CN109727196B - Image interpolation processing method - Google Patents
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Abstract
The image interpolation processing method comprises the steps of determining coordinate variables of an image and known function points of an interpolation frame, constructing a vandermonde matrix and a vandermonde vector space, calculating an inner product among base vectors of subspaces to obtain a degree tensor of the subspaces, obtaining a degree tensor of a dual space of the subspaces by a matrix inversion method, obtaining a base of the dual space through matrix operation, and transposing the base matrix of the dual space to obtain a pseudo-inverse matrix of the vandermonde matrix. Determining the position of the interpolated section in the interpolated frame point; and acquiring a vandermonde matrix with the same term number as the interpolation frame according to the abscissa of the insertion point set. The vandermonde matrix of the set of insertion points is multiplied by the pseudo-inverse matrix to obtain an interpolation matrix. And obtaining pixel values of new pixel points according to the interpolation matrix, moving the interpolation grid by grid according to the pixel grid in one dimension of the image pixels, converting the dimension, and interpolating in the other dimension after finishing the image conversion dimension until all the interpolation is finished. The image is more accurate and vivid, and is favorable for scientific research and exploration, public security case breaking and military reconnaissance imaging.
Description
Technical Field
The embodiment of the invention relates to the technical field of image processing, in particular to an image interpolation processing method.
Background
Existing interpolation techniques are commonly found in specialized image processing software (e.g., irfanView) and in the image processing toolbox in computer high-level languages (e.g., image Processing Toolbox in Matlab).
For IrfanView, although the source code of IrfanView cannot be really known, interpolation amplification is carried out twice in the operation process, and as the interpolation method destroys the inherent additive group property of the two-dimensional Euclidean plane, the pixel outer edge after the first interpolation amplification obviously exists in the image after the second interpolation amplification, and new noise is added. In the IrfanView operation process, because the two variables of x and y are mutually dependent, diagonal twill noise can appear after two interpolation amplification, and meanwhile, the inherent addition group property of the two-dimensional Euclidean plane is also destroyed. In addition, due to the high technical cost of IrfanView, pictures subjected to interpolation amplification in the running process cannot be stored in a disc due to the economic benefit of merchants.
In the specification of the advanced computer language Matlab and other summarized information, linear interpolation is the coarsest interpolation method commonly applied, while the most accurate interpolation is interpolation of a binary cube (Bi-cube) polynomial in a 16-pixel square, and although two-dimensional interpolation can be performed simultaneously, the simple and clear characteristic of the exchange group is destroyed.
In the conventional image processing literature, interpolation is often used for local repair or for enlarging a pattern. A technical scheme for increasing pixels by tens of times and hundreds of times for improving the picture accuracy in a large area does not exist. The Lagrange interpolation method in the existing mathematical textbooks and manuals only gives a few low-order interpolation coefficients, and the interpolation effect is not good. A new image processing solution is therefore needed.
Disclosure of Invention
Therefore, the embodiment of the invention provides an image interpolation processing method which can be used for image processing and video image processing, so that the images become more accurate and more continuous, and is favorable for development of scientific research exploration, public security investigation and military reconnaissance, and can also be applied to daily life photos.
In order to achieve the above object, the embodiment of the present invention provides the following technical solutions: an image interpolation processing method, comprising:
1) Determining the coordinate variable sequence and the coordinate system of the image to be interpolated;
2) Determining n+1 known function points as an interpolation frame, wherein the known function points comprise known coordinate points and known function values, and n is an order;
3) Construction of vandermonde matrix V by the known function points 1 ；
4) Using said vandermonde matrix V 1 N+1 rows of (2) as row vectors to form an n+1-order square matrix, wherein n is a natural number;
5) According to Guo Shidi, i.e. the composition consists of x=x 0 ，x 1 ，…，x n N+1 column vectors of the inverse matrix of the vandermonde square matrix generated at n+1 points are the respective coefficient vectors of the lagrangian interpolation polynomials at these points. Obtaining a coefficient matrix of an n+1-order Lagrangian interpolation polynomial by utilizing the inversion of a Van der Monte square matrix, wherein each column vector of the matrix is respectively formed by coefficient vectors of Lagrangian polynomials of each point;
6) Determining the positions of the interpolated section in the n+1 points of the interpolation frame;
7) Given the number of insertion points between two adjacent pixels, inserting d-1 new pixel points between the two adjacent pixels, wherein d represents the equal fraction of the insertion points to the insertion interval;
8) Determining the abscissa of the insertion point set;
9) Acquiring a set of points from the abscissaVandermonde matrix V of the same term number as the interpolation frame 2 ；
10 (d-1) × (n+1) interpolation matrix L dn ；
11 According to the interpolation matrix L dn Obtaining the pixel value z of d-1 new pixel points i Wherein i=1, …, d-1;
12 Constructing interpolation transformation G (x) =g (f (x)), wherein f (x) is a one-dimensional variable function, and G (x) is a function obtained after interpolation;
13 Starting from a first row or column in one dimension of the image pixel, shifting the pixel-wise interpolation forward in the first row or column;
14 Changing to the next row or column of the image pixels, repeating step 13) to complete the next row or column interpolation, and repeating steps 13) and 14) to complete the last row interpolation of the pixels of the image in the corresponding dimension;
15 Step 13) and step 14) are repeated to complete the interpolation of the other dimension after the image conversion dimension until the complete interpolation of the image is completed.
As a preferable scheme of the image interpolation processing method, the image is a two-dimensional image, a three-dimensional image or a high-dimensional image, and the dimension of the high-dimensional image is larger than three dimensions.
As a preferable embodiment of the image interpolation processing method, in the step 2), the known function value uses a gray value of the image pixel point, and the coefficient of the interpolation polynomial is constructed by the known function value.
As a preferable mode of the image interpolation processing method, in the step 6), n sections are provided in n+1 points, and when n is an odd number, a center section is provided in n sections.
As a preferred embodiment of the image interpolation processing method, in the step 10), the interpolation matrix has a formula of L dn ＝V 2 V 1 -1 。
As a preferable scheme of the image interpolation processing method, in the step 11), a pixel transplanter is adopted to carry out the pixel value z of d-1 new pixel points i Calculating;
this formula is z=l dn y, thisNamely an in-situ interpolation component of the pixel transplanter;
wherein z= (z) 1 ，…，z d-1 )
y＝(y 1 ，…，y n+1 )
y is the coordinate variable of the image.
In the step 13), the original pixel point of the image is re-interpolated at the original pixel point, and the interpolation matrix is moved forward from cell to cell in the same row or the same column until the last pixel by interpolation transformation G (x) =g (f (x)). Because of the motion function, the interpolation transform G (x) =g (f (x)) is what we call a pixel transplanter.
As a preferred embodiment of the image interpolation processing method, for color images, step 16) is also included,
step 16): repeating steps 13), 14) and 15) to successively perform interpolation according to the component order of the color image.
As a preferred scheme of the image interpolation processing method, for the high-dimensional image, steps 13), 14), 15) and 16) are repeatedly performed, and interpolation of the high-dimensional image is completed in sequence from one dimension to the next.
The embodiment of the invention has the following advantages: the method can be used for two-dimensional, three-dimensional and high-dimensional image processing, two-step interpolation is equivalent to one-step interpolation, the trace noise of the first interpolation cannot be seen after the second interpolation, as each-dimensional interpolation is independently carried out, the single-dimensional interpolation method does not have diagonal noise among different dimensions after the dimension-by-dimension interpolation, the interpolation of data and graphics of any dimension can be conveniently completed by the dimension-by-dimension interpolation method, the image can become more accurate, vivid and more insight into autumn milli and more in-vitro spider silk and horse trace, scientific research and exploration, public security scheme and development of military investigation are facilitated, the processing effect of the image is improved, and the interpolation image processing becomes more precise, more accurate and easier.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below. It will be apparent to those of ordinary skill in the art that the drawings in the following description are exemplary only and that other implementations can be obtained from the extensions of the drawings provided without inventive effort.
L65 and L83 related in the technical scheme represent interpolation series and are marked as follows:
l65: the Lagrangian interpolation method 6 equally divides 5 th order polynomial interpolation;
l83: lagrangian interpolation 8 equally divides the polynomial interpolation of 3 rd order, and so on;
FIG. 1 is a schematic flow chart of an image extrapolation processing method according to an embodiment of the present invention;
fig. 2 is a schematic diagram of step S16 of an image extrapolation processing method according to an embodiment of the present invention;
fig. 3 is a schematic diagram of step S17 of an image extrapolation processing method according to an embodiment of the present invention;
FIG. 4 is a schematic diagram showing contrast of the effect of the black-and-white illumination treatment according to the embodiment of the present invention;
FIG. 5 is an enlarged original image of a small bright hole in a street dance photograph provided in an embodiment of the present invention;
FIG. 6 is a small and bright hole plot of a Lagrangian interpolation street dance photo for center interval 6 equal division of a 5 th order polynomial provided in an embodiment of the present invention;
FIG. 7 is a view of green laser light provided in an embodiment of the present invention being emitted from outside into a room through a viewing aperture in a door;
FIG. 8 is an enlarged view of the surrounding aperture of the injection chamber and taken to form a low pixel count map about the aperture in accordance with an embodiment of the present invention;
FIG. 9 is a diagram of an interpolation process of an aperture image L65 of an injection chamber provided in an embodiment of the present invention;
Detailed Description
Other advantages and advantages of the present invention will become apparent to those skilled in the art from the following detailed description, which, by way of illustration, is to be read in connection with certain specific embodiments, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The technical scheme adopts the following theoretical basis:
the general geometric space can be described by the lie group and lie algebra structure. The basic structure of the prune group is a single-dimensional prune group in each dimension, and the generating element is the tangent vector of a single-dimensional curve to form a corresponding prune algebra element. The single-dimensional interpolation is then carried out by the full-dimensional interpolation, so that the interchangeability of the exchange group of the two-dimensional exchange lie algebra formed by Shan Weili algebra is greatly reserved.
Theoretical physicist and applied mathematicist Guo Dongsheng propose Guo Shidi an interpolation theorem.
Theorem 1: the composition is formed by x=x 0 ，x 1 ，…，x n N+1 column vectors of the inverse matrix of the vandermonde square matrix generated at n+1 points are the respective coefficient vectors of the lagrangian interpolation polynomials at these points.
The method makes calculation of Lagrange polynomial coefficient become parallel calculation and simple and feasible by using the theorem, and further makes Lagrange polynomial interpolation method become simple and feasible in scientific numerical calculation. By adopting the Guo Shidi interpolation theorem, the calculation accuracy can be improved by 100 hundred million (10) 10 ) The effect is multiplied.
Both interpolation and extrapolation of Lagrangian interpolation can be performed based on Guo Shidi-interpolation theorem and collectively referred to as Guo interpolation, where conventional extrapolation does not exceed the function subspace in which interpolation resides.
The technical scheme of the embodiment of the invention is based on the theoretical basis.
Specifically, referring to fig. 1, there is provided an image interpolation processing method, including the steps of:
s1: determining the sequence and the coordinate system of the coordinate variables of the image;
s2: determining n+1 known function points as an interpolation frame, wherein the known function points comprise known coordinate points and known function values, and n is an order;
s3: construction of vandermonde matrix V by the known function points 1 ；
S4: using said vandermonde matrix V 1 N+1 rows of (2) as row vectors to form an n+1-order square matrix, wherein n is a natural number;
s5: according to Guo Shidi, i.e. the composition consists of x=x 0 ，x 1 ，…，x n N+1 column vectors of an inverse matrix of the vandermonde square matrix generated by n+1 points are coefficient vectors of each Lagrangian interpolation polynomial at the points, the coefficient matrix of the n+1-order Lagrangian interpolation polynomial is obtained by inversion of the vandermonde square matrix, and each column vector of the matrix is respectively composed of coefficient vectors of Lagrangian polynomials of each point;
s6: determining the positions of the interpolated section in the n+1 points of the interpolation frame;
s7: given the number of insertion points between two adjacent pixels, inserting d-1 new pixel points between the two adjacent pixels, wherein d represents the equal fraction of the insertion points to the insertion interval;
s8: determining the abscissa of the insertion point set;
s9: acquiring a vandermonde matrix V with the same term number as the interpolation frame according to the abscissa of the insertion point set 2 ；
S10: constructing an interpolation matrix L of (d-1) × (n+1) dn ；
S11: according to the interpolation matrix L dn Obtaining the pixel value z of d-1 new pixel points i Wherein i=1, …, d-1;
s12: constructing interpolation transformation G (x) =g (f (x)), wherein f (x) is a one-dimensional variable function, and G (x) is a function obtained after interpolation;
s13: moving a pixel-by-pixel interpolation forward in a first row or column, starting from the first row or column in one dimension of the image pixel;
s14: changing to the next row or the next column of the image pixels, repeating the step S13 to finish the interpolation of the next row or the next column, and repeating the steps S13 and S14 to finish the interpolation of the last row of the pixels of the image in the corresponding dimension;
s15: and (3) converting to another dimension, and repeating the step S13 and the step S14 to finish interpolation of the other dimension after the dimension conversion of the image until the whole interpolation of the image is finished.
In one embodiment of the image interpolation processing method, the image is a two-dimensional image, a three-dimensional image or a high-dimensional image, and the dimension of the high-dimensional image is larger than three dimensions. The order and coordinate system of the coordinate variables of the image, such as h-w (height-width) or x-y (x-coordinate-y-coordinate), are determined in the image interpolation processing method. Interpolation is not limited to two-dimensional images, but may be arbitrary-degree-dimensional images. For example, a four-dimensional image, i.e., a spatially dynamic image, coordinates may be established as x-y-z-t. Because of adopting the single-dimensional interpolation method, the space with higher dimension can be sequentially subjected to interpolation in various dimensions without interference.
In one embodiment of the image interpolation processing method, in the step S2, the known function value uses a gray value of the pixel point of the image, and the coefficient of the interpolation polynomial is constructed by the known function value. Such as n +1 known function points. The known function values of these known function points will be used to construct the coefficients of the interpolation polynomial. The known function value of the known function point may be calculated by the inserted function or may be a predetermined value, such as a gray value of a pixel point.
In one embodiment of the image interpolation processing method, in the step S6, n intervals are included in n+1 points, and when n is an odd number, the n intervals include a center interval. For example, n=0 as a starting point, the central interval is between (n-1)/2 and (n+1)/2. For example, n+1 is 4, n is 3; there are 3 intervals between 4 points, (n-1)/2=1, (n+1)/2=2, with the central interval between n=1 and n=2. If n is even, there are 2 central intervals, and the interpolation should be in two intervals of n/2-1 to n/2, n/2 to n/2+1.
In one embodiment of the image interpolation processing method, in the step S10, the interpolation matrix has a formula of L dn ＝V 2 V 1 -1 . In the step S11, a pixel transplanter is adopted to carry out the pixel value z of d-1 new pixel points i Calculating;
this formula is z=l dn y, namely the in-situ interpolation component of the pixel transplanter;
wherein z= (z) 1 ，…，z d-1 )
y＝(y 1 ，…，y n+1 )
y is the coordinate variable of the image. Specifically, for example, a linear square is formed between two adjacent pixels, and the linear square is divided into d parts after interpolation (for a two-dimensional plane figure, a plane square is changed into d 2 And each square grid. D-1 new pixel points are inserted between two pixels. In visual terms, the two-dimensional image is a two-dimensional field mu, the interpolation matrix is a pixel transplanter which is fixed in place, the head-tail subtraction length of the interpolation matrix is n, the width of the interpolation matrix is 1, and the original pixels play a role in line fixing, row fixing and constant value reference.
Specifically, the correspondence from one number to one number is called a function, and the correspondence from one function to one function is called a transformation. f (x) is a function of a one-dimensional variable, such as a dimension of a two-dimensional picture, and the independent variable x is the position number of the pixel, and the definition field is 1 to N. Since interpolation is performed dimension by dimension, and the problem of multi-dimension simultaneous interpolation is not needed to be considered, g (x) is a function obtained after interpolation, d-1 points are inserted between two pixels, and the definition domain of the new function is 1 to d (N-1) +1. For the pixel value f (i) of the i-th pixel without interpolation, the d (i-1) +1-th pixel value g (d (i-1) +1) =f (i) of the new pixel function can be defined.
In one embodiment of the image interpolation processing method, in the step S13, the original pixel point of the image is re-interpolated at the original pixel point; the interpolation matrix is moved forward from cell to cell within the same row or column by interpolation transform G (x) =g (f (x)) until the last pixel.
Specifically, original values are reproduced at corresponding positions of original pixels by sweeping and reinserting the original pixels. The interpolation transformation G (x) =g (f (x)) is that the pixel transplanter placed on the conveyor belt moves the interpolation matrix forward in the same row, and cell by cell, until the last pixel.
In one embodiment of the image interpolation processing method, for color images, step S16 is further included,
step S16: the steps S13, S14, and S15 are repeatedly performed to successively complete interpolation in accordance with the component order of the color image. For color images, the color dimension has three components R, G and B, each being a two-dimensional image, which can be accomplished sequentially in component order according to steps S13, S14, S15.
In one embodiment of the image extrapolation processing method, for the high-dimensional image, the method further includes step S17, where step S17 repeatedly executes step S13, step S14, step S15, and step S16, and interpolation of the high-dimensional image is completed sequentially from one dimension to another.
The practical effects of the technical scheme in the embodiment of the invention are described below.
Referring to fig. 4, the technical scheme in the embodiment of the invention is applied to black and white photo processing. The picture was taken from an exemplary standard shot in Matlab, named Lena's head shot. The sides of the Lena cap are enlarged in order to show the function of the different interpolation. The upper diagram in fig. 4 is an enlarged original image, and the pixel square is visible due to the low pixel count. The middle graph in FIG. 4 uses an 8x8 interpolation between four points based on Lagrangian cubic polynomials, the interpolation being based on a function of 1, x 2 ，x 3 The image becomes smooth and continuous. The lowermost graph in FIG. 4 uses 8x8 function space interpolation method between four points based on the technical scheme of the invention, and the interpolation function base is x 0 ，…，x 6 The hat lines in the image become clearer.
The following verifies the effect of the technical solution of the present embodiment on processing life photographs.
Fig. 5 is an enlarged original image of a small bright hole, and a square pixel grid can be seen. Because the new picture of the enlarged screenshot becomes a low pixel count picture. Two light apertures in the figure show a square of pixels, except that there is no other anomaly.
A 5 th order polynomial lagrangian interpolation of the center interval 6 is performed with 6 points of known function values (5 intervals), see fig. 6, the graph is smoothed, but no new phenomenon occurs.
Fig. 7 is an overall picture of another person taking a door from 4-5 meters away from the room with a green laser entering the room through a viewing aperture in the door. Fig. 8 is an enlarged view around the aperture and taken as a low pixel count picture about the aperture, with the pixel square apparent. Fig. 9 is a graph of the results of L65 interpolation processing of pinhole images.
While the invention has been described in detail in the foregoing general description and specific examples, it will be apparent to those skilled in the art that modifications and improvements can be made thereto. Accordingly, such modifications or improvements may be made without departing from the spirit of the invention and are intended to be within the scope of the invention as claimed.
Claims (4)
1. An image interpolation processing method, comprising:
1) Determining the sequence and the coordinate system of the coordinate variables of the image;
2) Determining n+1 known function points serving as an interpolation frame, wherein the known function points comprise known coordinate points and known function values, n is an order, the known function values adopt gray values of image pixel points, coefficients of an interpolation polynomial are constructed through the known function values, and the known function values are obtained through calculation or a preset mode of the inserted function;
3) Construction of vandermonde matrix V by the known function points 1 ；
4) Using said vandermonde matrix V 1 N+1 rows of (2) are used as row vectors to form an n+1-order square matrix, wherein n is a natural number;
5) Obtaining a coefficient matrix of an n+1-order Lagrangian interpolation polynomial by utilizing the inversion of a Van der Monte square matrix, wherein each column vector of the matrix is respectively formed by coefficient vectors of Lagrangian polynomials of each point;
6) Determining the positions of the interpolated section in n+1 points of the interpolation frame;
7) Given the number of insertion points between two adjacent pixels, inserting d-1 new pixel points between the two adjacent pixels, wherein d represents the equal fraction of the insertion points to the insertion interval;
8) Determining the abscissa of the insertion point set;
9) Acquiring a vandermonde matrix V with the same term number as the interpolation frame according to the abscissa of the insertion point set 2 ；
10 Construction (d-1) × (n+1)Is an interpolation matrix L of (1) dn The formula of the interpolation matrix is L dn ＝V 2 V 1 -1 ；
11 According to the interpolation matrix L dn Obtaining the pixel value z of d-1 new pixel points i Where i=1, …, d-1, z=l dn y, where z= (z) 1 ，…，z d-1 )，y＝(y 1 ，…，y n+1 ) Y is a coordinate variable of the image;
12 Constructing interpolation transformation G (x) =g (f (x)), wherein f (x) is a one-dimensional variable function, and G (x) is a function obtained after interpolation;
13 Starting from the first row or column in one dimension of the image pixel, shifting the pixel-by-pixel interpolation forward in the first row or column, specifically: re-interpolating original pixel points of an image at original pixel points, and enabling an interpolation matrix to move forwards in the same row or the same column in a grid-by-grid manner through interpolation transformation G (x) =g (f (x)) until the last pixel;
14 Changing to the next row or column of the image pixels, repeating step 13) to complete the next row or column interpolation, and repeating steps 13) and 14) to complete the last row interpolation of the pixels of the image in the corresponding dimension;
15 Step 13) and step 14) are repeated to complete the interpolation of the other dimension after the image conversion dimension until the complete interpolation of the image is completed.
2. The image interpolation processing method according to claim 1, wherein the image is a two-dimensional image, a three-dimensional image, or a high-dimensional image, and the dimension of the high-dimensional image is larger than three dimensions.
3. The image interpolation processing method according to claim 1, wherein in the step 6), n sections exist in n+1 points, and when n is an odd number, n sections exist in a center section.
4. The image interpolation processing method according to claim 1, further comprising step 16,
step 16): repeating steps 13), 14) and 15) to successively perform interpolation according to the component order of the color image.
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