CN108765286B - Digital image interpolation algorithm for radiation modulation function fidelity - Google Patents
Digital image interpolation algorithm for radiation modulation function fidelity Download PDFInfo
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- CN108765286B CN108765286B CN201810477480.9A CN201810477480A CN108765286B CN 108765286 B CN108765286 B CN 108765286B CN 201810477480 A CN201810477480 A CN 201810477480A CN 108765286 B CN108765286 B CN 108765286B
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Abstract
The invention discloses a digital image interpolation algorithm for radiation modulation function fidelity. Performing sub-pixel interpolation processing on the digital image, and calculating the pixel value of a sub-pixel position by using a specially designed transverse and longitudinal interpolation coefficient formula for any sub-pixel position to perform digital image interpolation; the transverse interpolation coefficient and the longitudinal interpolation coefficient are calculated by the same method and are obtained by solving the multivariate constraint optimization problem. The interpolation coefficient of the invention has the interpolation function of the traditional interpolation algorithm, avoids the degradation of an image radiation modulation function (MTF) and ensures the detail resolution of the interpolated image. The invention overcomes the defect that the resolution of image details is reduced by the traditional interpolation algorithm.
Description
Technical Field
The invention belongs to the field of digital image processing, relates to an interpolation algorithm for operations such as digital image scaling, perspective transformation, geometric correction and the like, and particularly relates to a digital image interpolation algorithm for radiation modulation function fidelity.
Background
The radiation modulation function (MTF) is a function that depicts the detail resolution of an image at different spatial frequencies, which primarily considers the degree of reduction of the modulation of signals in an image compared to the modulation of signals in an ideal imaging system. In the fields of remote sensing, medical images and the like, a radiation modulation function (MTF) is an important index for evaluating the detail resolution of an image.
When operations such as digital image scaling, perspective transformation, geometric correction and the like are performed, interpolation is needed for the sub-pixel positions of the original image. The conventional interpolation method includes: nearest neighbor interpolation, linear interpolation, cubic variance value, B-spline interpolation. These commonly used interpolation methods essentially estimate the pixel values of the new location using a linear combination of adjacent pixel values, which is equivalent to an image filtering operation. The traditional interpolation algorithm can generate a low-pass filtering effect, so that the radiation modulation function (MTF) of the image is degraded, high-frequency information is lost, and the detail resolution of the image is reduced.
Disclosure of Invention
Aiming at the technical problems in the prior art, the invention provides a digital image interpolation algorithm with radiation modulation function fidelity, which can realize pixel interpolation at any position.
As shown in fig. 1, the technical solution of the present invention is:
performing sub-pixel interpolation processing on the digital image, and calculating pixel values of sub-pixel positions by using horizontal and vertical interpolation coefficients R (m, △ R) and R (n, △ c) for arbitrary sub-pixel positions by adopting the following formula:
wherein △ r, △ c respectively indicate the offset of the sub-pixel position with respect to the original image pixel F (r, c) in the image longitudinal and lateral directions, △ r, △ c ∈ [0,1 ],
the pixel value of a sub-pixel position is represented, F (R + m, c + n) represents the pixel value of a pixel adjacent to the sub-pixel position in an original image, R (m, △ R) and R (n, △ c) are a longitudinal interpolation coefficient and a transverse interpolation coefficient of the image respectively, m represents a serial number of the longitudinal interpolation coefficient, m is-2, -1, …,3, n represents a serial number of the transverse interpolation coefficient, and n is-2, -1, …, 3.
The sub-pixel position is the position between adjacent pixels, the pixel value is unknown, and the sub-pixel position is obtained by interpolation through the method.
The transverse interpolation coefficient R (n, △ c) and the longitudinal interpolation coefficient R (m, △ R) are calculated by the same method and are obtained by solving a multivariate constraint optimization problem.
The following description takes a longitudinal interpolation coefficient R (m, △ R) as an example, and the longitudinal interpolation coefficient R (m, △ R) is obtained by solving a multivariate constraint optimization problem represented by the following formula:
wherein MTF (u)i△ R) is a discrete Fourier transform of the longitudinal interpolation coefficient R (m, △ R) at spatial frequency uiThe amplitude of (i.e. MTF value, u)ii/N-0.5, i denotes the ordinal number of spatial frequencies in the discrete fourier transform, N denotes the total number of spatial frequencies in the discrete fourier transform;
and the following constraints are established:
a) and (3) constraint of an equation:
b) constraint of inequality
|p+△r|<0.05 (4)
Where p is equal to the value of the argument x when the longitudinal edge function E (x) is equal to 0.
The longitudinal edge function E (x) is expressed as
Wherein the content of the first and second substances,
the phase angle at spatial frequency u for the fourier transform of the longitudinal interpolation coefficient R (m, △ R), x represents the argument of the longitudinal edge function e (x).
The above formula shows the result of the vertical interpolation coefficient acting on the sign function.
The following description takes a transverse interpolation coefficient R (n, △ c) as an example, and the transverse interpolation coefficient R (n, △ c) is obtained by solving a multivariate constraint optimization problem represented by the following formula:
wherein MTF (u)i△ c) is the discrete Fourier transform of the transverse interpolation coefficient R (n, △ c) at spatial frequency uiThe amplitude of (i.e. MTF value, u)ii/N-0.5, i denotes the ordinal number of spatial frequencies in the discrete fourier transform, N denotes the total number of spatial frequencies in the discrete fourier transform;
and the following constraints are established:
a) and (3) constraint of an equation:
b) constraint of inequality
|q+△c|<0.05 (8)
Where q is equal to the value of the argument y for a lateral edge function E (y) equal to 0.
The transverse edge function E (y) is expressed as:
wherein the content of the first and second substances,
the phase angle at spatial frequency u for the fourier transform of the lateral interpolation coefficient R (n, △ c), y represents the argument of the lateral edge function.
The above formula shows the result of the action of the transverse interpolation coefficient on the sign function.
The interpolation coefficient of the invention has the interpolation function of the traditional interpolation algorithm, simultaneously avoids the degradation of an image radiation modulation function (MTF), ensures the detail resolution of the interpolated image, and solves the defect that the traditional interpolation algorithm reduces the detail resolution of the image.
Compared with the prior art, the invention has the advantages that:
the method eliminates the low-pass filtering effect of the traditional interpolation algorithm, avoids the degradation of MTF (modulation transfer function) of the image and ensures the detail resolution of the interpolated image.
The image interpolation essentially uses interpolation coefficients to filter the original image, so the MTF of the interpolated image is equal to the MTF of the original image multiplied by the MTF of the interpolation coefficients, fig. 2 and fig. 3 show the MTF of the interpolation coefficients corresponding to a series of offsets △ r, and it can be seen that compared with the cubic difference method, the MTF of the interpolation coefficients at each frequency is close to 1, so the interpolated image can keep the original MTF.
Secondly, the interpolation algorithm of the invention has high geometric accuracy. When constructing a multivariate constraint optimization problem about interpolation coefficients, constraints on geometric accuracy are added. Fig. 4 is a schematic diagram showing the comparison of the geometric accuracy of the present invention and the cubic interpolation method, and it can be seen that the geometric accuracy of the present invention is superior to the cubic interpolation algorithm.
Drawings
FIG. 1 is a schematic illustration of digital image interpolation;
FIG. 2 is a graph of MTF for the interpolation coefficients of the present invention;
FIG. 3 is a schematic MTF of cubic interpolation coefficients;
FIG. 4 is a schematic diagram of the geometric accuracy comparison of the present invention with cubic interpolation;
FIG. 5 is an artwork of an embodiment of the present invention;
fig. 6 is a schematic diagram illustrating a comparison between an interpolation map obtained by interpolating a local 1 in an original image by two interpolation methods and the original image;
fig. 7 is a schematic diagram illustrating a comparison between an original image and a restored image obtained by interpolating and restoring a local portion 2 in the original image by two interpolation methods;
FIG. 8 is an error map of the restoration image of the present invention;
fig. 9 is an error map of the cubic interpolation restoration image.
Detailed Description
An exemplary implementation of the present invention will now be described in detail by taking an image shift as an example, and the effect advantage of the present invention will be explained using a cubic interpolation method as a comparison.
The embodiment of the invention is as follows:
the embodiment realizes the sub-pixel translation of an 8-bit gray image through interpolation. In order to verify the fidelity performance of the radiation modulation function, the embodiment translates the whole original image upwards by 0.375 pixel to obtain an interpolation image, then translates the obtained interpolation image downwards by 0.375 pixel to obtain a restored image, and objectively evaluates the fidelity performance of the radiation modulation function of the interpolation algorithm through comparing the restored image with the original image.
The specific steps for performing the sub-pixel translation of the image according to the method of the invention are as follows:
step 1: the original image F (r, c) is translated upwards by 0.375 pixel to obtain an interpolation image G (r, c), and the mapping relation between the interpolation image and the original image is constructed:
when only longitudinal translation is performed, equation (11) can be simplified as:
it is therefore only necessary to solve the interpolation coefficient R (m,0.375), m being-2, -1, …, 3.
Step 2: constructing a multivariate constraint optimization problem about interpolation coefficients R (m,0.375), m is-2, -1, …, 3:
wherein N is 64, ui＝i/N-0.5。
And establishing the constraint conditions as follows:
(1) constraint of polynomial
(2) Constraint of geometric accuracy
|p+△c|<0.05 (14)
Where p is equal to the value of the argument x when the function shown in equation (15) is equal to 0.
Wherein the content of the first and second substances,
the phase angle at spatial frequency u is the discrete fourier transform of the interpolated coefficient R (m, 0.375).
And step 3: solving the multivariate constraint optimization problem constructed in the step 2 to obtain an interpolation coefficient:
and 4, step 4: and (3) combining the vertical type (10) and the formula (11) and substituting the interpolation coefficient to obtain an interpolation image:
fig. 5 is an original image F (r, c) according to an embodiment of the present invention, fig. 6 is a schematic diagram illustrating a comparison between an interpolation map obtained by interpolating a local portion 1 in the original image by two interpolation methods and the original image, and comparing the original image and the interpolation map of the present invention, it can be seen that the luminance distribution in the interpolation map of the present invention is shifted in the longitudinal direction, that is, the interpolation method of the present invention shifts the original image upward by 0.375 pixel.
And 5: the entire interpolated image G (r, c) is shifted down by 0.375 pixels to obtain a restored image F' (r, c). Constructing a mapping relation between the restored image and the interpolated image:
from formula (11):
referring to step 2, the interpolation coefficient R (m,0.625) can be solved, where m is-2, -1, …, 3.
Equation (20) shows the specific value of the interpolation coefficient R (m, 0.625):
the interpolation coefficient R (m,0.625) is substituted for the expression (19) to obtain a restored image. Fig. 7 is a schematic diagram illustrating a comparison between an original image and a restoration image obtained by interpolating and restoring the local area 2 in the original image by two interpolation methods. Comparing the original image with the original image restored by the cubic interpolation method, it can be seen that after the cubic interpolation method is used for carrying out two times of interpolation operation, the detail part of the original image is lost, the edge becomes fuzzy, namely the cubic interpolation method has the effect of low-pass filtering, and the radiation modulation function is degraded; comparing the original image with the restored image of the invention, it can be seen that the detail of the original image is still retained after the two times of interpolation operation is executed by using the invention, i.e. the interpolation method of the invention has the beneficial effect of radiation modulation function fidelity.
The similarity between the restored image and the original image can be objectively evaluated by calculating the peak signal-to-noise ratio (PSNR). By calculation, in this embodiment, the PSNR of the restoration image of the present invention is 58.3034, and the PSNR of the restoration image of the cubic interpolation method is 35.3382, so the restoration effect of the present invention is better.
The original image and the restored image are subtracted to obtain an error map of the restored image, fig. 8 and 9 are the error map of the restored image and the error map of the restored image by the cubic interpolation method, respectively, referring to the legend, gray represents an error of 0, white and black represent a positive error and a negative error, respectively, comparing fig. 8 and 9, it can be seen that the restored error of the invention is smaller, and through calculation, the mean value of the error map of the restored image of the invention is-4.5776 × 10-05Standard deviation of 0.3196, mean value of error map of cubic interpolation restored image of 7.4768 × 10-04The standard deviation is 4.3715, it can be seen that the mean of the two error maps is close to 0, and the standard deviation of the error map of the invention is obviously smaller than that of the error map of the cubic interpolation method, i.e. the recovery error of the invention is smaller.
Claims (2)
1. A digital image interpolation method for radiation modulation function fidelity is characterized in that:
performing sub-pixel interpolation processing on the digital image, and calculating pixel values of sub-pixel positions by using longitudinal and transverse interpolation coefficients R (m, Δ R) and R (n, Δ c) according to the following formula for any sub-pixel position to perform digital image interpolation:
where Δ r, Δ c represent the amount of shift of the sub-pixel position from the original pixel F (r, c) in the image longitudinal and lateral directions, respectively, Δ r, Δ c ∈ [0,1),
a pixel value representing a sub-pixel position, F (R + m, c + n) representing a pixel value of a pixel adjacent to the sub-pixel position in the original image, R (m, Δ R) and R (n, Δ c) being a longitudinal interpolation coefficient and a lateral interpolation coefficient of the image, respectively; m represents the serial number of the longitudinal interpolation coefficient, m is-2, -1, …,3, n represents the serial number of the transverse interpolation coefficient, n is-2, -1, …, 3;
the longitudinal interpolation coefficient R (m, Δ R) is obtained by solving a multivariate constrained optimization problem represented by the following formula:
wherein MTF (u)iΔ R) is the discrete Fourier transform of the longitudinal interpolation coefficient R (m, Δ R) at the spatial frequency uiAmplitude of (i.e. value of radiation modulation function, u)ii/N-0.5, i denotes the ordinal number of spatial frequencies in the discrete fourier transform, N denotes the total number of spatial frequencies in the discrete fourier transform;
and the following constraints are established:
a) and (3) constraint of an equation:
b) constraint of inequality
|p+Δr|<0.05
Wherein p is equal to the value of the argument x when the longitudinal edge function E (x) is equal to 0;
the longitudinal edge function E (x) is expressed as
Wherein the content of the first and second substances,
the phase angle at spatial frequency u for the fourier transform of the longitudinal interpolation coefficient R (m, Δ R), x representing the argument of the longitudinal edge function e (x);
the transverse interpolation coefficient R (n, Δ c) is obtained by solving a multivariate constrained optimization problem represented by the following formula:
wherein MTF (u)iΔ c) is the discrete Fourier transform of the transverse interpolation coefficient R (n, Δ c) at the spatial frequency uiAmplitude of (i.e. value of radiation modulation function, u)ii/N-0.5, i denotes the ordinal number of spatial frequencies in the discrete fourier transform, N denotes the total number of spatial frequencies in the discrete fourier transform;
and the following constraints are established:
a) and (3) constraint of an equation:
b) constraint of inequality
|q+Δc|<0.05
Wherein q is equal to the value of the argument y for a lateral edge function E (y) equal to 0;
the transverse edge function E (y) is expressed as:
wherein the content of the first and second substances,
is a transverse interpolation coefficient R: (n, Δ c) at spatial frequency u, y represents the argument of the lateral edge function.
2. The method of claim 1 for interpolating a digital image with fidelity to a radiation modulation function, wherein: the transverse interpolation coefficient R (n, delta c) and the longitudinal interpolation coefficient R (m, delta R) are calculated by the same method and are obtained by solving a multivariate constraint optimization problem.
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