure 4.2. We assume the same response function (not
changing the shape) along the entire energy scale. So the response
matrix is composed of mutually shifted response functions by one
channel, however of the same shape.
The result after deconvolution is given in Figure 4.3. It substantially
improves the resolution in the spectrum.
We have developed a new high resolution deconvolution algorithm. We have
observed that the Gold deconvolution converges to its stable state
(solution). It is useless to increase the number of iterations, the
result obtained does not change. To continue in decreasing the width of
peaks we have found that when the solution reaches its stable state it
is necessary to stop iterations, then to change the vector in a way and
repeat again the Gold deconvolution . We have found that to change the
particular solution we need to apply non-linear boosting function to it.
The power function proved to give the best results. At the beginning the
function calculates exact solution of the Toeplitz system of linear
equations.
$$ x^{(0)} = [x_e^2(0),x_e^2(1),...,x_e^2(N-1),]^T$$
where :
$$ x_e=H^{'-1}y^{'}$$
Then it applies the Gold deconvolution algorithm to the solution and
carries out preset number of iterations. Then the power function with
the exponent equal to the boosting coefficient is applied to the
deconvolved data. These data are then used as initial estimate of the
solution of linear system of equations and again the Gold algorithm is
employed. The whole procedure is repeated number\_of\_repetitions times.
The form of the high-resolution deconvolution function is
```{.cpp}
char *Deconvolution1HighResolution(float *source,
const float *resp,
int size,
int number-of-iterations,
int number-of-repetitions,
double boost);
```
This function calculates deconvolution from source spectrum according
to response spectrum
The result is placed in the vector pointed by source pointer.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`resp`** pointer to the vector of response spectrum
- **`size`** length of source and response spectra
- **`number_of_iterations`** for details we refer to manual
- **`number_of_repetitions`** for details we refer to manual
- **`boost`** boosting factor, for details we refer to manual
The result obtained using the data from Figures 4.1, 4.2 and applying the
high resolution deconvolution is given in Figure 4.4. It decomposes the
peaks even more (practically to 1, 2 channels).
Another example of synthetic data is given in Figure 4.5. We have
positioned the two peaks very close to each other (2 channels). The
method of high resolution deconvolution has decomposed these peaks
practically to one channel. The original data are shown as polyline, the
data after deconvolution as bars. The numbers at the bars denote the
heights of bars and the numbers in parenthesis the original area of
peaks. The area of original peaks is concentrated into one channel.
Up to now we assumed that the shape of the response is not changing.
However the method of Gold decomposition can be utilized also for the
decomposition of input data (unfolding) with completely different
responses. The form of the unfolding function is as follows
```{.cpp}
char *Deconvolution1Unfolding(float *source,
const float **resp,
int sizex,
int sizey,
int number-of-iterations);
```
This function unfolds source spectrum according to response matrix
columns. The result is placed in the vector pointed by source
pointer.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`resp`** pointer to the matrix of response spectra
- **`sizex`** length of source spectrum and \# of rows of response matrix
- **`sizey`** \# of columns of response matrix
- **`number_of_iterations`**
Note!!! sizex must be \>= sizey. After decomposition the resulting
channels are written back to the first sizey channels of the source
spectrum.
The example of the response matrix composed of the responses of
different chemical elements is given in Figure 4.6
The original spectrum before unfolding is given in Figure 4.7. The obtained
coefficients after unfolding, i.e., the contents of the responses in the
original spectrum is presented in the Figure 4.8.
Another example where we have used unfolding method is the
decomposition of continuum of gamma-ray spectra. Using simulation and
interpolation techniques we have synthesized the response matrix (size
3400x3400 channels) of Gammasphere spectrometer (Figure 4.9). Its detail is
presented in Figure 4.10. The original spectrum of Co56 before and after
continuum decomposition are presented in Figures 4.11, 4.12, respectively.
## 2-DIMENSIONAL SPECTRA ##
We have extended the method of Gold deconvolution also for
2-dimensional data. Again the goal of the deconvolution methods is to
improve the resolution in the spectrum and to decompose multiplets. In
detail the method of optimized 2-dimensional deconvolution is described
in [8].
Mathematical formulation of 2-dimensional convolution system is as
follows
$$ y(i_1,i_2) = \sum_{k_1=0}^{N_1-1}\sum_{k_2=0}^{N_2-1}h(i_1-k_1,i_2-k_2)x(k_1,k_2), i_1=0,1,2,...,N_1-1, i_2=0,1,2,...,N_2-1 $$
Assuming we know the output
spectrum **y** and the response spectrum **h,**the task is to calculate
the matrix **x**.
The basic function has the form
``` {.cpp}
char *Deconvolution2(float **source,
const float **resp,
int sizex,
int sizey,
int niter);
```
This function calculates deconvolution from source spectrum according to
response spectrum. The result is placed in the matrix pointed by source
pointer.
Function parameters:
- **`source`** pointer to the matrix of source spectrum
- **`resp`** pointer to the matrix of response spectrum
- **`sizex`** x length of source and response spectra
- **`sizey`** y length of source and response spectra
- **`number_of_iterations`** for details see [8]
The example of 2-dimensional spectrum before deconvolution is presented
in Figure 4.13. In the process of deconvolution we have used the response
matrix (one peak shifted to the beginning of the coordinate system)
given in Figure 4.14. Employing the Gold deconvolution algorithm implemented
in the decon2 function we get the result shown in Figure 4.15. One can
notice that the peaks became narrower thus improving the resolution in
the spectrum.
Analogously to 1-dimensional case we have developed
high-resolution 2-dimensional deconvolution. From the beginning the
exact solution of the cyclic convolution 2-dimensional system is
calculated. Then we apply the repeated deconvolution with boosting in
the same way like in 1- dimensional case. The form of the
high-resolution 2-dimensional deconvolution function is:
``` {.cpp}
char *Deconvolution2HighResolution(float **source,
const float **resp,
int sizex,
int sizey,
int number_of_iterations,
int number_of_repetitions,
double boost);
```
This function calculates deconvolution from source spectrum
according to response spectrum
The result is placed in the matrix pointed by source pointer.
Function parameters:
- **`source`** pointer to the matrix of source spectrum
- **`resp`** pointer to the matrix of response spectrum
- **`sizex`** x length of source and response spectra
- **`sizey`** y length of source and response spectra
- **`number_of_iterations`**
- **`number_of_repetitions`**
- **`boost`** boosting factor
When we apply this function to the data from Figure 4.13 using response
matrix given in Figure 4.14 we get the result shown in Figure 4.16. It is
apparent that the high-resolution deconvolution decomposes the input
data even more than the original Gold deconvolution.
# FITTING #
A lot of algorithms have been developed (Gauss-Newton, Levenber-Marquart
conjugate gradients, etc.) and more or less successfully implemented
into programs for analysis of complex spectra. They are based on matrix
inversion that can impose appreciable convergence difficulties mainly
for large number of fitted parameters. Peaks can be fitted separately,
each peak (or multiplets) in a region or together all peaks in a
spectrum. To fit separately each peak one needs to determine the fitted
region. However it can happen that the regions of neighboring peaks are
overlapping (mainly in 2-dimensional spectra). Then the results of
fitting are very poor. On the other hand, when fitting together all
peaks found in a spectrum, one needs to have a method that is stable
(converges) and fast enough to carry out fitting in reasonable time. The
gradient methods based on the inversion of large matrices are not
applicable because of two reasons
a. calculation of inverse matrix is extremely time consuming;
b. due to accumulation of truncation and rounding-off errors the
result can become worthless.
We have implemented two kinds of fitting functions. The first approach
is based on the algorithm without matrix inversion [9]-awmi algorithm.
It allows to fit large blocks of data and large number of parameters.
The other one is based on calculation of the system of linear equations
using Stiefel-Hestens method [10]. It converges faster than awmi
algorithm, however it is not suitable to fit large number of parameters.
## 1-DIMENSIONAL SPECTRA##
- the quantity to be minimized in the fitting procedure for one
dimensional spectrum is defined as :
$$ \chi^2 = \frac{1}{N-M}\sum_{i=1}^{N}\frac{[y_i-f(i,a)]^2}{y_i} $$
where $i$ is the channel in the fitted spectrum, $N$ is the number of
channels in the fitting subregion, $M$ is the number of free parameters,
$y_i$ is the content of the i-th channel, $a$ is a vector of the
parameters being fitted and $f(i,a)$ is a fitting or peak shape function.
Instead of the weighting coefficient $y_i$ in the denominator of the above
given formula one can use also the value of $f(i,a)$. It is suitable for
data with poorstatistics [11], [12].
The third statistic to be optimized, which is implemented in the fitting
functions is Maximum Likelihood Method. It is of the choice of the user
to select suitable statistic..
- after differentiating $\chi^2$ we obtain the following $M$ simultaneous
equations
$$ \sum_{i=1}^{N}
\frac{y_i-f(i,a^{(t)})}{y_i}
\frac{\partial f(i,a^t)}{\partial a_k}=
\sum_{j=1}^{M}\sum_{i=1}^{N}
\frac{\partial f(i,a^{(t)})}{\partial a_j}
\frac{\partial f(i,a^{(t)})}{\partial a_k}
\Delta a_j^{(t)} $$
- in $\gamma$-ray spectra we have to fit together tens, hundreds of peaks
simultaneously that represent sometimes thousands of parameters.
- the calculation of the inversion matrix of such a size is
practically impossible.
- the awmi method is based on the assumption that off-diagonal
terms in the matrix A are equal to zero.
$$
\Delta a_{k}^{(t+1)} = \alpha^{(t)}
\frac{
\sum_{i=1}^{N} \frac{e_{i}^{(t)}}{y_i}\frac{\partial f(i,a^{(t)})}{\partial a_k}
}{
\sum_{i=1}^{N} \left[ \frac{\partial f(i,a^{(t)})}{\partial a_k}\right]^2\frac{1}{y_i}
}
$$
where the error in the channel $i$ is $e_{i}^{(t)} = y_i-f(i,a^{(t)}); k=1,2,...,M$ and
$\alpha^{(t)}=1$ if the process is convergent or $\alpha^{(t)}=0.5 \alpha^{(t-1)}$
if it is divergent. Another possibility is to optimize this coefficient.
the error of $k$-th parameter estimate is
$$
\Delta a_k^{(e)}=
\sqrt{\frac
{\sum_{i=1}^{N}\frac{e_i^2}{y_i}}
{\sum_{i=1}^{N} \left[ \frac{\partial f(i,a^{(t)})}{\partial a_k}\right]^2\frac{1}{y_i}}
}
$$
algorithm with higher powers w=1,2,3...
$$
\Delta a_{k,w}^{(t+1)}=
\alpha^{(t)}
\frac
{\sum_{i=1}^{N} \frac{e_i}{y_i}\left[ \frac{\partial f(i,a^{(t)})}{\partial a_k}\right]^{2w-1}}
{\sum_{i=1}^{N} \left[ \frac{\partial f(i,a^{(t)})}{\partial a_k}\right]^{2w}\frac{1}{y_i}}
$$
we have implemented the nonsymmetrical semiempirical peak shape function.
it contains the symmetrical Gaussian as well as nonsymmetrical terms.
$$
f(i,a) =
\sum_{i=1}^{M} A(j)
\left\{
exp\left[\frac{-(i-p(j))^2}{2\sigma^2}\right]
+\frac{1}{2}T.exp\left[\frac{(i-p(j))}{B\sigma}\right]
.erfc\left[\frac{(i-p(j))}{\sigma}+\frac{1}{2B}\right]
+\frac{1}{2}S.erfc\left[\frac{(i-p(j))}{\sigma}\right]
\right\}
$$
where $T,S$ are relative amplitudes and $B$ is a slope.
Detailed description of the algorithm is given in [13].
The fitting function implementing the algorithm without matrix inversion
has the form
``` {.cpp}
char* Fit1Awmi(float *source,
TSpectrumOneDimFit *p,
int size);
```
This function fits the source spectrum. The calling program should fill
in input parameters of the one\_dim\_fit structure The fitted parameters
are written into structure pointed by one\_dim\_fit structure pointer
and fitted data are written into source
spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`p`** pointer to the one\_dim\_fit structure pointer
- **`size`** length of source spectrum
The one\_dim\_fit structure has the form:
``` {.cpp}
class TSpectrumOneDimFit{
public:
int number_of_peaks; // input parameter, should be >0
int number_of_iterations; // input parameter, should be >0
int xmin; // first fitted channel
int xmax; // last fitted channel
double alpha; // convergence coefficient, input parameter, it should be
// positive number and <=1
double chi; // here the function returns resulting chi-square
int statistic_type; // type of statistics, possible values
// FIT1_OPTIM_CHI_COUNTS (chi square statistics with
// counts as weighting coefficients),
// FIT1_OPTIM_CHI_FUNC_VALUES (chi square statistics
// with function values as weighting coefficients)
// FIT1_OPTIM_MAX_LIKELIHOOD
int alpha_optim; // optimization of convergence coefficients, possible values
// FIT1_ALPHA_HALVING,
// FIT1_ALPHA_OPTIMAL
int power; // possible values FIT1_FIT_POWER2,4,6,8,10,12
int fit_taylor; // order of Taylor expansion, possible values
// FIT1_TAYLOR_ORDER_FIRST, FIT1_TAYLOR_ORDER_SECOND
double position_init[MAX_NUMBER_OF_PEAKS1]; // initial values of
// peaks positions, input parameters
double position_calc[MAX_NUMBER_OF_PEAKS1]; // calculated values
// of fitted positions, output parameters
double position_err[MAX_NUMBER_OF_PEAKS1]; // position errors
bool fix_position[MAX_NUMBER_OF_PEAKS1]; // logical vector which allows to fix
// appropriate positions (not fit). However they
// are present in the estimated functional
double amp_init[MAX_NUMBER_OF_PEAKS1]; // initial values of peaks
// amplitudes, input parameters
double amp_calc[MAX_NUMBER_OF_PEAKS1]; // calculated values of
// fitted amplitudes, output parameters
double amp_err[MAX_NUMBER_OF_PEAKS1]; // amplitude errors
bool fix_amp[MAX_NUMBER_OF_PEAKS1]i; // logical vector, which allows to fix
// appropriate amplitudes (not fit). However they
// are present in the estimated functional
double area[MAX_NUMBER_OF_PEAKS1]; // calculated areas of peaks
double area_err[MAX_NUMBER_OF_PEAKS1]; // errors of peak areas
double sigma_init // sigma parameter, see peak shape function
double sigma_calc;
double sigma_err;
bool fix_sigma;
double t_init // t parameter, , see peak shape function
double t_calc;
double t_err;
bool fix_t;
double b_init // b parameter, , see peak shape function
double b_calc;
double b_err;
bool fix_b;
double s_init; // s parameter, , see peak shape function
double s_calc;
double s_err;
bool fix_s;
double a0_init; // backgroud is estimated as a0+a1*x+a2*x*x
double a0_calc;
double a0_err;
bool fix_a0;
double a1_init;
double a1_calc;
double a1_err;
bool fix_a1;
double a2_init;
double a2_calc;
double a2_err;
bool fix_a2;
};
```
As an example we present simple 1-dimensional synthetic spectrum with 5
peaks. The fit obtained using above given awmi fitting function is given
in Figure 5.1. The chi-square achieved in this fit was 0.76873. The input
value of the fit (positions of peaks and their amplitudes) were
estimated using peak searching function.
Let us go to more complicated fit with lot of overlapping peaks Figure 5.2.
The initial positions of peaks were determined from original data, using
peak searching function. The fit is not very good, as there are some
peaks missing.
However to analyze the spectrum we can proceed in completely different
way employing the sophisticated functions of background elimination and
deconvolution. First let us remove background from the original raw
data. We get spectrum given in Figure 5.3.
Then we can apply the Gold deconvolution function to these data. We
obtain result presented in Figure 5.4.
Using peaks searching method, looking just for local maxima, (sigma=0)
with appropriate threshold (50), we can estimate initial positions of
peaks for fitting function. After the fit of the original experimental
spectrum (with background) we obtain the result shown in Figure 5.5. Now
the fitted function corresponds much better to experimental values
We have implemented also the fitting function with matrix inversion
based on Stiefel-Hestens method of the solution of the system of linear
equations. The form of the function is as follows
```{.cpp}
char *Fit1Stiefel(float *source,
TSpectrumOneDimFit* p,
int size);
```
This function fits the source spectrum. The calling program should fill
in input parameters of the one\_dim\_fit structure The fitted parameters
are written into structure pointed by one\_dim\_fit structure pointer
and fitted data are written into source
spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum
- **`p`** pointer to the one\_dim\_fit structure pointer
- **`size`** length of source spectrum
The structure one\_dim\_fit is the same as in awmi function. The
parameters power, fit\_taylor are not applicable for this function
The results for small number of fitted parameters are the same as with
awmi function. However it converges faster. The example for data given
in Figure 5.1 is given in the following table:
+-------------------+----------+-------------+
| \# of iterationsi | Chi awmi | Chi-Stiefel |
+-------------------+----------+-------------+
| 1 | 924 | 89.042 |
+-------------------+----------+-------------+
| 5 | 773.15 | 0.96242 |
+-------------------+----------+-------------+
| 10 | 38.13 | 0.77041 |
+-------------------+----------+-------------+
| 50 | 0.90293 | 0.76873 |
+-------------------+----------+-------------+
| 100 | 0.76886 | 0.76873 |
+-------------------+----------+-------------+
| 500 | 0.76873 | 0.76873 |
+-------------------+----------+-------------+
## 2-DIMENSIONAL SPECTRA ##
it is straightforward that for two dimensional spectra one can write
$$
\Delta a_k^{(t+1)}=\alpha^{(t)}
\frac
{\sum_{i_1=1}^{N_1}\sum_{i_2=1}^{N_2}\frac{e_{i_1,i_2}^{(t)}}{y_{i_1,i_2}}
\frac{\partial f(i_1,i_2,a^{(t)})}{\partial a_k}}
{\sum_{i_1=1}^{N_1}\sum_{i_2=1}^{N_2}
\left[\frac{\partial f(i_1,i_2,a^{(t)})}{\partial a_k} \right]^2
\frac{1}{y_{i_1,i_2}}}
$$
analogously for two dimensional peaks we have chosen the peak shape
function of the following form
$$
f(i_1,i_2,a) = \sum_{j=1}^{M}\left\{
\begin{array}{l}
A_{xy}(j) exp\left\{-\frac{1}{2(1-\rho^2)}\left[
\frac{(i_1-p_x(j))^2}{\sigma_x^2}
-\frac{2\rho(i_1-p_x(j))(i_2-p_y(j))}{\sigma_x\sigma_y}
+\frac{(i_2-p_y(j))^2}{\sigma_y^2}
\right]\right\} \\
+A_x(j) exp\left[-\frac{(i_1-p_{x_1}(j))^2}{2\sigma_x^2} \right]
+A_y(j) exp\left[-\frac{(i_2-p_{y_1}(j))^2}{2\sigma_y^2} \right]
\end{array}
\right\}+b_0+b_1i_1+b_2i_2
$$
The meaning of the parameters is analogous to 1-dimensional case. Again
all the details can be found in [13]
The fitting function implementing the algorithm without matrix inversion
for 2-dimensional data has the form
```{.cpp}
char* Fit2Awmi(float **source,
TSpectrumTwoDimFit* p,
int sizex,
int sizey);
```
This function fits the source spectrum. The calling program should
fill in input parameters of the two\_dim\_fit structure The fitted
parameters are written into structure pointed by two\_dim\_fit structure
pointer and fitted data are written back into source spectrum.
Function parameters:
- **`source`** pointer to the matrix of source spectrum
- **`p`** pointer to the two\_dim\_fit structure pointer, see manual
- **`sizex`** length x of source spectrum
- **`sizey`** length y of source spectrum
The two\_dim\_fit structure has the form
``` {.cpp}
class TSpectrumTwoDimFit{
public:
int number_of_peaks; // input parameter, shoul be>0
int number_of_iterations; // input parameter, should be >0
int xmin; // first fitted channel in x direction
int xmax; // last fitted channel in x direction
int ymin; // first fitted channel in y direction
int ymax; // last fitted channel in y direction
double alpha; // convergence coefficient, input parameter, it should be positive
// number and <=1
double chi; // here the function returns resulting chi square
int statistic_type; // type of statistics, possible values
// FIT2_OPTIM_CHI_COUNTS (chi square statistics with
// counts as weighting coefficients),
// FIT2_OPTIM_CHI_FUNC_VALUES (chi square statistics
// with function values as weighting
// coefficients),FIT2_OPTIM_MAX_LIKELIHOOD
int alpha_optim; // optimization of convergence coefficients, possible values
// FIT2_ALPHA_HALVING, FIT2_ALPHA_OPTIMAL
int power; // possible values FIT21_FIT_POWER2,4,6,8,10,12
int fit_taylor; // order of Taylor expansion, possible values
// FIT2_TAYLOR_ORDER_FIRST,
// FIT2_TAYLOR_ORDER_SECOND
double position_init_x[MAX_NUMBER_OF_PEAKS2]; // initial values of x
// positions of 2D peaks, input parameters
double position_calc_x[MAX_NUMBER_OF_PEAKS2]; // calculated values
// of fitted x positions of 2D peaks, output parameters
double position_err_x[MAX_NUMBER_OF_PEAKS2]; // x position errors of 2D peaks
bool fix_position_x[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows to fix appropriate
// x positions of 2D peaks (not fit).
// However they are present in the estimated functional
double position_init_y[MAX_NUMBER_OF_PEAKS2]; // initial values of y
// positions of 2D peaks, input parameters
double position_calc_y[MAX_NUMBER_OF_PEAKS2]; // calculated values
// of fitted y positions of 2D peaks, output parameters
double position_err_y[MAX_NUMBER_OF_PEAKS2]; // y position errors of 2D peaks
bool fix_position_y[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows to fix appropriate
// y positions of 2D peaks (not fit).
// However they are present in the estimated functional
double position_init_x1[MAX_NUMBER_OF_PEAKS2]; // initial values of x
// positions of 1D ridges, input parameters
double position_calc_x1[MAX_NUMBER_OF_PEAKS2]; // calculated values of
// fitted x positions of 1D ridges, output parameters
double position_err_x1[MAX_NUMBER_OF_PEAKS2]; // x position errors of 1D ridges
bool fix_position_x1[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows to fix appropriate
// x positions of 1D ridges (not fit).
// However they are present in the estimated functional
double position_init_y1[MAX_NUMBER_OF_PEAKS2]; // initial values of y
// positions of 1D ridges, input parameters
double position_calc_y1[MAX_NUMBER_OF_PEAKS2]; // calculated values
// of fitted y positions of 1D ridges, output parameters
double position_err_y1[MAX_NUMBER_OF_PEAKS2]; // y position errors of 1D ridges
bool fix_position_y1[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows to fix
// appropriate y positions of 1D ridges (not fit).
// However they are present in the estimated functional
double amp_init[MAX_NUMBER_OF_PEAKS2]; // initial values of 2D peaks
// amplitudes, input parameters
double amp_calc[MAX_NUMBER_OF_PEAKS2]; // calculated values of
// fitted amplitudes of 2D peaks, output parameters
double amp_err[MAX_NUMBER_OF_PEAKS2]; // amplitude errors of 2D peaks
bool fix_amp[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows
// to fix appropriate amplitudes of 2D peaks (not fit).
// However they are present in the estimated functional
double amp_init_x1[MAX_NUMBER_OF_PEAKS2]; // initial values of 1D
// ridges amplitudes, input parameters
double amp_calc_x1[MAX_NUMBER_OF_PEAKS2]; // calculated values of
// fitted amplitudes of 1D ridges, output parameters
double amp_err_x1[MAX_NUMBER_OF_PEAKS2]; // amplitude errors of 1D ridges
bool fix_amp_x1[MAX_NUMBER_OF_PEAKS2]; // logical vector which alloes to fix
// appropriate amplitudes of 1D ridges (not fit).
// However they are present in the estimated functional
double amp_init_y1[MAX_NUMBER_OF_PEAKS2]; // initial values of 1D
// ridges amplitudes, input parameters
double amp_calc_y1[MAX_NUMBER_OF_PEAKS2]; // calculated values of
// fitted amplitudes of 1D ridges, output parameters
double amp_err_y1[MAX_NUMBER_OF_PEAKS2]; // amplitude errors of 1D ridges
bool fix_amp_y1[MAX_NUMBER_OF_PEAKS2]; // logical vector which allows to fix
// appropriate amplitudes of 1D ridges (not fit).
// However they are present in the estimated functional
double volume[MAX_NUMBER_OF_PEAKS1]; // calculated volumes of peaks
double volume_err[MAX_NUMBER_OF_PEAKS1]; // errors of peak volumes
double sigma_init_x; // sigma x parameter
double sigma_calc_x;
double sigma_err_x;
bool fix_sigma_x;
double sigma_init_y; // sigma y parameter
double sigma_calc_y;
double sigma_err_y;
bool fix_sigma_y;
double ro_init; // correlation coefficient
double ro_calc;
double ro_err;
bool fix_ro;
double txy_init; // t parameter for 2D peaks
double txy_calc;
double txy_err;
bool fix_txy;
double sxy_init; // s parameter for 2D peaks
double sxy_calc;
double sxy_err;
bool fix_sxy;
double tx_init; // t parameter for 1D ridges (x direction)
double tx_calc;
double tx_err;
bool fix_tx;
double ty_init; // t parameter for 1D ridges (y direction)
double ty_calc;
double ty_err;
bool fix_ty;
double sx_init; // s parameter for 1D ridges (x direction)
double sx_calc;
double sx_err;
bool fix_sx;
double sy_init; // s parameter for 1D ridges (y direction)
double sy_calc;
double sy_err;
bool fix_sy;
double bx_init; // b parameter for 1D ridges (x direction)
double bx_calc;
double bx_err;
bool fix_bx;
double by_init; // b parameter for 1D ridges (y direction)
double by_calc;
double by_err;
bool fix_by;
double a0_init; // backgroud is estimated as a0+ax*x+ay*y
double a0_calc;
double a0_err;
bool fix_a0;
double ax_init;
double ax_calc;
double ax_err;
bool fix_ax;
double ay_init;
double ay_calc;
double ay_err;
bool fix_ay;
};
```
The example of the original spectrum and the fitted spectrum is given in
Figures. 5.6, 5.7, respectively. We have fitted 5 peaks. Each peak was
represented by 7 parameters, which together with sigmax, sigmay and b0
resulted in 38 parameters. The chi-square after 1000 iterations was
0.6571.
The awmi algorithm can be applied also to large blocks of data and large
number of peaks. In the next example we present spectrum with identified
295 peaks. Each peak is represented by 7 parameters, which together with
sigmax, y and b0 resulted in 2068 fitted parameters. The original
spectrum and fitted function are given in Figures. 5.8, 5.9, respectively.
The achieved chi-square was 0.76732.
We have implemented the fitting function with matrix inversion based on
Stiefel-Hestens method of the solution of the system of linear equations
also for 2-dimensional data. The form of the function is as follows
```{.cpp}
char* Fit2Stiefel(float **source,
TSpectrumTwoDimFit* p,
int sizex,
int sizey);
```
This function fits the source spectrum. The calling program should
fill in input parameters of the two\_dim\_fit structure The fitted
parameters are written into structure pointed by two\_dim\_fit structure
pointer and fitted data are written back into source
spectrum.
Function parameters:
- **`source`** pointer to the matrix of source spectrum
- **`p`** pointer to the two\_dim\_fit structure pointer, see manual
- **`sizex`** length x of source spectrum
- **`sizey`** length y of source spectrum
The structure two\_dim\_fit is the same as in awmi function. The
parameters power, fit\_taylor are not applicable for this function
The results for small number of fitted parameters are the same as with
awmi function. However it converges faster. The example for data given
in Figure 5.6 (38 parameters) is presented in the following table:
+------------------+----------+-------------+
| \# of iterations | Chi awmi | Chi-Stiefel |
+------------------+----------+-------------+
| 1 | 24.989 | 10.415 |
+------------------+----------+-------------+
| 5 | 20.546 | 1.0553 |
+------------------+----------+-------------+
| 10 | 6.256 | 0.84383 |
+------------------+----------+-------------+
| 50 | 1.0985 | 0.64297 |
+------------------+----------+-------------+
| 100 | 0.657 | 1 0.64297 |
+------------------+----------+-------------+
| 500 | 0.651 | 94 0.64297 |
+------------------+----------+-------------+
Again Stiefel-Hestens method converges faster. However its calculation
is for this number of parameters approximately 3 times longer. For
larger number of parameters the time needed to calculate the inversion
grows with the cube of the number of fitted parameters. For example the
fit of large number of parameters (2068) for data in Figure 5.8 using awmi
algorithm lasted about 12 hours (using 450 MHz PC). The calculation
using matrix inversion method is not realizable in reasonable time.
# TRANSFORMS #
## 1-DIMENSIONAL SPECTRA ##
Orthogonal transforms can be successfully used for the
processing of nuclear spectra . They can be used to remove high
frequency noise, to increase signal-to-background ratio as well as to
enhance low intensity components [14]. We have implemented also the
function for the calculation of the commonly used orthogonal transforms
- Haar
- Walsh
- Cos
- Sin
- Fourier
- Hartley
Between these transform one can define so called generalized mixed
transforms that are also implemented in the transform function
- Fourier-Haar
- Fourier-Walsh
- Walsh-Haar
- Cos-Walsh
- Cos-Haar
- Sin-Walsh
- Sin-Haar
The suitability of the application of appropriate transform depends on
the character of the data, i.e., on the shape of dominant components
contained in the data. The form of the transform function is as follows:
```{.cpp}
char *Transform1(const float *source,
float *dest,
int size,
int type,
int direction,
int degree);
```
This function transforms the source spectrum. The calling program
should fill in input parameters. Transformed data are written into dest
spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum, its length should
be equal to size parameter except for inverse FOURIER, FOUR-WALSH,
FOUR-HAAR transform. These need 2\*size length to supply real and
imaginary coefficients.
- **`dest`** pointer to the vector of dest data, its length should be equal to
size parameter except for direct FOURIER, FOUR-WALSh, FOUR-HAAR. These
need 2\*size length to store real and imaginary coefficients
- **`size`** basic length of source and dest spectra
- **`type`** type of transform
- TRANSFORM1\_HAAR
- TRANSFORM1\_WALSH
- TRANSFORM1\_COS
- TRANSFORM1\_SIN
- TRANSFORM1\_FOURIER
- TRANSFORM1\_HARTLEY
- TRANSFORM1\_FOURIER\_WALSH
- TRANSFORM1\_FOURIER\_HAAR
- TRANSFORM1\_WALSH\_HAAR
- TRANSFORM1\_COS\_WALSH
- TRANSFORM1\_COS\_HAAR
- TRANSFORM1\_SIN\_WALSH
- TRANSFORM1\_SIN\_HAAR
- **`direction`** transform direction (forward, inverse)
- TRANSFORM1\_FORWARD
- TRANSFORM1\_INVERSE
- **`degree`** applies only for mixed transforms
Let us illustrate the applications of the transform using an example. In
Figure 6.1 we have spectrum with many peaks, complicated background and
high level of noise.
In Figures. 6.2, 6.3, 6.4 we present this spectrum transformed using Haar,
Walsh and Cosine transforms, respectively.
Haar transforms (Figure 6.2) creates clusters of data. These coefficients
can be analyzed and then filtered, enhanced etc. On the other hand Walsh
transform (Figure 6.3) concentrates the dominant components near to zero of
the coordinate system. It is more suitable to process data of
rectangular shape (e.g. in the field of digital signal processing).
Finally Cosine transform concentrates in the best way the transform
coefficients to the beginning of the coordinate system. From the point
of view of the variance distribution it is sometimes called suboptimal.
One can notice that approximately one half of the coefficients are
negligible. This fact can be utilized to the compression purposes (in
two or more dimensional data), filtering (smoothing) etc.
We have implemented several application functions utilizing
the properties of the orthogonal transforms. Let us start with zonal
filtration function. It has the form.
```{.cpp}
char *Filter1Zonal(const float *source,
float *dest,
int size,
int type,
int degree,
int xmin,
int xmax,
float filter-coeff);
```
This function transforms the source spectrum. The calling program should
fill in input parameters. Then it sets transformed coefficients in the
given region (xmin, xmax) to the given filter\_coeff and transforms it
back Filtered data are written into dest spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum, its length should be size
- **`dest`** pointer to the vector of dest data, its length should be size
- **`size`** basic length of source and dest spectra
- **`type`** type of transform
- TRANSFORM1\_HAAR
- TRANSFORM1\_WALSH
- TRANSFORM1\_COS
- TRANSFORM1\_SIN
- TRANSFORM1\_FOURIER
- TRANSFORM1\_HARTLEY
- TRANSFORM1\_FOURIER\_WALSH
- TRANSFORM1\_FOURIER\_HAAR
- TRANSFORM1\_WALSH\_HAAR
- TRANSFORM1\_COS\_WALSH
- TRANSFORM1\_COS\_HAAR
- TRANSFORM1\_SIN\_WALSH
- TRANSFORM1\_SIN\_HAAR
- **`degree`** applied only for mixed transforms
- **`xmin`** low limit of filtered region
- **`xmax`** high limit of filtered region
- **`filter_coeff`** value which is set in filtered region
An example of the filtration using Cosine transform is given in the Figure 6.5.
It illustrates a part of the spectrum from Figure 6.1 and two spectra
after filtration preserving 2048 coefficients and 1536 coefficients. One
can observe very good fidelity of the overall shape of both spectra with
the original data. However some distortion can be observed in details of
the second spectrum after filtration preserving only 1536 coefficients.
The useful information in the transform domain can be compressed into
one half of the original space.
In the transform domain one can also enhance (multiply with
the constant \> 1) some regions. In this way one can change
peak-to-background ratio. This function has the form
```{.cpp}
char *Enhance1(const float *source,
float *dest,
int size,
int type,
int degree,
int xmin,
int xmax,
float enhance-coeff);
```
This function transforms the source spectrum. The calling program
should fill in input parameters. Then it multiplies transformed
coefficients in the given region (xmin, xmax) by the given
enhance\_coeff and transforms it back Processed data are written into
dest spectrum.
Function parameters:
- **`source`** pointer to the vector of source spectrum, its length should be size
- **`dest`** pointer to the vector of dest data, its length should be size
- **`size`** basic length of source and dest spectra
- **`type`** type of transform
- TRANSFORM1\_HAAR
- TRANSFORM1\_WALSH
- TRANSFORM1\_COS
- TRANSFORM1\_SIN
- TRANSFORM1\_FOURIER
- TRANSFORM1\_HARTLEY
- TRANSFORM1\_FOURIER\_WALSH
- TRANSFORM1\_FOURIER\_HAAR
- TRANSFORM1\_WALSH\_HAAR
- TRANSFORM1\_COS\_WALSH
- TRANSFORM1\_COS\_HAAR
- TRANSFORM1\_SIN\_WALSH
- TRANSFORM1\_SIN\_HAAR
- **`degree`** applied only for mixed transforms
- **`xmin`** low limit of filtered region
- **`xmax`** high limit of filtered region
- **`enhance_coeff`** value by which the filtered region is multiplied
An example of enhancement of the coefficients from region 380-800 by the
constant 2 in the Cosine transform domain is given in the Figure 6.6. The
determination of the region is a matter of analysis in the appropriate
transform domain. We assumed that low frequency components are placed in
the low coefficients. As it can be observed the enhancement changes the
peak-to-background ratio.
## 2-DIMENSIONAL SPECTRA ##
Analogously to1-dimensional data we have implemented the
transforms also for 2-dimensional data. Besides of the classic
orthogonal transforms
- Haar
- Walsh
- Cos
- Sin
- Fourier
- Hartley
- Fourier-Haar
- Fourier-Walsh
- Walsh-Haar
- Cos-Walsh
- Cos-Haar
- Sin-Walsh
- Sin-Haar
```{.cpp}
char *Transform2(const float **source,
float **dest,
int sizex,
int sizey,
int type,
int direction,
int degree);
```
This function transforms the source spectrum. The calling program should
fill in input parameters. Transformed data are written into dest
spectrum.
Function parameters:
- **`source`** pointer to the matrix of source spectrum, its size should
be sizex\*sizey except for inverse FOURIER, FOUR-WALSH, FOUR-HAAR
transform. These need sizex\*2\*sizey length to supply real
and imaginary coefficients.
- **`dest`** pointer to the matrix of destination data, its size should
be sizex\*sizey except for direct FOURIER, FOUR-WALSh, FOUR-HAAR.
These need sizex\*2\*sizey length to store real and imaginary
coefficients
- **`sizex,sizey`** basic dimensions of source and dest spectra
- **`type`** type of transform
- TRANSFORM2\_HAAR
- TRANSFORM2\_WALSH
- TRANSFORM2\_COS
- TRANSFORM2\_SIN
- TRANSFORM2\_FOURIER
- TRANSFORM2\_HARTLEY
- TRANSFORM2\_FOURIER\_WALSH
- TRANSFORM2\_FOURIER\_HAAR
- TRANSFORM2\_WALSH\_HAAR
- TRANSFORM2\_COS\_WALSH
- TRANSFORM2\_COS\_HAAR
- TRANSFORM2\_SIN\_WALSH
- TRANSFORM2\_SIN\_HAAR
- **`direction`** transform direction (forward, inverse)
- **`degree`** applies only for mixed transforms
An example of the 2-dimensional Cosine transform of data from Figure 5.6 is
given in Figure 6.7. One can notice that the data are concentrated again
around the beginning of the coordinate system. This allows to apply
filtration, enhancement and compression techniques in the transform
domain.
In some cases when the spectrum is smooth the cosine transforms are very
efficient. In Figures 6.8, 6.9 we show original spectrum and transformed
coefficients using Cosine transform, respectively.
Analogously to 1-dimensional case we have implemented also
the functions for zonal filtration, Gauss filtration and enhancement.
The zonal filtration function using classic transforms has the form
```{.cpp}
char *Filter2Zonal(const float **source,
float **dest,
int sizex,
int sizey,
int type,
int degree,
int xmin,
int xmax,
int ymin,
int ymax,
float filter-coeff);
```
This function transforms the source spectrum. The calling program should
fill in input parameters. Then it sets transformed coefficients in the
given region to the given
filter\_coeff and transforms it back Filtered data are written into dest
spectrum.
Function parameters:
- **`source`** pointer to the matrix of source spectrum, its size should be sizex\*sizey
- **`dest`** pointer to the matrix of destination data, its size should be sizex\*sizey
- **`sizex,sizey`** basic dimensions of source and dest spectra
- **`type`** type of transform
- TRANSFORM2\_HAAR
- TRANSFORM2\_WALSH
- TRANSFORM2\_COS
- TRANSFORM2\_SIN
- TRANSFORM2\_FOURIER
- TRANSFORM2\_HARTLEY
- TRANSFORM2\_FOURIER\_WALSH
- TRANSFORM2\_FOURIER\_HAAR
- TRANSFORM2\_WALSH\_HAAR
- TRANSFORM2\_COS\_WALSH
- TRANSFORM2\_COS\_HAAR
- TRANSFORM2\_SIN\_WALSH
- TRANSFORM2\_SIN\_HAAR
- **`degree`** applies only for mixed transforms
- **`xmin`** low limit x of filtered region
- **`xmax`** high limit x of filtered region
- **`ymin`** low limit y of filtered region
- **`ymax`** high limit y of filtered region
- **`filter_coeff`** value which is set in filtered region
The enhancement function using transforms has the form
```{.cpp}
char *Enhance2(const float **source,
float **dest,
int sizex,
int sizey,
int type,
int degree,
int xmin,
int xmax,
int ymin,
int ymax,
float enhance-coeff);
```
This function transforms the source spectrum. The calling program should
fill in input parameters. Then it multiplies transformed coefficients in
the given region by the given enhance\_coeff and transforms it
back
Function parameters:
- **`source`** pointer to the matrix of source spectrum, its size should be sizex\*sizey
- **`dest`** pointer to the matrix of destination data, its size should be sizex\*sizey
- **`sizex,sizey`** basic dimensions of source and dest spectra
- **`type`** type of transform
- TRANSFORM2\_HAAR
- TRANSFORM2\_WALSH
- TRANSFORM2\_COS
- TRANSFORM2\_SIN
- TRANSFORM2\_FOURIER
- TRANSFORM2\_HARTLEY
- TRANSFORM2\_FOURIER\_WALSH
- TRANSFORM2\_FOURIER\_HAAR
- TRANSFORM2\_WALSH\_HAAR
- TRANSFORM2\_COS\_WALSH
- TRANSFORM2\_COS\_HAAR
- TRANSFORM2\_SIN\_WALSH
- TRANSFORM2\_SIN\_HAAR
- **`degree`** applies only for mixed transforms
- **`xmin`** low limit x of filtered region
- **`xmax`** high limit x of filtered region
- **`ymin`** low limit y of filtered region
- **`ymax`** high limit y of filtered region
- **`enhance-coeff`** value which is set in filtered region
# VISUALIZATION #
## 1-DIMENSIONAL SPECTRA ##
The 1-dimensional visualization function displays spectrum
(or part of it) on the Canvas of a form. Before calling the function one
has to fill in one\_dim\_pic structure containing all parameters of the
display. The function has the form
```{.cpp}
char *display1(struct one-dim-pic* p);
```
This function displays the source spectrum on Canvas. All parameters are
grouped in one\_dim\_pic structure. Before calling display1 function the
structure should be filled in and the address of one\_dim\_pic passed as
parameter to display1 function. The meaning of appropriate parameters is
apparent from description of one\_dim\_pic structure. The constants ,
which can be used for appropriate parameters are defined in procfunc.h
header file.
``` {.cpp}
struct one_dim_pic {
float *source; // spectrum to be displayed
TCanvas *Canvas; // Canvas where the spectrum will be displayed
int size; // size of source spectrum
int xmin; // x-starting channel of spectrum
int xmax; // x-end channel of spectrum
int ymin; // base counts
int ymax; // count full scale
int bx1; // position of picture on Canvas, min x
int bx2; // position of picture on Canvas, max x
int by1; // position of picture on Canvas, min y
int by2; // position of picture on Canvas, max y
int display_mode; // spectrum display mode (points, polyline, bars, rainbow, steps, bezier)
int y_scale; // y scale (linear, log, sqrt)
int levels; // # of color levels for rainbow display mode, it does not apply
// for other display modes
float rainbow1_step; // determines the first color component step for neighbouring
// color levels, applies only for rainbow display mode
float rainbow2_step; // determines the second component color step for
// neighbouring color levels, applies only for rainbow display mode
float rainbow3_step; // determines the third component color step for
// neighbouring color levels, applies only for rainbow display mode
int color_alg; // applies only for rainbow display mode (rgb smooth algorithm, rgb
// modulo color component, cmy smooth algorithm, cmy modulo color
// component, cie smooth algorithm, cie modulo color component, yiq
// smooth algorithm, yiq modulo color component, hsv smooth
// algorithm, hsv modulo color component [15]
int bar_thickness; // applies only for bar display mode
int bar_empty_flag; // (empty bars, full bars) applies only for bar display mode
int border_color; // color of background of the picture
int full_border; // decides whether background is painted
int raster_en_dis; // decides whether the axes and rasters are shown
int raster_long; // decides whether the rasters are drawn as long lines
int raster_color; // color of the rasters
char *raster_description_x; // x axis description
char *raster_description_y; // y axis description
int pen_color; // color of spectrum
int pen_dash; // style of pen
int pen_width; // width of line
int chanmark_style; // style of channel marks
int chanmark_width; // width of channel marks
int chanmark_height; // height of channel marks
int chanmark_en_dis; // decides whether the channel marks are shown
int chanmark_color; // color of channel marks
// auxiliary variables, transform coefficients, for internal use only
double mx;
double my;
double px;
double py;
// auxiliary internal variables, working place
double gbezx,gbezy;
TPoint bz[4];
};
```
The examples using different display parameters are shown in the next
few Figures
## 2-DIMENSIONAL SPECTRA ##
The 2-dimensional visualization function displays spectrum
(or part of it) on the Canvas of a form. Before calling the function one
has to fill in two\_dim\_pic structure containing all parameters of the
display. The function has the form
```{.cpp}
char *display2(struct two-dim-pic* p);
```
This function displays the source two dimensional spectrum on Canvas.
All parameters are grouped in two\_dim\_pic structure. Before calling
display2 function the structure should be filled in and the address of
two\_dim\_pic passed as parameter to display2 function. The meaning of
appropriate parameters is apparent from description of one\_dim\_pic
structure. The constants , which can be used for appropriate parameters
are defined in procfunc.h header file.
``` {.cpp}
struct two_dim_pic {
float **source; // source spectrum to be displayed
TCanvas *Canvas; // Canvas where the spectrum will be displayed
int sizex; // x-size of source spectrum
int sizey; // y-size of source spectrum
int xmin; // x-starting channel of spectrum
int xmax; // x-end channel of spectrum
int ymin; // y-starting channel of spectrum
int ymax; // y-end channel of spectrum
int zmin; // base counts
int zmax; // counts full scale
int bx1; // position of picture on Canvas, min x
int bx2; // position of picture on Canvas, max x
int by1; // position of picture on Canvas, min y
int by2; // position of picture on Canvas, max y
int mode_group; // display mode algorithm group (simple modes-
// PICTURE2_MODE_GROUP_SIMPLE, modes with shading
// according to light-PICTURE2_MODE_GROUP_LIGHT, modes with
// shading according to channels counts-
// PICTURE2_MODE_GROUP_HEIGHT, modes of combination of
// shading according to light and to channels counts-
// PICTURE2_MODE_GROUP_LIGHT_HEIGHT)
int display_mode; // spectrum display mode (points, grid, contours, bars, x_lines,
// y_lines, bars_x, bars_y, needles, surface, triangles)
int z_scale; // z scale (linear, log, sqrt)
int nodesx; // number of nodes in x dimension of grid
int nodesy; // number of nodes in y dimension of grid
int count_reg; // width between contours, applies only for contours display mode
int alfa; // angles of display,alfa+beta must be less or equal to 90, alpha- angle
// between base line of Canvas and left lower edge of picture picture
// base plane
int beta; // angle between base line of Canvas and right lower edge of picture base plane
int view_angle; // rotation angle of the view, it can be 0, 90, 180, 270 degrees
int levels; // # of color levels for rainbowed display modes, it does not apply for
// simple display modes algorithm group
float rainbow1_step; // determines the first component step for neighbouring color
// levels, applies only for rainbowed display modes, it does not apply
// for simple display modes algorithm group
float rainbow2_step; // determines the second component step for neighbouring
// color levels, applies only for rainbowed display modes, it does not
// apply for simple display modes algorithm group
float rainbow3_step; // determines the third component step for neighbouring
// color levels, applies only for rainbowed display modes, it does not
// apply for simple display modes algorithm group
int color_alg; // applies only for rainbowed display modes (rgb smooth algorithm,
// rgb modulo color component, cmy smooth algorithm, cmy modulo
// color component, cie smooth algorithm, cie modulo color component,
// yiq smooth algorithm, yiq modulo color component, hsv smooth
// algorithm, hsv modulo color component, it does not apply for simple
// display modes algorithm group [15]
float l_h_weight; // weight between shading according to fictive light source and
// according to channels counts, applies only for
// PICTURE2_MODE_GROUP_LIGHT_HEIGHT modes group
int xlight; // x position of fictive light source, applies only for rainbowed display
// modes with shading according to light
int ylight; // y position of fictive light source, applies only for rainbowed display
// modes with shading according to light
int zlight; // z position of fictive light source, applies only for rainbowed display
// modes with shading according to light
int shadow; // determines whether shadow will be drawn (no shadow, shadow),
// for rainbowed display modes with shading according to light
int shading; // determines whether the picture will shaded, smoothed (no shading,
// shading), for rainbowed display modes only
int bezier; // determines Bezier interpolation (applies only for simple display
// modes group for grid, x_lines, y_lines display modes)
int border_color; // color of background of the picture
int full_border; // decides whether background is painted
int raster_en_dis; // decides whether the rasters are shown
int raster_long; // decides whether the rasters are drawn as long lines
int raster_color; // color of the rasters
char *raster_description_x; // x axis description
char *raster_description_y; // y axis description
char *raster_description_z; // z axis description
int pen_color; // color of spectrum
int pen_dash; // style of pen
int pen_width; // width of line
int chanmark_en_dis; // decides whether the channel marks are shown
int chanmark_style; // style of channel marks
int chanmark_width; // width of channel marks
int chanmark_height; // height of channel marks
int chanmark_color; // color of channel marks
int chanline_en_dis; // decides whether the channel lines (grid) are shown
// auxiliary variables, transformation coefficients for internal use only
double kx;
double ky;
double mxx;
double mxy;
double myx;
double myy;
double txx;
double txy;
double tyx;
double tyy;
double tyz;
double vx;
double vy;
double nu_sli;
// auxiliary internal variables, working place
double z,zeq,gbezx,gbezy,dxspline,dyspline;
int xt,yt,xs,ys,xe,ye,priamka,z_preset_value;
unsigned short obal[MAXIMUM_XSCREEN_RESOLUTION];
unsigned short obal_cont[MAXIMUM_XSCREEN_RESOLUTION];
TPoint bz[4];
};
```
The examples using different display parameters are shown in the next few Figures.
# References # {.unnumbered}
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[11] T. Awaya: A new method for curve fitting to the data with low
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NIM A 457 (2001) 384-401.
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[15] D. Hearn, M. P. Baker: Computer Graphics, Prentice-Hall
International, Inc., 1994.
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