JPH06110917A - Vector calculator - Google Patents

Abstract

PURPOSE:To prolong a vector length, and to attain a Fourier transform by one-dimensionally rearranging input data with a data arraying means, and rearranging data after a butterfly arithmetic operation with a rearranging means. CONSTITUTION:This calculator is equipped with a data arraying means 1 which arrays the input data in a prescribed sequence, butterfly arithmetic means 2 which operates a butterfly arithmetic operation, and rearranging means 3 which rearranges the data after the butterfly arithmetic operation Then, when the multi-dimensional data are inputted, the multi-dimensional data are one- dimensionally arranged by the data arraying means 1. Then, a processing equivalent to a one-dimensional fast Fourier transform(FFT) is operated until the Pn-th stage, and the transformed data are rearranged. This processing is operated (n) times, and finally the processing equivalent to the one-dimensional FFT is repeated until the P1st stage. That is, the vector length can be prolonged by lengthening a loop length by one-dimensionally arranging the multi- dimensional data, and the arithmetic speed can be fastened by calculating the multi-dimensional FFT by operating one time of the processing equivalent to the one-dimensional FFT.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a vector computer for performing a Fourier transform, and more particularly to a technique for high speed operation.

[0002]

2. Description of the Related Art Conventionally, Fourier transform has been widely used in the field of image processing and physical property calculation, and the advent of a vector computer capable of performing fast Fourier transform has been desired.

Generally, the multidimensional discrete Fourier transform is defined by the following equation.

[0004]

[Equation 1] Here, n is the number of dimensions, and N _i (i = 1, ..., N) is the number of data for each dimension. Expression (1) is modified as follows.

[0005]

[Equation 2] From the equation (2), the multidimensional discrete Fourier transform is j ₁ , ...,
performs Fourier transform on fixing the j _n-1 j n, then j _1, ..., to secure the j _n-2, j n converts respect j _n-1, the one-dimensional discrete Fourier transform as that dimension It turns out that it only has to be repeated several times. The same applies to the case of FFT (Fast Fourier Transform), and the multidimensional FFT can be calculated by repeating the one-dimensional FFT several times.

The number of repetitions N _i (i = 1, ..., N)
There is also a method of exchanging the outer loop of FFT with the loop in the FFT algorithm. Here, in order to simplify the mechanism of the one-dimensional FFT, the case of N = 8 will be described. The one-dimensional discrete Fourier transform is defined by the following equation (3).

[0007]

[Equation 3] If the calculation is performed according to the definition of the equation (3), it is necessary to perform 8 × 8 = 64 operations to convert eight pieces of data. However, in FFT, 8 × 3 = 24 calculations are required.
In equation (3)

[Equation 4] Then, the state of the conversion is shown in FIG. 8 data f
(J) {j = 0, ..., 7} is converted in three steps as shown in FIG. As can be seen from FIG. 18, f (0) and f (0)
G (0) and g (4) are generated from (4). f (0)
Since f (4) and f (4) are not necessary to obtain values other than g (0) and g (4), g (0) and g (4) are f (0) and f (4).
Can be overwritten on. This operation is called a butterfly operation because an intersecting form can be obtained by representing this data flow in a diagram. Also, from g (0) and g (2) to h
(0) and h (4) are generated, but g (0) and g (2)
Can be overwritten in the same way, since is not necessary for calculations other than h (0) and h (4).

FIG. 19 shows how the data is converted by this overwriting. This method has the advantage of not taking up extra storage capacity, but the last available c (k) is not in the correct order. Therefore, usually, the data is exchanged at the end.

Looking at the state of conversion in FIG. 18, c (k) are arranged in the correct order. Therefore, a Stockham algorithm for supercomputers that uses a working array without overwriting and does not require the final rearrangement will be described.

N (= 2) in the 1st stage of the intermediate stage of FFT
^{The p} ) pieces of data are defined by the following equation (5).

[0011]

[Equation 5] And this is the initial data. When l = p, j = 0

[Equation 6] And this is the desired Fourier transform itself. Furthermore, the following recurrence formula holds between X _l and X _{l + 1} .

[0012]

[Equation 7] This conversion is illustrated in FIG. If this operation is started from the vertical vector of one column and repeated until the horizontal vector of one row is obtained, it means that the Fourier transform can be performed. FIG. 21 shows the flow of FFT when N = 8. From FIG. 21, the initial data are f (0), f (1), ..., F (N-1) in this order.
Since the results are stored in the order of c (0), c (1), ..., C (N-1), it can be seen that the last data exchange is not necessary.

This method can be vectorized, and the vector length increases as the number N of data increases, so that it is effective for execution in a supercomputer. As described above, the multidimensional FFT can be calculated by repeating the one-dimensional FFT several times. Therefore, in the case of a multidimensional FFT, increasing the number of data N _i in each dimension is effective because the vector length increases.

[0014]

As described above, the conventional vector computer has a problem that it takes a long time to perform the calculation because the vector length when performing the Fourier transform is short.

The present invention has been made to solve such conventional problems, and an object of the present invention is to provide a vector computer capable of performing Fourier transform by increasing the vector length. .

[0016]

In order to achieve the above object, the present invention provides, in a Becult computer for calculating a multidimensional Fourier transform, an array means for arranging multidimensional data one-dimensionally and a butterfly operation in the array. It is characterized in that it has a butterfly operation means for performing the operation and a rearrangement means for rearranging the data after the butterfly operation is completed, and performs the Fourier transform by repeating the butterfly operation and the data rearrangement one or more times.

[0017]

In the present invention configured as described above, the input data is rearranged one-dimensionally by the data arrangement means, and the data after the butterfly operation is rearranged by the rearrangement means. Therefore, it becomes possible to increase the vector length, which allows the Fourier transform to be performed at high speed.

[0018]

Embodiments of the present invention will be described below with reference to the drawings. FIG. 1 is a schematic block diagram showing a configuration of a portion related to Fourier transform calculation in a vector computer to which the present invention is applied. As shown in the figure, this vector computer has a data arranging means 1 for arranging input data in an order as will be described later, a butterfly calculating means 2 for performing a butterfly operation, and a rearranging means 3 for rearranging the data after the butterfly operation. is doing.

Next, the procedure for performing the Fourier transform using the vector computer thus constructed will be described with reference to the flow chart shown in FIG. When the multidimensional data is input, the data array means 1 arranges the multidimensional data one-dimensionally (step ST1). For example,
Input data is the following 3D data

F (j ₁ , j ₂ , j ₃ ,) {j ₁ = 0, ..., N ₁ -1, j ₂ = 0, ..., N ₂ -1, j ₃ = 0, ..., N ₃ −1} (10),

F (0,0,0), f (1,0,0), ... f (N ₁ -1,0,0), f (0,1,0), f (1,1,0) 0), ..., f (N 1 -1,1,0), ..., f (N 1 -1, N 2 -1, and stores the N ₃ -1) ... 1-dimensional array in the order of (11). However, N ₁ = 2 ^p1 , N ₂
= 2 ^p2 and N ₃ = 2 ^p3 . And the state of storage is shown in FIG.
As shown in.

Next, the same processing as the one-dimensional FFT is performed in the Pth
Perform up to _n steps (step ST2). The state of the data at the end of this process is as shown in FIG. Then, the converted data is rearranged (step ST
3). In the processing here, as shown in FIG. 5, the data arranged in the column order is arranged in the row order. Thereafter, the same processing as the one-dimensional FFT is performed up to P _n-1 steps (step ST
4). The state of the data at this time is as shown in FIG. The above processing is performed n times, and finally the same processing as the one-dimensional FFT is repeated up to the P ₁ stage.

In the case of the three-dimensional data expressed by the equation (10), the butterfly operation similar to the one-dimensional FFT is performed up to the P ₃ -th step on the data of length N ₁ × N ₂ × N _3. . Then, let the result be f '. N ₁ = N ₂ = N ₃
FIG. 8 shows the state of conversion in the case of = 4. In FIG. 8, let us look at the state of conversion in the first stage and the second stage. The data after the second stage conversion, X ₂ (0,0), X ₂ (0,
Let's look at the flow of how 1), X ₂ (0, 2), and X ₂ (0, 3) are generated. X ₂ (0,0) and X ₂
(0,2) is calculated from X ₁ (0,0) and X ₁ (16,0),
X ₂ (0,1) and X ₂ (0,3) are X ₁ (0,1) and X
It is generated from ₁ (16, ₁ ). And X ₁ (0,
0) and X ₁ (0,1) are X ₀ (0,0) and X ₀ (32,
0), X ₁ (16,0) and X ₁ (16,1) are X ₀
They are generated from (16,0) and X ₀ (48,0), respectively. Where f (0,0,0) = X ₀ (0,0) f (0,0,1) = X ₀ (16,0) f (0,0,2) = X ₀ (32,0) f (0,0,3) = X ₀ (48,0), and if we look only at this part of the transformation up to the second stage, we see j
_This is the same as performing the Fourier transform in the j ₃ direction when ₁ = j ₂ = 0. Similarly, X ₂ (1,0),
X ₂ (1,1), X ₂ (1,2), X ₂ (1,3) are initial data f (1,0,0), f (1,0,1), f
(1,0,2) and f (1,0,3) are j ₁ = 1 and j ₂ =
In the case of 0, the Fourier transform in the j ₃ direction is performed. Therefore, it is understood that the transformation up to the second stage is the same as the Fourier transformation in the j ₃ direction for all j ₁ and j ₂ . The state of the data at this point is shown in FIG.

Since the data after the second stage conversion are stored in the array in the order of the arrows in FIG. 10, they are next rearranged in the order of the arrows in FIG. Then this time the data arrangement

F ′ (0,0,0), f ′ (0,0,1), ..., f ′ (0,0, N ₃ −1), f ′ (1,0,0), f '(1,0,1), ..., f ' (1,0, N 3 -1), ..., f '(N 1 -1, N 2 -1, N 3 -1) ... in the order of (12) Become. And as before, one-dimensional F up to the second stage
If FT is performed (result is f ″), then all j ₁
And j ₃ are the same as the Fourier transform in the j ₂ direction.

Finally, the rearrangement is performed again, and this time,

F ″ (0,0,0), f ″ (0,1,0), ..., f ″ (0, N ₂ −1,0,) f ″ (0,0,1), f "(0,1,1), ..., f " (0, N 2 -1,1), ..., f "(N 1 -1, N 2 -1, N 3 -1) ... in the order of (13) After storing it in an array and performing the same process, the Fourier transform in the j ₁ direction is performed.
T ends.

It should be noted that FIG. 7 shows the procedure of the conventional multidimensional FFT, and it can be understood that the loop length is increased and the operation speed is increased according to this embodiment.

In general, the n-dimensional FFT is similar to the three-dimensional case, and the data is arranged one-dimensionally and the butterfly operation similar to the one-dimensional FFT is performed halfway. Then, if the rearrangement of the data in the middle of the FFT is performed n times, the n-dimensional FFT is completed. At this time, since data is accessed discontinuously, there is a possibility that bank conflicts will occur, but here, a data access method similar to the hyperplane method is used to prevent the occurrence of bank conflicts.

A three-dimensional FFT routine was created according to the algorithm described above, and the execution times of some test data were compared with the conventional method. It is shown in Table 1.

[0027]

[Table 1] Here, the time when processed by the algorithm examined this time is shown as 1. When the number of data increases, the difference between the two decreases, but the speedup is realized. Further, FIG. 12 shows a characteristic curve of the number of input data and the processing time ratio of the conventional method and the present invention, and it can be seen that the Fourier transform can be performed in the processing time of 30% or less of the conventional processing time.

Next, an example of accelerating the reconstruction operation of the X-ray CT scanner using the vector computer according to the present invention will be described. FIG. 13 is a block diagram showing a schematic configuration of an X-ray CT scanner. An X-ray generator 11 that irradiates an object with X-rays, and a high-voltage generator 12 that supplies a high voltage to an X-ray tube.
And a bed 13 on which the subject is placed and enters the gantry
A scanning system 14 for scanning X-rays, a detector 15 for detecting X-rays transmitted through the subject, and the high voltage generator 1
2, a computer system 16 for overall control of the bed 13, the scanning system 14 and for reconstructing the signals collected by the detector to display a CT image on a monitor 17,
It is composed of a T18 and a disk 19.

The procedure for reconstructing an image in such a configuration will be described with reference to the flowchart shown in FIG. First, as shown in FIG.
When irradiating toward 2, the detector 21 displays, for example, the curve S1
X-ray detection data as shown in FIG. Then, by repeating this operation while rotating as shown in FIG. 15, X-ray data from various angles are obtained (FIG. 16, step ST11).

Thereafter, this data is digitized (step ST12) and temporarily stored in the memory (step S).
T13). Then, the fast Fourier transform is performed using the method of the present invention (step ST14), and after the interpolation processing (step ST15), the inverse Fourier transform is performed (step ST1).
6). After that, a D / A converter is used for analogization to obtain a CT image (step ST17). Then, this is displayed on the monitor (step ST18).

Thus, the CT image can be reconstructed at high speed.

FIG. 17 is a flow chart for solving a partial differential equation using the vector computer of the present invention.
As the processing procedure, first, the boundary condition is made into a function (step ST21), and fast Fourier transform is performed to obtain the boundary condition in the wave space (steps ST22 and ST23).

Further, the partial differential equation is converted into an algebraic equation in the wave space (step ST24), and the algebraic equation is solved under the boundary condition obtained in step 23 (step ST).
25). After that, this result is subjected to inverse FFT (step S
T26), obtain the solution of the partial differential equation.

Thus, the partial differential equation can be solved at high speed.

Also, DSA (Dyanamic Simulated Ann) recently performed by R. Car, M. Parrinello and others.
The present invention can also be applied to physical property calculations such as the ealing) method (Phys. Rev. Lett. 55 (1985) p2471). For example DS
In the method A, the electronic state in the substance is calculated in the k-space (reciprocal lattice vector space, wave vector space) by utilizing the feature that the crystal lattice of the substance is regularly arranged. Therefore, when the force applied to the ions is calculated from the electronic state, it is necessary to perform a three-dimensional inverse Fourier transform on the function expressing the electronic state developed in the k-space to convert it into a function in the real space. In this way, in the physical property calculation such as DSA, the function in k-space is converted into the function in real space,
Since many multi-dimensional FFT processings for correcting a function in the real space into a function in the k-space are used, the physical properties can be calculated at high speed by applying the present invention.

In this way, by using the vector computer of the present invention, FF such as image reconstruction and partial differential equation solving method can be achieved.
The calculation speed can be increased in various fields using T.

[0037]

As described above, in the vector computer of the present invention, in order to increase the vector length, multidimensional data are arranged one-dimensionally to increase the loop length, and the one-dimensional FFT is performed.
Since the multidimensional FFT is calculated by performing the same processing once as in (1), the operation speed can be significantly increased. This is extremely useful because the operation speed can be increased in each field using the Fourier transform.

[Brief description of drawings]

FIG. 1 is a block diagram showing a schematic configuration according to an embodiment of a vector computer to which the present invention is applied.

FIG. 2 is a flowchart showing the operation of FFT processing using the vector computer of this embodiment.

FIG. 3 is an explanatory diagram showing an array of multidimensional data.

FIG. 4 is an explanatory diagram showing a state at the end of the P _n-th stage of butterfly calculation.

FIG. 5 is an explanatory diagram showing rearrangement of data.

FIG. 6 is an explanatory diagram showing a state at the end of the P _n−1 stage of butterfly calculation.

FIG. 7 is a flowchart showing a procedure of a conventional multidimensional FFT.

FIG. 8 is an explanatory diagram showing a flow of butterfly calculation.

FIG. 9 is an explanatory diagram showing a state of data at the end of the second stage.

FIG. 10 is an explanatory diagram showing the order in which the data after the second stage conversion is stored in the array.

FIG. 11 is an explanatory diagram showing the order of storing the data after the second stage conversion in a one-dimensional array.

FIG. 12 is a characteristic diagram showing the relationship between the number of input data and the processing time ratios of the related art and the present invention.

FIG. 13 is a block diagram showing a schematic configuration of an X-ray CT scanner.

FIG. 14 is an explanatory diagram showing an example of collecting detection data by irradiating X-rays.

FIG. 15 is an explanatory diagram showing an example in which a collection angle of X-ray data is changed.

FIG. 16 is a flowchart showing a procedure for reconstructing an X-ray CT image.

FIG. 17 is a flowchart showing a calculation procedure of a partial differential equation.

FIG. 18 is an explanatory diagram showing how one-dimensional FFT is converted when N = 8.

FIG. 19 is a one-dimensional FFT by overwriting when N = 8.
2 is a flowchart of.

FIG. 20 is a conversion diagram of an FFT for vector computers.

FIG. 21 is a one-dimensional F for vector computers when N = 8.
It is a flow chart of FT.

[Explanation of symbols]

1 data arranging means 2 butterfly computing means 3
Rearrangement means 11 X-ray generator 15 Detector 16 Computer system

Claims (1)

[Claims] 1. A Bekult computer for calculating a multidimensional Fourier transform, array means for arranging multidimensional data in a one-dimensional manner, and butterfly operation means for performing a butterfly operation in the array.
And a rearrangement means for rearranging the data after the butterfly calculation, and performing the Fourier transform by repeating the butterfly calculation and the rearrangement of the data one or more times.