Interactive visualization of multidimensional coincidence spectra

1754-0410-1-7 1754-0410 Research article Interactive visualization of multidimensional coincidence spectra Morháč Miroslav Miroslav.Morhac@savba.sk Matoušek Vladislav Vladislav.Matousek@savba.sk

Department of Nuclear Physics, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 845 11 Bratislava, Slovakia

PMC Physics A 1754-0410 2007 1 1 7 http://www.physmathcentral.com/1754-0410/1/7 10.1186/1754-0410-1-7 22 6 2007 23 11 2007 23 11 2007 2007 Morháč et al; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper presents direct visualization techniques of multidimensional nuclear spectra as well as visualization techniques based on projections of embedded subspaces. While the first group of graphical models is limited to four dimensions, the second one can be theoretically extended to any dimension. The presented algorithms of visualization have been implemented in nuclear data acquisition, processing and visualization system developed at the Institute of Physics, Slovak Academy of Sciences. The paper focuses on presentation of nuclear spectra. However the majority of algorithms can be successfully applied for visualization of scalar arrays of other data types.

PACS Code: 29.85.+c, 07.05.Rm

1. Introduction

The power of computers to collect, store and manipulate experimental data has increased dramatically. In today's nuclear physics experiments the number of detectors being included in the measurements is going up to one hundred or more. The results of such measurements, however, generate such large data sets as to be nearly incomprehensible. Scanning these large sets of numbers to determine trends and relationships is a tedious and ineffective process. To address this problem the physicists have turned to visualization of experimental data. If the data are converted to a visual form, the trends are often immediately apparent. Without visualization much of the increased power of computers would be wasted because experiments are poor at gaining insight from data presented in numerical form.

The goal of visualization of experimental data is an improved understanding of the result of the information gathered during experiment. It is one of the most powerful and direct ways how the huge amount of information can be conveyed in a form comprehensible to a human eye. As a result, development of visualization algorithms takes on great significance, offering a promising technology for transforming an indigestible mass of numbers into a medium, which experimenters can understand, interpret and explore.

The visualization techniques presented in this work make it possible to display either raw experimental spectra, processed data or to make slices of the same or lower dimensionality in an interactive way. They allow obtaining an imagination about event distribution and correlations in coincidence spectra up to five-dimensional space.

The objective of the paper is to present visualization techniques and graphical models to display multidimensional nuclear spectra (histograms), which were implemented in the nuclear data acquisition, processing and visualization system 12. Though the software package is designed mainly for use with nuclear data, any kind of data can be processed as well. Other systems for nuclear spectra graphical representation were presented in 34.

2. Direct visualization techniques of scalar fields

A scalar variable is a single quantity, in the case of nuclear spectra – counts, which can be represented as a function of independent variables – particle energies. Most scalar visualization techniques use a consistent approach across one-, two-, or three-dimensional fields. More recent techniques, e.g. of the visualization of three-dimensional fields, attempt to show the full three-dimensional variations of a scalar variable within a volume field. These techniques include isosurfaces, particle clouds, volume slicing and sampling planes 567.

The sophisticated visualization algorithms are presented in 8. The paper presents conventional as well as newly developed visualization techniques and graphical models. The structure and complexity of the algorithms lend themselves for implementation in on-line live mode during the data acquisition or processing. The pictures can be simultaneously updated.

One can select various attributes of the display, e.g. color of the spectrum, the limits of the displayed part of the spectrum, window, marker, type of scale, and various display modes, slices, to rotate two-, or more-dimensional data. In the above-mentioned paper, we have developed the direct visualization algorithms up to four-dimensional data.

2.1. Two-dimensional spectra

Two-parameter coincidence nuclear spectrum (histogram) is represented by a matrix of data with two independent variables (parameters) and one dependent variable, (counts), i.e., c = f(x, y). To project the three-dimensional scene onto a two-dimensional screen the axonometric transformation is employed. To display three-dimensional data on screen we have employed the following model

x ′ = t x x ⋅ i + t x y ⋅ j + v x y ′ = t y x ⋅ i + t y y ⋅ j + t y z ⋅ c + v y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGceaqabeaacuWG4baEgaqbaiabg2da9iabdsha0naaBaaaleaacqWG4baEcqWG4baEaeqaaOGaeyyXICTaemyAaKMaey4kaSIaemiDaq3aaSbaaSqaaiabdIha4jabdMha5bqabaGccqGHflY1cqWGQbGAcqGHRaWkcqWG2bGDdaWgaaWcbaGaemiEaGhabeaaaOqaaiqbdMha5zaafaGaeyypa0JaemiDaq3aaSbaaSqaaiabdMha5jabdIha4bqabaGccqGHflY1cqWGPbqAcqGHRaWkcqWG0baDdaWgaaWcbaGaemyEaKNaemyEaKhabeaakiabgwSixlabdQgaQjabgUcaRiabdsha0naaBaaaleaacqWG5bqEcqWG6bGEaeqaaOGaeyyXICTaem4yamMaey4kaSIaemODay3aaSbaaSqaaiabdMha5bqabaaaaaa@6526@

where t_xx, t_xy, t_yx, t_yy, t_yz, v_x, v_yare transform coefficients reflecting translation in both original two-dimensional scalar field (in x, y dimensions as well as in counts) and in the position on screen, scaling, rotation around z-axis and elevation of the view. The position of a point on the screen is x', y' and

x = x_min+ k_x·i; y = y_min+ k_y·j; i ∈ < 0, n_x>; j ∈ < 0, n_y>

k x = x m a x − x m i n n x ; k y = y m a x − y m i n n y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aaSbaaSqaaiabdIha4bqabaGccqGH9aqpjuaGdaWcaaqaaiabdIha4naaBaaabaacbiGae8xBa0Mae8xyaeMae8hEaGhabeaacqGHsislcqWG4baEdaWgaaqaaiab=1gaTjab=LgaPjab=5gaUbqabaaabaGaemOBa42aaSbaaeaacqWG4baEaeqaaaaakiabcUda7iabdUgaRnaaBaaaleaacqWG5bqEaeqaaOGaeyypa0tcfa4aaSaaaeaacqWG5bqEdaWgaaqaaiab=1gaTjab=fgaHjab=Hha4bqabaGaeyOeI0IaemyEaK3aaSbaaeaacqWFTbqBcqWFPbqAcqWFUbGBaeqaaaqaaiabd6gaUnaaBaaabaGaemyEaKhabeaaaaaaaa@5544@

n_x, n_yare numbers of nodes of a regular grid. The model proposed in such a way allows:

• to choose and display any part of the spectrum by setting x_min, x_max, y_min, y_maxto appropriate values

• to set any range of displayed counts – c_min, c_max

• to place the display of spectrum anywhere on the screen

• to rotate and elevate the view of the spectrum

• to change the density of display nodes. This is important when displaying accumulated spectra in on-line mode, i.e., during the acquisition of spectra.

To illustrate the capabilities of the proposed visualization algorithms we introduce several examples. In Fig. 1 we present two-dimensional spectrum shown in contours display mode. The same spectrum shown in triangle display mode in log scale can be seen in Fig. 2. To identify interesting locations in spectra together with displayed spectrum one can display one-dimensional slices and to move with them in both directions (see Fig. 3).
Figure 1
Two-dimensional spectrum shown in contours display mode

Two-dimensional spectrum shown in contours display mode.
Figure 2
Two-dimensional spectrum shown in triangle display mode (logscale)

Two-dimensional spectrum shown in triangle display mode (logscale).
Figure 3
Two-dimensional spectrum shown in points display mode with raster and slices

Two-dimensional spectrum shown in points display mode with raster and slices.

Sophisticated surface display mode with shading according to heights of peaks is shown in Fig. 4. Shading according to the position of fictive light source is shown in Fig. 5. One can change the position of the light source thus giving the possibility to achieve special effects.
Figure 4
An example of surface display mode with shading according to heights

An example of surface display mode with shading according to heights.
Figure 5
An example of surface display mode with shading according to fictive light source

An example of surface display mode with shading according to fictive light source.

Furthermore, there exists possibility to combine both shading methods. In Fig. 6 we present the display mode with mixed surface shading (according to height and light position) with ratio 50:50. One can include also the display of shadows according to the light source (Fig. 7).
Figure 6
An example of surface display mode with combined shading algorithms (given in Fig. 4 and Fig. 5)

An example of surface display mode with combined shading algorithms (given in Fig. 4 and Fig. 5).
Figure 7
Surface display mode (like in Fig. 5) with shadows

Surface display mode (like in Fig. 5) with shadows.

All display parameters can be changed according to the needs of the experimenters. Informative way of the display is rectangular view with contour shading (positron annihilation spectrum) shown in Fig. 8. One can define Regions Of Interests (ROIs) in the spectrum. Every ROI has its own display parameters independent of the main spectrum and other ROIs. In Fig. 9, one can see two ROIs in the two-dimensional – ray spectrum displayed in different display modes and color shadings. Similarly, one can display also the peaks in the spectrum found in the process of peak identification (Fig. 10). There are many other display combinations possible. Their presentation however, goes beyond the scope of this work.
Figure 8
Rectangular view of positron annihilation spectrum with contour shading

Rectangular view of positron annihilation spectrum with contour shading.
Figure 9
Two-dimensional γ-X-ray spectrum with two ROIs

Two-dimensional γ-X-ray spectrum with two ROIs.
Figure 10
Two-dimensional γ-γ-coincidence spectrum with displayed peaks

Two-dimensional γ-γ-coincidence spectrum with displayed peaks.

2.2. Three-dimensional spectra

Analogously to the previous section three-parameter γ-ray coincidence nuclear spectrum is three-dimensional scalar field with three independent parameters x, y, z (particle energies) and one dependent variable – counts c = f(x, y, z). As with two-parameter scalar fields one can idealize the display of three-parameter scalar field using discrete symbols at specific locations in space, or use techniques that show the variations in the three-parameter space. Hence each channel is defined by three parameters – coordinates x, y, z in original space, which determine the position of a channel. First let us consider a model where the channel is shown as a sphere (other marks as square, triangle, star etc. also can be used) with the size proportional to the event counts it contains. Then the position of the channel x, y, z on the screen is

x ′ = t x x ⋅ i + t x y ⋅ j + t x z ⋅ k + v x ; y ′ = t y x ⋅ i + t y y ⋅ j + t y z ⋅ k + v y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6F5C@

where t_xx, t_xy, t_xz, t_yx, t_yy, t_yz, v_x, v_y, v_z, are display transform coefficients reflecting translations in both original three-dimensional scalar field (in x, y, z dimensions as well as in counts) and in the position on screen, scaling, rotation around axes x, y, z and

x = x_min+ k_x·i; y = y_min+ k_y·j; z = z_min+ k_z·k

where

k x = x max ⁡ − x min ⁡ n x ; k y = y max ⁡ − y min ⁡ n y ; k z = z max ⁡ − z min ⁡ n z ; MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6B9F@

n_x, n_y, n_zare numbers of nodes of regular grid. The model proposed in such a way allows:

• to choose and display any part of three-dimensional array – x_min, x_max, y_min, y_max, z_min, z_max

• to choose any range of displayed counts – c_min, c_max

• to place the picture anywhere on the screen

• to rotate the spectrum around the axes x, y, z

• to change the density of display nodes.

The particle gradient display modes where the channels are shown as spheres with either diameter or color proportional to their contents are shown in Fig. 11 and Fig. 12, respectively. From the figures, one can localize interesting parts (peaks) in the spectrum. Sometimes however, to identify these parts, it is preferable to display only a slice in the spectrum, to move with it and interactively find appropriate channels. One- and two-dimensional slice in three-dimensional spectrum is shown in Fig. 13 and 14, respectively.
Figure 11
Three-dimensional γ-γ-γ-ray coincidence spectrum with channels shown as spheres with diameters proportional to counts

Three-dimensional γ-γ-γ-ray coincidence spectrum with channels shown as spheres with diameters proportional to counts.
Figure 12
Three-dimensional γ-γ-γ-ray coincidence spectrum with channels shown as spheres with colors proportional to counts

Three-dimensional γ-γ-γ-ray coincidence spectrum with channels shown as spheres with colors proportional to counts.
Figure 13
One-dimensional slice in three-dimensional spectrum

One-dimensional slice in three-dimensional spectrum.
Figure 14
Two-dimensional slice in three-dimensional spectrum

Two-dimensional slice in three-dimensional spectrum.

One can use even more sophisticated surface display mode. The defined surface separates the events with higher counts (statistics) from those with lower counts. Moreover, to achieve smooth surface one can interpolate the three-dimensional space using B-spline technique. In Fig. 15 and Fig. 16, we see the three-dimensional γ-γ-γ – ray coincidence spectrum and positron annihilation spectrum, respectively.
Figure 15
Three-dimensional γ-γ-γ-ray coincidence spectrum shown in smoothed surface display mode

Three-dimensional γ-γ-γ-ray coincidence spectrum shown in smoothed surface display mode.
Figure 16
Three-dimensional positron annihilation spectrum shown in smoothed surface display mode

Three-dimensional positron annihilation spectrum shown in smoothed surface display mode.

Finally, there exists a possibility to display three-dimensional spectrum in volume rendering mode. From the color contours on the sides of cube, one can get an imagination about positions of interesting peaks in three-dimensional space. An example of such a display mode is given in Fig. 17. Again, one can employ interpolation of the three-dimensional space using B-splines of various degrees.
Figure 17
Three-dimensional γ-γ-γ-ray coincidence spectrum shown in volume rendering mode

Three-dimensional γ-γ-γ-ray coincidence spectrum shown in volume rendering mode.

2.3. Four-dimensional spectra

Now the counts is a function of four parameters (particle energies), i.e., c = f(x, y, z, v). Let us imagine that instead of one channel belonging to one point of 3-D space in three-parameter nuclear spectrum visualization now this point represents a slice in the fourth parameter, i.e.,

c_{i, j, k}(v) = f(x_i, y_j, z_k, v)

We depict each slice as a closed polygon with the center positioned in analogy with three-dimensional data at the location

x ′ i , j , k = t x x ⋅ i + t x y ⋅ j + t x z ⋅ k + v x y ′ i , j , k = t y x ⋅ i + t y y ⋅ j + t y z ⋅ k + v y . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7B66@

For the positions of vertices of the polygon i, j, k on screen we have

x ″ i , j , k ( v ) = x ′ i , j , k − r m a x ⋅ c i , j , k ( v ) − c m i n c m a x − c m i n ⋅ c o s ( 2 π ( v − v m i n ) v m a x − v m i n + 1 + φ 0 ) , y ″ i , j , k ( v ) = y ′ i , j , k + r m a x ⋅ c i , j , k ( v ) − c m i n c m a x − c m i n ⋅ s i n ( 2 π ( v − v m i n ) v m a x − v m i n + 1 + φ 0 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@EFF8@

where the fourth parameter v ∈ <v_min, v_max>, r_max(constant value) is maximum distance of a polygon vertex from its center, φ₀is starting angle of the display of the first vertex of the polygon and c_min, c_maxdetermine the range of displayed counts. The principle of 4D display is illustrated in Fig. 18.
Figure 18
Principle of 4D display

Principle of 4D display.

In Fig. 19 we show four-dimensional display of the synthetic Gaussian with the center at x = y = z = v = 8. Three parameters determine the position of the center of the slice. The channels of the slice are shown as bars (drawn in red color) starting in the center of the slice with lengths proportional to their contents. The channels are displayed starting from 9 o'clock position in clockwise direction. The example of a chunk (16 × 16 × 16 × 16 channels) of smoothed experimental γ-γ-γ-γ-ray spectrum through the use of this display algorithm is shown in Fig. 20.
Figure 19
Display of four-dimensional synthetic Gaussian

Display of four-dimensional synthetic Gaussian.
Figure 20
Smoothed experimental four-parameter spectrum

Smoothed experimental four-parameter spectrum.

Analogously to 2 and 3D data to find interesting parts of the spectra one can display slices of various dimensionality. In Fig. 21 we present 3D slices with fixed y and z variables, respectively. Changing the values of y and z one can move with the slices. Subsequently in Fig. 22 we give an example of three two-dimensional slices in four-dimensional spectrum with fixed xy, xz and yz variables. In Fig. 23 we illustrate similar situation when we fix the forth variable. We show the two-dimensional slice with fixed zv variables. Finally in Fig. 24 we introduce the display of three one-dimensional slices with fixed xyv, xzv and yzv variables, respectively.
Figure 21
Two three-dimensional slices in four-dimensional spectrum

Two three-dimensional slices in four-dimensional spectrum.
Figure 22
Three two-dimensional slices in four-dimensional spectrum with fixed xy, xz, yz variables

Three two-dimensional slices in four-dimensional spectrum with fixed xy, xz, yz variables.
Figure 23
Two-dimensional slice in four-dimensional spectrum with fixed zv variables

Two-dimensional slice in four-dimensional spectrum with fixed zv variables.
Figure 24
Three one-dimensional slices in four-dimensional spectrum with fixed xyv, xzv, yzv variables

Three one-dimensional slices in four-dimensional spectrum with fixed xyv, xzv, yzv variables.

To illustrate the display of ridges in four-dimensional space we show the examples of synthetic spectrum before and after background elimination in Fig. 25 and Fig. 26, respectively.
Figure 25
Synthetic four-dimensional spectrum before background elimination

Synthetic four-dimensional spectrum before background elimination.
Figure 26
Four-dimensional peaks after background elimination from the data from 25

Four-dimensional peaks after background elimination from the data from 25.

In Fig. 27, we present four-fold coincidence positron annihilation spectrum with interpolated channels.
Figure 27
Four-fold coincidence positron annihilation spectrum

Four-fold coincidence positron annihilation spectrum.

In pies display mode one can change the color (level of shading) while keeping the radius of circle constant. According to the resolution in the fourth independent variable the circle is divided to channels (pies) with colors proportional to the contents of channels. The size of the circle is proportional to the sum of counts in the fourth dimension ∑v=vmin⁡vmax⁡f(xi,yj,zk,v) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaWaaabCaeaacqWGMbGzcqGGOaakcqWG4baEdaWgaaWcbaGaemyAaKgabeaakiabcYcaSiabdMha5naaBaaaleaacqWGQbGAaeqaaOGaeiilaWIaemOEaO3aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcqWG2bGDcqGGPaqkaSqaaiabdAha2jabg2da9iabdAha2naaBaaameaacyGGTbqBcqGGPbqAcqGGUbGBaeqaaaWcbaGaemODay3aaSbaaWqaaiGbc2gaTjabcggaHjabcIha4bqabaaaniabggHiLdaaaa@4C6D@ An example of synthetic 4D spectrum in pies display mode is shown in Fig. 28.
Figure 28
Four-dimensional synthetic spectrum displayed in pies display mode

Four-dimensional synthetic spectrum displayed in pies display mode.

Finally in Fig. 29 we present four-dimensional spectrum in isosurface mode. Analogously to three-dimensional data, the surface separates the channels with higher counts from those with lower counts. In this case, however, the color of the surface is defined by the position of the first occurrence of the channel with the same or higher value than the defined boundary value.
Figure 29
Four-dimensional spectrum shown in isosurface display mode

Four-dimensional spectrum shown in isosurface display mode.

3. Technique of successive projections of embedded subspaces

The dimensionality of above-presented visualization techniques is limited to four. However, with increasing dimensionality of nuclear spectra the requirements in developing of multidimensional scalar visualization techniques becomes striking. In principle, the above-mentioned algorithms can be used even for higher dimensions by employing a new technique of embedded subspaces. Using this technique we divide the multidimensional space into outer subspace and one or more successive inner (embedded) subspaces, all of dimensionalities more convenient to human imagination.

The goal is to propose a technique that allows one to localize and scan interesting parts (peaks) in multidimensional spectra. Moreover it should permit to find correlations in the data, mainly among neighboring points, and thus to discover prevailing trends around multidimensional peaks.

The proposed technique makes benefit of specific character and features of nuclear spectra. It utilizes the fact that the interesting objects (peaks) have shape of quasi Gaussians. Further, in enormous multidimensional space the events are distributed very sparsely, which allows to preserve main features of data even after reducing the dimensionality by employing projection functional. Successive decreasing the dimensionality makes it possible to determine the positions of appropriate multidimensional peaks.

Without loss of generality, we shall assume the reduction of the space up to two-dimensional one. Other alternatives are also possible, but the display of two-dimensional array using perpendicular view allows utilizing screen area the most efficiently. Let us start with three-dimensional spectrum f(x, y, z). Let us apply a projection functional reducing dimensionality by one to two-dimensional array, e.g.

f⁽¹⁾(x, y) = F[f(x, y, z)].

In place of the functional one can use, e.g. sum of channels contents in a slice

f ( 1 ) ( x , y ) = ∑ z = z min ⁡ z max ⁡ f ( x , y , z ) , MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aaWbaaSqabeaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGGOaakcqWG4baEcqGGSaalcqWG5bqEcqGGPaqkcqGH9aqpdaaeWbqaaiabdAgaMjabcIcaOiabdIha4jabcYcaSiabdMha5jabcYcaSiabdQha6jabcMcaPaWcbaGaemOEaONaeyypa0JaemOEaO3aaSbaaWqaaiGbc2gaTjabcMgaPjabc6gaUbqabaaaleaacqWG6bGEdaWgaaadbaGagiyBa0MaeiyyaeMaeiiEaGhabeaaa0GaeyyeIuoakiabcYcaSaaa@516A@

or maximum in a slice

f⁽¹⁾(x, y) = max{f(x, y, z)},

where

z ∈ <z_min, z_max>

or any other suitable operation. Let us display each channel i, j in the form of a mark with size proportional to f⁽¹⁾(i, j). Again, because of the most efficient way of utilizing the screen, in place of the mark we choose a rectangle. The rectangle represents a "window" into the subspace. Inside of the rectangle, we can display the slice f(i, j, z), z ∈ <z_min, z_max>. From the distribution of rectangles, one can find out the positions of interesting peaks, then focus the view to an appropriate region or to zoom a slice to full screen size, respectively.

Let us proceed to four-dimensional data f(x₁, x₂, x₃, x₄). In place of the functional, we shall use the sums of channels in appropriate two-dimensional regions

f ( 1 ) ( x 1 , x 2 ) = ∑ x 3 = x 3 min ⁡ x 3 max ⁡ ∑ x 4 = x 4 min ⁡ x 4 max ⁡ f ( x 1 , x 2 , x 3 , x 4 ) . MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aaWbaaSqabeaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGGOaakcqWG4baEdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabdIha4naaBaaaleaacqaIYaGmaeqaaOGaeiykaKIaeyypa0ZaaabCaeaadaaeWbqaaiabdAgaMjabcIcaOiabdIha4naaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaemiEaG3aaSbaaSqaaiabikdaYaqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaeG4mamdabeaakiabcYcaSiabdIha4naaBaaaleaacqaI0aanaeqaaOGaeiykaKcaleaacqWG4baEdaWgaaadbaGaeGinaqdabeaaliabg2da9iabdIha4naaBaaameaacqaI0aancyGGTbqBcqGGPbqAcqGGUbGBaeqaaaWcbaGaemiEaG3aaSbaaWqaaiabisda0iGbc2gaTjabcggaHjabcIha4bqabaaaniabggHiLdaaleaacqWG4baEdaWgaaadbaGaeG4mamdabeaaliabg2da9iabdIha4naaBaaameaacqaIZaWmcyGGTbqBcqGGPbqAcqGGUbGBaeqaaaWcbaGaemiEaG3aaSbaaWqaaiabiodaZiGbc2gaTjabcggaHjabcIha4bqabaaaniabggHiLdGccqGGUaGlaaa@7133@

Then inside of each rectangle belonging to the channel i₁, i₂we display two-dimensional slice f(i₁, i₂, x₃, x₄), using any of the two-dimensional above presented graphical models.

In the case of five-dimensional spectrum, we can apply one-, or two-step reduction of dimensionality, i.e., either

f ( 1 ) ( x 1 , x 2 ) = ∑ x 3 = x 3 min ⁡ x 3 max ⁡ ∑ x 4 = x 4 min ⁡ x 4 max ⁡ ∑ x 5 = x 5 min ⁡ x 5 max ⁡ f ( x 1 , x 2 , x 3 , x 4 , x 5 ) , MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8836@

or

f ( 1 ) ( x 1 , x 2 , x 3 , x 4 ) = ∑ x 5 = x 5 min ⁡ x 5 max ⁡ f ( x 1 , x 2 , x 3 , x 4 , x 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6766@

f ( 2 ) ( x 1 , x 2 ) = ∑ x 3 = x 3 min ⁡ x 3 max ⁡ ∑ x 4 = x 4 min ⁡ x 4 max ⁡ f ( 1 ) ( x 1 , x 2 , x 3 , x 4 ) . MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@740E@

In the first case in each rectangle window belonging to the channel i₁, i₂one can display three-dimensional slice f(i₁, i₂, x₃, x₄, x₅) using any of the three-dimensional graphical models. In the second one, in each rectangle belonging to the channel i₁, i₂one can display two-dimensional distribution of f⁽¹⁾(i₁, i₂, x₃, x₄) again in the form of rectangles. Then in each rectangle belonging to the channel i₁, i₂, i₃, i₄, one can display the one-dimensional slice f(i₁, i₂, i₃, i₄, x₅). Employing this algorithm and using successive zooming one can localize the positions of five-dimensional peaks.

Though realizing the technical limitations, apparently the technique of embedded subspaces lends itself to generalization for p – dimensional nuclear spectra employing several level merging and projections. Without loss of generality, we shall assume that p is even. Analogously to the above-given relations one can write

f ( 1 ) ( x 1 , x 2 , ... , x p − 2 ) = ∑ x p − 1 = x ( p − 1 ) min ⁡ x ( p − 1 ) max ⁡ ∑ x p = x p min ⁡ x p max ⁡ f ( x 1 , x 2 , ... , x p ) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@85D7@

f ( 2 ) ( x 1 , x 2 , ... , x p − 4 ) = ∑ x p − 3 = x (