JP2686744B2 - Image reconstruction device - Google Patents

Description

TECHNICAL FIELD The present invention relates to an image reconstruction method using Fourier transform such as an X-ray CT apparatus and an NMR-CT apparatus, and an apparatus thereof. More specifically, the present invention relates to an image reconstruction device that halves the amount of calculation of inverse Fourier transform. (Prior Art) A conventional two-dimensional image reconstruction method will be described using an NMR-CT apparatus. As is well known, the NMR-CT apparatus uses the magnetic resonance (NMR) phenomenon to measure the density distribution and relaxation time of specific atomic nuclei such as protons and obtain a tomographic image of the human body or the like. A method for measuring this NMR phenomenon will be described with reference to FIG. FIG. 6 shows conventional NMR-C.
It is the schematic of the T apparatus. First, a static magnetic field H _O that is uniform in the Z direction
The subject 2 is inserted and installed in the main magnetic field coil 1 which is generating. The gradient magnetic field coil 4 superimposes a linear gradient magnetic field Gz on the static magnetic field H _O and selects a slice plane to be excited in the subject 2. Next, an RF wave rotating in the XY plane is applied from the RF coil 3 to the nuclei of this slice plane, which are in Larmor precession at an angular velocity ωo, to induce an NMR phenomenon. And
The NMR signal from the entire slice plane is read out by the gradient magnetic field G
By x, position information in the X direction is assigned to a frequency (readout time) and collected. This signal is sampled at a predetermined frequency and then detected to form complex number data. Positional information in the Y direction is given a phase shift of spins in the Y direction by a gradient magnetic field (warp gradient) Gy, and while repeatedly changing the magnitude of the warp gradient, the NMR signals are repeatedly collected. It is given as phase information. N times obtained by performing the sampling number N times and collecting NMR signals N times
× N complex number data (two-dimensional complex spectrum)
Is represented on the Fourier plane as shown in FIG. 7 (a). FIG. 7 is a diagram showing a conventional image reconstruction method. In FIG. 7, the horizontal axis (j axis) is the read time axis, and the vertical axis (i axis) is the warp axis (warp gradient). Size), and the shaded area represents the distribution of data. Next, the two-dimensional complex spectrum data on the Fourier plane is subjected to N-dimensional one-dimensional inverse Fourier transform N times in the j-axis direction as shown in FIG. 7B. still,
The arrow in FIG. 7 indicates the direction of the one-dimensional inverse Fourier transform. Thereafter, as shown in FIG. 7 (c), the one-dimensional inverse Fourier transform of N points in the i-axis direction is repeated N times on the created intermediate complex data to obtain XY
Create N × N real number image data on a plane. Such an NMR-CT has a problem that it takes a lot of time to collect the NMR signals. In addition, the two-dimensional inverse Fourier transform for reconstructing the collected two-dimensional complex spectrum into N × N real number image data increases the amount of calculation as the number of data increases, and the fast Fourier transform decision is performed. Even if (FFT) is used, there is a problem that the calculation requires a considerable amount of time. Therefore, there is a method in which data is collected only on the half plane and the conjugate symmetry of the data is used to calculate and derive the data on the remaining half planes. This method utilizes the fact that the complex scutrum data has the origin conjugate symmetry on the Fourier plane because it is a two-dimensional Fourier transform image of a real number image. FIG. 8 is a diagram showing such a conventional image reconstruction method. The symbols in FIG. 8 are used in the same meanings as in FIG. 7, and therefore the description thereof is omitted here. Fig. 8 (a)
Is a measurement of the above-mentioned NMR signal as a free induction decay (FID) signal, and the data on the negative read time axis can be calculated from the data on the positive read time axis by using the above symmetry. It is possible (Japanese Patent Laid-Open No. 61-194339). Similarly, FIG. 8B shows the NMR signal measured as a spin echo, and the data when the negative warp is applied can be generated from the data when the positive warp is applied. (Half encoding method (Japanese Patent Laid-Open No. 62-8747
issue)). (Problems to be Solved by the Invention) However, in the above method, although the data collection time is reduced by half, the calculation amount of the Fourier transform, that is, the calculation time is not shortened. There is also a problem that the amount of memory required to store and store the data being calculated becomes enormous as the number of data increases. Therefore, an object of the present invention is to solve the above problems and provide an image reconstructing apparatus that reduces the amount of calculation of inverse Fourier transform by half. (Means for Solving the Problems) The present invention for achieving the above-mentioned object is performed by performing an inverse Fourier transform on an m-dimensional complex Fourier spectrum obtained by an image capturing apparatus to obtain N ^m ( (N, m is an integer of 2 or more) is an image reconstructing device for creating real image data, wherein (N / 2 + 1) × N ^(m-1) number of the m-dimensional complex Fourier spectra By performing (m-1) -dimensional complex inverse Fourier transform of N points in any of the (m-1) -axis directions in (N / 2 + 1) times, intermediate complex data (F _k, where k = 0 , 1,…, N / 2
-1) for generating an intermediate complex data and a data converting means for generating a complex sequence (G _k ) for the intermediate complex data (F _k ) by an operation for data conversion based on the following equation. _{Te, Re (G k) = 1} /2 · {Re (F k) + Re (F N / 2-k)} Im (G k) = 1/2 · {Im (F k) -Im (F N / _2−k )} where F _k = Re (F _k ) + jIm (F _k ), j = (− 1) ^1/2 , where Re is a real part and Im is an imaginary part. Complex number data (g _k) is obtained by performing N / 2-point one-dimensional complex inverse Fourier transform on the complex number sequence (G _k ) in the direction of the axis other than the axis on which the m-dimensional complex inverse Fourier transform is performed. ), And the real number part of the complex number data (g _k ) is an even number in order to generate N real number image data (f _i, where i = 0, 1, ..., N−1). Th real image data allocates the f _2k), a data assignment means for assigning the imaginary part of the complex data (g _k) to an odd-numbered real image data (f _{2k + 1),} conversion of the data conversion unit, 1 of the complex number data creating means And a control means for controlling the dimensional complex inverse Fourier transform and the allocation of the data allocation means to be repeated N ^(m-1) times. (Operation) In order to create N ^m pieces of real image data, almost half [(N / 2 + 1) × N ^(m-1) pieces of collected data are used as they are for image reconstruction. ) The number of inverse Fourier transforms is halved from N to (N / 2 + 1), and the number of subsequent one-dimensional inverse Fourier transforms is halved from N points to N / 2 points, halving the overall calculation amount. It In addition, the amount of memory required to store the collected data and the intermediate complex data can be reduced by half accordingly. (Embodiment) First, the principle of the image reconstructing method of the present invention will be described. Conventionally, there is a method of halving the amount of calculation when performing real-valued one-dimensional fast Fourier transform ("The F
ast Fourier Transform "P167−169 Prentice−Hall EO
aran Brigham). In this method, the real sequence f _i ; (i = 0,1,
, N-1) is subjected to fast Fourier transform (FFT), first, f _i is set as g _i = f _2i + jf _{2i + 1} ; i = 0,1, ... N / 2-1 (1) , the real part of even-numbered f _i, making complex sequence g _i half the number of the f _i assigned the odd f _i the imaginary unit. This g _i
Fast Fourier transform of the result G _k ; (K = 0,1, ... N
/ 2-1) is Becomes Where j = (− 1) _1/2 , Represents every other sum from m = 0 to m = 2, m = 4, ..., M = N−2. On the other hand, when f _i is fast Fourier transformed, the result F _k is Therefore, from (2) and (3), Becomes _{However, F k = Re (F k} ) + jIm (F k); (K = 0,1,
, N / 2−1), and Re (F _k ) is the real part of F _k , Im (F _k ).
Is the imaginary part. Here, from (4) and (5), F _k is K =
0,1, ..., N / 2-1 is obtained, but (2) is K = N /
Even if it is 2 or more, it is formally established, so that F _k is K = 1,2 ..., N-1
Asked about. In this way, N real values f _i
After replacing with two complex values g _i and Fourier transform,
The calculation amount can be cut in half. Next, the principle of the image construction method of the present invention in which this method is applied to one-dimensional inverse Fourier transform from intermediate complex data to real image data will be described. First, equation (4), (5) the solving Conversely, represent G _k at F _k, the real part of G _k Re (G _k)
And the imaginary part Im (G _k ) becomes, respectively. Re (G _k ) = 1/2 · [Re (F _k ) + Re (F _{N / 2−k} )] − 1/2 · [Re (F _k ) −Re (F _{N / 2−k} )] × sin 2πk / N −1/2 ・ [Im (F _k ) + Im (F _{N / 2−k} )] × cos2πk / N (6) Im (G _k ) = 1/2 ・ [Im (F _k ) −Im (F _{N / 2−k} )] −1 / 2 · [Im (F _k ) + Im (F _{N / 2−k} )] × sin 2πk / N −1 / 2 · [Re (F _k ) −Re (F _{N / 2-−k} )] × cos2πk / N (7) Therefore, N / 2 + 1 intermediate complex data F _k ; (K = 0,1, ..., N
/ 2) is converted by equations (6) and (7) to obtain N / 2
Complex number sequence G _k = Re (G _k ) + jIm (G _k ); (K = 0,1, ...,
N / 2-1) make, inverse fast Fourier transform G _k (IFFT). Since the result of Fourier transform of g _i is G _k as described above, the result of fast inverse Fourier transform of G _k is g _i = f _2i + jf
_{2i + 1} ; (i = 0, 1, ... N / 2-1). Here, if the real part and the imaginary part of g _i are separated and rearranged, the N real-valued sequences f _i (i = 0, 1, ... N-1) to be obtained can be derived with a half amount of calculation. Next, a method and apparatus for actually reconstructing an image using the principle described above will be described. FIG. 1 is a diagram showing a two-dimensional image reconstruction method of NMR-CT according to an embodiment of the present invention. Since the symbols in FIG. 1 have the same meanings as in FIGS. 7 and 8, the description thereof is omitted here. In FIG. 1, (a) shows N × N (N is an integer of 2 or more) pieces of two-dimensional complex Fourier spectrum data (hereinafter referred to as complex spectrum) having Fourier distribution on the Fourier plane. It is a thing. The complex spectrum like this is
Since it is a three-dimensional Fourier transform image, it has origin conjugate symmetry on the Fourier plane. Therefore, only the data for the half plane has the information of the data for all the planes. And, the same thing as this holds even after the one-dimensional inverse Fourier transform is performed on the data in one direction. That is, if F _ij = F _-ij ^* (8), then F _j ^-1 [F _{-i j} ] = F _j ^-1 [F _ij ^* ] = {F _j [F _ij ]} ^* = {F _j ^-1 [F _ij ]} ^* (9). However, F _ij is the complex spectrum of the real image,
i is a read axis, j is a warp axis, F _j ^-1 is a one-dimensional inverse Fourier transform in the j axis direction, and ^* represents a complex conjugate operation. Therefore, the result of the one-dimensional inverse Fourier transform of the origin conjugate symmetric data with respect to one axis direction, that is, the intermediate complex data has a complex conjugate symmetry relationship with respect to the axis, and only the intermediate complex data for half planes or all planes. Have data information. Therefore, as shown in FIG. 1B, first, on the Fourier plane, the N-dimensional one-dimensional inverse Fourier transform of N points in the readout axis direction is performed only for the complex spectrum in the upper half plane region by N / 2.
Do +1 times. Therefore, the intermediate complex data of the half plane area created by this calculation has information of the data of all planes. Since the one-dimensional inverse Fourier transform is performed only on the complex spectrum of the upper half plane, the amount of calculation in this transform can be halved. Next, one-dimensional inverse Fourier transform is performed in the i-axis direction on the intermediate complex data on the upper half plane. At this time, N / 2 of the k-th column (the reading axis direction is the row and the warp axis direction is the column)
Assuming that +1 pieces of intermediate complex data are F _k ; (K = 0, 1, ..., N / 2), the conversion of equations (6) and (7) is performed on F _k . Next, the converted N / 2 complex number sequence G _k ; (K = 0,1, ..., N / 2−
In contrast to 1), as shown in Fig. 1 (c), N / 2 in the i-axis direction
Perform a one-dimensional inverse Fourier transform of the points. From this result, as described above, the result of the one-dimensional inverse Fourier transform of F _k , that is,
N real values f _i are determined. By repeating this processing N times, the target N × N real number image data can be created as shown in FIG. According to this method, since the intermediate complex data F _k on the upper half plane is converted into N / 2 G _k and the one-dimensional inverse Fourier transform is performed, the calculation amount in this conversion can be reduced by half. Next, an apparatus for actually reconstructing an image will be described according to the method described above. FIG. 2 is a block diagram of an NMR-CT two-dimensional image reconstructing apparatus according to an embodiment of the present invention. In FIG. 2, 5a is a memory for storing the detected complex spectrum, 6a is a one-dimensional complex inverse Fourier transformer for converting the complex spectrum into intermediate complex data, and 5b is one-dimensional in one axial direction on the Fourier plane. Memory for storing intermediate complex data that has been subjected to inverse Fourier transform, 6b
Is a one-dimensional complex inverse Fourier transformer for converting the intermediate complex data to real image data, 5c is a memory for storing the real image data, and 7 is a controller for controlling the one-dimensional complex inverse Fourier transformer 6a and the Fourier transformer 6b. is there. In FIG. 2, thin lines represent control flow and double lines represent data flow. The operation of the image reconstructing apparatus having the above configuration will be described with reference to FIGS. 2 and 3. FIG. 3 is a flowchart showing an image reconstruction method according to an embodiment of the present invention. The left side of the flowchart in FIG.
The operation is performed by the Fourier transformer 6a, and the operation on the right side is performed by the Fourier transformer 6b. First, the controller 7 controls the Fourier transformer 6a to read the two-dimensional complex Fourier spectrum of the i-th row stored in the memory 5a,
Perform N-dimensional one-dimensional complex inverse Fourier transform in the j-axis direction,
The result is written in the memory 5b as the intermediate complex data of the i-th row. This is repeated with respect to i from 0 to N / 2 N / 2 + 1 times to create intermediate complex data in the area of the half plane. Next, the controller 7 controls the primary element Fourier transformer 6b to read the intermediate complex data F _j and F _{N / 2-j} of the jth column and the N / 2-jth column stored in the memory 5b. , After the transformation of equations (6) and (7), i for the new complex sequence G _j
Perform N / 2-point one-dimensional complex inverse Fourier transform in the axial direction,
The created N / 2 complex number data gi are rearranged as N real number image data f _i , and are written in the memory 5c as the j-th column data. This is for j, -N / 2 + 1 to N / 2
Up to N times are repeated to create N × N real number image data, and the process is terminated. In this case, the amount of calculation is halved as compared with the case of performing a two-dimensional inverse Fourier transform of the complex spectrum (N rows × N columns) of all planes on the Fourier plane, and at the same time, the storage of the memory 5a and the memory 5b that is required accordingly. Since the capacity of half plane is enough, it is halved. In the above embodiment, the method and apparatus for reconstructing an image using the data of the upper half plane on the Fourier plane have been shown. Similarly, as shown in FIG. FID signal when the gradient is positive, or data on the left half plane as shown in Fig. 4 (b) (when the readout gradient is negative)
An image can be reconstructed from the FID signal) and the data of the lower half plane (echo signal when the warp gradient is negative) as shown in FIG. However, when the image is reconstructed from the data of the left half plane and the lower half plane, the one-dimensional complex spectrum at the Nyquist frequency is periodically shifted as shown in FIGS. 4 (b) and 4 (c). And then need to connect with other data. FIG. 5 is a diagram showing an image reconstruction method according to another embodiment of the present invention. The symbols in FIG. 5 are used in the same meanings as in FIG. 1, and the description thereof is omitted here. Fifth
Figure (a) is a half-dimensional three-dimensional complex spectrum,
An example of reconstructing a three-dimensional real number image from this data will be described. At this time, as shown in FIG. 5B, a normal N × N point two-dimensional inverse Fourier transform is performed on the k-i plane, and this is repeated N / 2 + 1 times from j = 0 to N / 2. Next, after converting the created semi-solid intermediate complex data in the j direction in the same manner as described above using the equations (6) and (7), as shown in FIG. N / 2-point fast inverse Fourier transform is repeated N × N times, and the result is rearranged into real number data in the same manner as described above to obtain N × N × N real number image data as shown in FIG. 5D. To get Thus, any semi-stereo data (N × N ×
From (N / 2 + 1) pieces, N × N × N pieces of real number image data can be reconstructed. Therefore, even if the dimension is increased, the amount of calculation and the amount of memory required for it can be halved by expanding and using the same method and apparatus as those in the above-mentioned embodiment. That is, when converting an m-dimensional complex spectrum into N ^m real number image data, (m-1) -dimensional inverse Fourier transform of N ^(m-1) points is repeated N / 2 + 1 times for the complex spectrum. After that, for the created intermediate complex data, after conversion of equations (6) and (7),
The N / 2-point one-dimensional inverse Fourier transform may be repeated N ^(m-1) times, and the result may be rearranged as N ^m pieces of real image data. The present invention is not limited to the above embodiment. Various modifications are possible within the scope of the claims. For example, the remaining half-plane or half-solid collection data (complex spectrum) that was not used for image reconstruction in the above-described embodiment is folded back to the origin symmetry after the complex conjugate calculation, and the arithmetic mean with the data at the symmetry point is added. After taking, the image reconstruction calculation by the method of the present invention may be performed to improve the S / N ratio of the image.
Further, in the conversion of the equations (6) and (7), the values of sin and cos may be tabulated in advance, and this processing may be sped up. Further, the Fourier transformer may be configured by hardware dedicated to Fourier transform or software by a computer. Further, it may be used for image reconstruction such as X-ray CT which becomes real number image data after Fourier transform. (Effects of the Invention) As described above, according to the image reconstruction device of the present invention, the amount of calculation of the inverse Fourier transform can be reduced by half, and as a result, the calculation time of the inverse Fourier transform can be shortened. ..

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a diagram showing an image reconstruction method of NMR-CT according to an embodiment of the present invention, and FIG. 2 is a block showing an image reconstruction device of an embodiment of the present invention. FIG. 3 is a flowchart showing an image reconstruction method according to an embodiment of the present invention,
FIG. 4 is a diagram showing an image reconstruction method of an embodiment of the present invention, FIG. 5 is a diagram showing an image reconstruction method of another embodiment of the present invention, and FIG. 6 is a conventional NMR- FIG. 7 is a schematic view showing a CT apparatus, FIG. 7 is a view showing a conventional image reconstruction method,
FIG. 8 is a diagram showing a conventional image reconstruction method. 1 ... Static magnetic field magnet, 2 ... Gradient magnetic field coil 3 ... RF coil, 4 ... Subject, 5a, 5b, 5c ... Memory, 6a, 6b ... One-dimensional complex inverse Fourier transformer, 7 ... controller.

Claims (1)

(57) [Claims] It is an image reconstruction device that creates N ^m (N, m is an integer of 2 or more) real number image data by performing an inverse Fourier transform on the m-dimensional complex Fourier spectrum obtained by the image capturing device. Then, for any of the (N / 2 + 1) × N ^{(m-1) m-} dimensional complex Fourier spectra, any (m
-1) Repeating the (m-1) -dimensional complex inverse Fourier transform of N points in the direction of the (-1) axis (N / 2 + 1) times to obtain intermediate complex data (F _k, where k = 0, 1, ..., N / 2-1) for creating intermediate complex data, and data converting means for creating a complex sequence (G _k ) on the intermediate complex data (F _k ) by an operation for data conversion based on the following equation. a _{is, Re (G k) = 1} /2 · {Re (F k) + Re (F N / 2-k)} Im (G k) = 1/2 · {Im (F k) -Im (F _{N / 2−k} )} where F _k = Re (F _k ) + jIm (F _k ), j = (− 1) ^1/2 , where Re is a real part and Im is an imaginary part. And the complex number data (G _k ) by performing one-dimensional complex inverse Fourier transform of N / 2 points in the direction of the axis other than the axis on which the m-dimensional complex inverse Fourier transform is performed. complex data to create a g _k) And creating means, N real numbers image data (f _i where, i = 0,1, ..., N -1) to create the even-numbered real image data to the real part of the complex number data (g _k) ( f _2k ), and
Data allocating means for allocating the imaginary part of the complex number data (g _k ) to the odd-numbered real number image data (f _{2k + 1} ); the transformation of the data transforming means; the one-dimensional complex inverse Fourier transform of the complex number data creating means; An image reconstructing apparatus comprising: a control unit that performs control for repeating the allocation of the data allocation unit N ^(m-1) times.