CN109887051B - Linear interpolation method in CT image reconstruction back projection process - Google Patents
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Abstract
The invention relates to a linear interpolation method in the CT image reconstruction back projection process, which is characterized in that a circular disc with a changeable radius and taking the pixel point P as a circle center is defined for the pixel point P, a radius coefficient is selected according to the image field and the number of the pixel points, so that the circular disc meets the resolution requirement and the data processing capability, the channel boundary covered by the circular disc after being projected to a detector is calculated and obtained, the weighted average of the image values of all channels covered by the circular disc after being projected to the detector is carried out as the value of the pixel point, and the weighted coefficient is according to the covered proportion of the corresponding channel. The invention has good signal-to-noise ratio and resolution ratio, has good elasticity and adaptability to different situations, and when the pixel size is reduced, the interpolation method is degraded to be similar to a linear interpolation method, so that the invention has better signal-to-noise ratio when a large matrix is used for reconstructing CT images, and can be better suitable for the development of CT image reconstruction technology.
Description
Technical Field
The invention relates to a linear interpolation method in a CT image reconstruction back projection process.
Background
The back projection (back projection) is one of the main processes in the CT image reconstruction algorithm, and the existing back projection method is generally divided into two types, namely a ray-based back projection method (ray-driven back projection) and a pixel-based back projection method (pixel-driven back projection), but in practice, the pixel-based back projection method is mostly used in terms of simplicity and convenience in implementation.
The conventional pixel-based back projection method is to assume that a given pixel point has no size and is simply a point, and consider the connection of the x-ray source point and the given pixel point P to the detectionThe intersection point of the detector may typically be between two detector units, and in order to obtain data at the pixel point corresponding to the detector, interpolation methods are typically required, wherein the two most common interpolation methods are nearest-neighbor (nearest neighbour) interpolation and linear (linear) interpolation, and wherein the nearest-neighbor interpolation is performed with the detector unit γ nearest to the intersection point i The value g (gamma) i ) As the value of the pixel point (the calculation principle can be seen in fig. 3), a linear interpolation algorithm is applied to two detector units gamma adjacent to the intersection point i And gamma i+1 The value g (gamma) i ) And g (gamma) i+1 ) As the value g of the pixel point P p . The most important characteristic of the two interpolation methods is that the size of the pixel point is not considered, and as a result, the connection line between the light source and the pixel point can only fall between two detector units at most, and the noise characteristic cannot meet the increasingly improved requirements of people.
To improve the back projection technique, us patent 7,227,982 proposes a new distance-based back projection technique. In contrast to past pixel-based backprojection techniques, this technique no longer considers the pixel to be of no size, but instead assumes that the pixel is a square with a side delta. Assuming that the field of view of the CT image to be reconstructed is FOV (mm), and N pixels are respectively arranged in the horizontal (x-direction) and vertical (y-direction) directions in the field of view, the square side length delta is calculated by the following method:
equation (1) shows that the length of a square is actually the side length of a grid in image space. The technology projects the pixel boundary and the detector unit onto a common axis, and the overlapping part of the two projections is used as the weight of the back projection, so that the CT image obtained by the back projection method has higher signal-to-noise ratio (signal-to-noise ratio), namely the noise of the image is lower, but the method is complex to realize, has higher difficulty in implementation and affects popularization and use in practice.
Another approach is also proposed in us patent 8,116,426 in order to improve the back projection technique. Similar to us patent 7,227,982, this approach also considers the size of the pixels, and for an image having n×n pixels with FOV (mm) in view, each pixel is considered as a disk with radius Δs, where Δs is calculated by the following equation:
since this method is closely related to the present invention, this method is described in detail herein (its computational principle can be seen in fig. 2).
In fig. 2, a given pixel point P has coordinates (x, y) and is considered as a disk with a radius Δs, and the point P is located at the center of the disk. The disc radius deltas is half the pixel size, calculated by equation (2). During the back projection, the method takes into account the number of channels that this disc with radius Δs occupies for projection onto the detector.
First, the channel number gamma corresponding to the P point on the detector needs to be calculated by using the following formula:
c＝int(γ) (4)
in the formula (3), R is the distance from the position S of the light source to the rotation center O, that is, r= |so|, and β is the angle formed by the line segment SO and the y-axis. Equation (4) is to take the floating point channel value γ obtained in equation (3) as an integer. Note that equation (3) is numbered with the detector channel number corresponding to SO being 0 (the number at the intersection of the line segment SO extension and the detector arc in fig. 2 being 0), SO the channel number has a positive number and a negative number. If other numbering methods are adopted to correspondingly adjust the formula (3), the formula (3) should be adjusted according to the corresponding relationship between the numbering rules and the numbering rules represented by the formula (3), for example, a start number gamma can be added on the right side of the formula (3) 0 ，γ 0 For the detector channel number corresponding to SO, or without the addition of the start numberIn the form, gamma is considered as the relative channel number relative to the number at the intersection of the line segment SO extension and the detector arc, i.e., the actual channel number minus gamma 0 The relative channel number of (2) is used as the channel number of equation (3).
Secondly consider the number of channels occupied by a disk of radius Δs projected onto the detector.
Using gamma i And gamma j To represent the boundary channel value of the disk projected onto the detector, at gamma i And gamma j The channels in between all belong to the projection area of the disk. Gamma is calculated using the following formula i And gamma j
L in formulas (5) and (6) represents the distance from the light source S to the P point, i.e., l= |sp|, and its calculation formula is
Definition of the definition
ΔC＝γ j -γ i (8)
ΔC represents the number of channels occupied by a disk of radius ΔS projected onto the detector.
To obtain the detector data value g corresponding to the disk P Using g (-) to represent the value corresponding to a certain detector channel, interpolating by the following method to obtain g P ：
c i ＝int(γ i ) (9)
c j ＝int(γ j ) (10)
w i ＝c i+1 -γ i (11)
w j ＝γ j -c j (12)
If c i ＝c＝c j Then
g P ＝g(c) (13)
Otherwise
Formulas (9) and (10) are floating point channel numbers γ calculated by formulas (5) and (6) i And gamma j Converting into integer channel number values. The boundary channel gamma is calculated by the formulas (11) and (12) i And gamma j Weights in the weighted average.
Compared with other prior art, the image reconstruction method of the U.S. patent 8,116,426 regards the pixel points as discs with the radius being half of the size of the grid, has simple implementation process and easy use, but the elasticity of the pixel model is insufficient. Firstly, the pixel points are regarded as a disc with the radius being half of the size of a grid, once the size of the grid is fixed, the radius of the disc is fixed (see formula (2)), and elasticity is lacked, so that the pixel points are too large in size under the condition that the field of view is large and the number of pixels is relatively unchanged, and the number of channels which are projected onto a detector is too large due to the fact that the pixel points are too large in size, so that the reconstructed image is low in noise, but low in resolution and lost in detail; secondly, the method will fall back to nearest neighbor interpolation as the mesh size becomes smaller, as can be seen from the limit case Δs=0, when Δs=0, formulas (5) and (6) give γ i ＝γ j =γ, thereby c i ＝c＝c j It is seen from equations (4) and (15) that this is actually the nearest neighbor interpolation method, and thus an image obtained using this method may have a large noise in the case where the pixel size becomes small. However, at least two cases exist for decreasing the pixel size, and as can be seen from the formula (2), the first case is that if the number N of pixels of the image is unchanged, the field of view size FOV becomes smaller, which causes the pixel size to become smaller; the second case is that if the field of view size FOV is unchanged and the number of pixels N becomes largeThe pixel size will also be smaller, and the second situation is currently becoming a trend in CT image reconstruction, i.e. reconstructing a CT image with a large matrix (e.g. a 1K or 2K matrix), and thus, us patent 8,116,426 is difficult to adapt to the development requirements of CT image reconstruction techniques.
Disclosure of Invention
In order to solve the technical problems, the invention provides a linear interpolation method in the CT image reconstruction back projection process, which is simple and easy to implement, has good elasticity and can not cause excessive noise under the condition of smaller pixel size.
The technical scheme of the invention is as follows: a linear interpolation method in the CT image reconstruction back projection process defines a circular disc with a changeable radius by taking the pixel point P as a circle center, wherein the radius delta S' of the circular disc is calculated according to the following formula:
wherein the method comprises the steps of
ρ is not less than 0, the FOV is the field of view, N is the number of pixels in the FOV,
calculating and obtaining the boundary of the detector channel covered by the disk after being projected to the detector, and calculating the value g of the pixel point P according to the value g (-) corresponding to each detector channel covered by the projection by the following formula P ，
If c' i ＝c' j Then
Otherwise (i.e. c' i +1≤c' j )
Wherein the method comprises the steps of
c' i ＝int(γ' i ) (22)
c' j ＝int(γ' j ) (23)
w' i ＝γ' i -c' i (24)
w' j ＝γ' j -c' j (25)
Wherein,,
x and y are the x-axis y-axis coordinates of the pixel point P, respectively;
r is the distance from the position S of the light source to the rotation center O, i.e. r= |so|;
beta is the angle formed by the line segment SO and the y-axis;
l is the distance from the light source S to the P point, i.e., l= |sp|;
gamma is the calculated value of the relative channel number corresponding to the pixel point P, and the channel with the calculated value of the relative channel number gamma is called channel gamma;
γ i and gamma j The relative channel number of the boundary covered by the disk after being projected onto the detector is calculated to be between gamma i And gamma j The channels between the two (if any) are all channels covered by the disk after being projected to the detector, and the calculated value of the relative channel number is gamma i And gamma j The channels of (a) are respectively called channel gamma i And channel gamma j ；
c' i And c' j Respectively is to gamma i And gamma j Taking an integer;
w' i and w' j Respectively boundary channels gamma i And gamma j Weight coefficient of (2) at gamma i And gamma j The channels in between (if any) are all covered channels, and the weight coefficients are all 1.
The radius coefficient ρ may be set or changed depending on the image field of view and the number of pixels.
In general, ρ >0 may be set.
0< ρ <1 can be set, with the corresponding disk radius between 0 and half the image grid size (side length).
ρ >1 can be set, the corresponding disc radius exceeds half the image grid size, and overlapping areas exist for discs of different pixels.
The relative channel number is the channel number corresponding to the channel number at the intersection point of the line segment SO extension line and the detector arc, and is equal to the channel number under the condition that the channel number at the intersection point of the line segment SO extension line and the detector arc is zero or the channel number at the intersection point of the line segment SO extension line and the detector arc is subtracted from the actual channel number.
The beneficial effects of the invention are as follows: the pixel point is regarded as a disc with a variable radius, the radius coefficient rho is selected according to the image field of view and the number of pixel points, so that the pixel point meets the resolution requirement and the data processing capability, the complex processing process caused by regarding the pixel point as a rectangle is avoided, meanwhile, the image reconstruction has good signal-to-noise ratio and resolution, the pixel point has good elasticity and adaptability to different situations, when the pixel size is reduced, the interpolation method is degraded to be similar to a linear interpolation method, and is not degraded to be similar to nearest neighbor interpolation as in the method of US patent 8,116,426, the better signal-to-noise ratio is ensured when a CT image is reconstructed by using a large matrix (such as a 1K matrix or a 2K matrix), and the method is better applicable to the development of CT image reconstruction technology.
Drawings
FIG. 1 is a schematic view of a back projection model and computational principle, showing a disk radius less than half the image mesh size (side length);
FIG. 2 is a schematic view of a back projection model and computational principle, showing a disk radius equal to half the image mesh size (side length);
FIG. 3 is a schematic view of a back projection model and calculation principle, which corresponds to the limit case when the radius of the disk is zero, i.e. the limit case when the radius of the disk is zero (infinite approach to zero) can be considered as the model regardless of the pixel size;
FIG. 4 is a comparison of image quality without regard to pixel size (prior art) and with regard to pixel size (present invention) during back projection in the same situation, wherein the left side image is an image without regard to pixel size and the right side image is an image with regard to pixel size;
fig. 5 is a comparison of image quality for a selection of different radius coefficients during the back projection in the same case, where the left image is an image with a radius coefficient of 1 (i.e., the radius of the disk is equal to half the image grid side), and the right image is an image with a radius coefficient of 0.5 (i.e., the radius of the disk is equal to one quarter the image grid side).
Detailed Description
Referring to fig. 1-3, in order to increase the elasticity of the pixel model, the pixel point is regarded as a disc with a variable radius based on the prior art, and the radius is as follows:
wherein ρ is not less than 0.
When ρ=0, the pixel model is to consider the pixel point as a point of no size; when ρ=1, the pixel points are seen as a disk with a radius half the size of the image grid; when 0< ρ <1, the radius of the disk is between 0 and half the image mesh size; when ρ >1, it means that the radius of the disk is larger than half the mesh size, i.e. there is an overlap between adjacent pixels. The pixel model depicted in fig. 2 and 3 is a specific embodiment of the present invention in two cases, corresponding to the case where the pixel point is seen as a circular disk with a radius half the size of the image grid and the pixel point is seen as one point in the prior art, respectively. Whereas the embodiment shown in fig. 1 is the case when 0< p <1, the radius of the disk is between 0 and half the size of the image grid.
After such a pixel model is designed, the channel boundary covered by the disk after being projected to the detector is calculated, and the specific formula is as follows:
from the above, γ i ≤γ j ，γ i And gamma j The boundary channel values respectively representing the projection of the disc onto the detector correspond to the boundary positions of the disc projected on the detector,
to obtain the value g of P point (detector data value or image pixel value) P The following interpolation formula is used:
c' i ＝int(γ' i )
c' j ＝int(γ′ j )
w' i ＝γ' i -c' i
w' j ＝γ' j -c' j
such as
c' i ＝c' j
Then
Otherwise, i.e
The subscripts i, j denote the variables of the two boundary channels for the respective parameters, unless specifically stated or otherwise explicitly indicated.
In formula (16), although there is c' i ＝c' j But in general w' i ≠w' j Thus, denominator (1-w' i )+w' j Not equal to 1, equation (16) is in fact a variant of linear interpolation. Equation (17) is a generalization of equation (16).
Now consider a case where the limit Δs ' =0, i.e., ρ=0, where γ ' is calculated ' i ＝γ' j And c' i ＝c' j Further w' i ＝w' j In this case, equation (16) is degenerated into the calculation equation of the linear interpolation method:
g P ＝(1-w′ i )g(c′ i )+w′ i g(c′ i +1)
from the above analysis, it can be seen that the interpolation method of the present invention approaches a linear interpolation method as the pixel size becomes smaller, unlike the interpolation method of us patent 8,116,426, which approaches a nearest neighbor interpolation method, and therefore, the noise of the reconstructed image of the present invention is significantly lower than that of the reconstructed image of us patent 8,116,426 as the pixel size becomes smaller.
Experiments prove that the image with higher signal-to-noise ratio can be obtained by considering the size of the pixel in back projection. In the back projection process of the left image in fig. 4, the pixel size is not considered, that is, the pixel is considered as a physical point without size in the back projection process, the right image adopts a model considering the pixel size, which is an image obtained by using the interpolation method of the present invention, and the noise characteristic of the right image is improved compared with that of the left image, which indicates that the signal to noise ratio can be improved by considering the pixel model according to the model of the present invention in the back projection process.
Experiments also prove that the pixel model has stronger elasticity, and the balance of noise and resolution can be obtained by adjusting the size of the pixels in back projection so as to obtain better detection results. Fig. 5 shows a comparison of image quality for different pixel sizes during the back projection process, where the pixel model used for the left image is one half the radius of the pixel disk as the image grid size (ρ=1), the pixel model used for the right image is one quarter the radius of the pixel disk as the image grid size (ρ=0.5), and as a result, the resolution at the arrows of the left image and the right image is different, the black small dot structure at the arrows of the left image is blurred, and the black small dot structure can be seen for the right image, i.e. the resolution is improved.
Claims (5)
1. A linear interpolation method in the CT image reconstruction back projection process defines a circular disc with a changeable radius by taking the pixel point P as a circle center, wherein the radius delta S' of the circular disc is calculated according to the following formula:
wherein the method comprises the steps of
ρ is not less than 0, the FOV is the field of view, N is the number of pixels in the FOV,
calculating and obtaining the boundary of the detector channel covered by the disk after being projected to the detector, and calculating the value g of the pixel point P according to the value g (-) corresponding to each detector channel covered by the projection by the following formula P ，
If c' i ＝c' j Then
Otherwise
Wherein the method comprises the steps of
c′ i ＝int(γ′ i )
c′ j ＝int(γ′ j )
w′ i ＝γ′ i -c′ i
w′ j ＝γ′ j -c′ j
Wherein the method comprises the steps of
x and y are the x-axis y-axis coordinates of the pixel point P,
r is the distance from the position S of the light source to the rotation center O,
beta is the angle of the segment SO with the y-axis.
2. A method as claimed in claim 1, characterized in that the radius coefficient ρ is set or changed in dependence on the image field of view and the number of pixels.
3. A method as claimed in claim 1 or 2, characterized in that ρ >0 is set.
4. A method as claimed in claim 1 or 2, characterized in that 0< ρ <1 is set.
5. A method as claimed in claim 1 or 2, characterized in that ρ >1 is set.
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