CN106651775B - Optimization method of logarithmic polar coordinate transformation based on digital image processing - Google Patents

Optimization method of logarithmic polar coordinate transformation based on digital image processing Download PDF

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CN106651775B
CN106651775B CN201611167716.6A CN201611167716A CN106651775B CN 106651775 B CN106651775 B CN 106651775B CN 201611167716 A CN201611167716 A CN 201611167716A CN 106651775 B CN106651775 B CN 106651775B
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师丽娜
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Xi'an Hui Guang Photoelectric Technology Co Ltd
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Abstract

An optimization method of logarithmic polar coordinate transformation based on image processing comprises the following steps: acquiring a digital image I to be processed, and calculating the maximum radius r of the digital image I to be processedmax(ii) a Establishing a digital image rectangular coordinate system to be processed, and respectively determining equal mapping circles, concave areas and peripheral areas in the digital image rectangular coordinate system to be processed; then N concentric circles in the digital image I to be processed are obtained; carrying out logarithm polar coordinate transformation on a digital image rectangular coordinate system to be processed to obtain an image J after the logarithm polar coordinate transformation, and respectively determining the maximum mapping distance rho of the J with the horizontal size of an angular sampling rate m and the vertical size of JmaxFinally, sequentially calculating the m multiplied by r in the concave area of the image J and the equal mapping circle area after the conversion of the log-polar coordinates corresponding to the concave area and the equal mapping circle in the digital image rectangular coordinate system to be processed0Gray values of the pixel points and mx (rho) in the peripheral area in the image J after log-polar coordinate transformation corresponding to the peripheral area in the digital image rectangular coordinate system to be processedmax‑r0) The gray value of each pixel point.

Description

Optimization method of logarithmic polar coordinate transformation based on digital image processing
Technical Field
The invention belongs to the technical field of image processing, and particularly relates to an optimization method of logarithmic polar coordinate transformation based on digital image processing, which is suitable for the fields of computer vision, image understanding, target recognition, image registration and the like.
Background
The traditional calculation method for logarithmic polar coordinate transformation comprises three steps, namely, firstly, determining the radius of a mapping circle and the like, and then adopting different mapping methods according to different characteristics of a concave area and the periphery. However, the conventional calculation method only ensures that the farthest point of the original image under the rectangular coordinate system is mapped to the farthest point in the log polar coordinate image; such a simple correspondence would make it impossible for points near the iso-map circle in the rectangular coordinate system to map to the iso-map circle region in the log polar image, resulting in a large error.
Disclosure of Invention
In view of the above drawbacks of the prior art, the present invention provides an optimization method for log-polar transformation based on digital image processing, which provides a calculation method capable of ensuring both the farthest point mapping and the mapping around the equal mapping circle, aiming at the disadvantage of mapping the peripheral region of the digital image in the conventional method, so that the result of the whole log-polar transformation is more reasonable and accurate.
In order to achieve the purpose, the invention is realized by adopting the following technical scheme.
An optimization method of log-polar transformation based on image processing comprises the following steps:
step 1, acquiring a digital image I to be processed, wherein the size of the digital image I to be processed is X multiplied by Y, and X, Y are integers which are larger than 0 respectively; calculating the maximum radius r of the digital image I to be processedmax；
Establishing a digital image rectangular coordinate system to be processed, and determining an equal mapping circle in the digital image rectangular coordinate system to be processed; the region inside the mapping circle in the digital image rectangular coordinate system to be processed is a concave region, and the region outside the mapping circle is a peripheral region;
step 2, carrying out logarithmic polar coordinate transformation on a digital image rectangular coordinate system to be processed to obtain an image J after the logarithmic polar coordinate transformation, and respectively determining the horizontal size of the image J after the logarithmic polar coordinate transformation as an angular sampling rate m and the vertical size of the image J after the logarithmic polar coordinate transformation as the farthest mapping distance rho of the image J after the logarithmic polar coordinate transformationmax；
Wherein the maximum radius r of the digital image I to be processedmaxThe mapping distance in the log-polar-transformed image J is the farthest mapping distance ρ in the log-polar-transformed image JmaxM represents the angular sampling rate of the digital image rectangular coordinate system to be processed;
step 3, calculating to obtain m multiplied by r in the concave area of the image J and the equal mapping circle area after the concave area and the equal mapping circle corresponding to the concave area and the equal mapping circle in the digital image rectangular coordinate system to be processed are transformed by the log-polar coordinates0Gray value of individual pixel point, r0Representing the radius of an equal mapping circle in a digital image rectangular coordinate system to be processed;
step 4, calculating to obtain mx (rho) in the peripheral area in the image J after the logarithm polar coordinate corresponding to the peripheral area in the digital image rectangular coordinate system to be processed is transformedmax-r0) The gray value of each pixel point; further obtaining a final image after logarithmic polar coordinate transformation, wherein the logarithmic polar coordinate transformationThe transformed final image includes a concave region and m × r regions within the equal mapping circle0Gray value of each pixel point, and mx (ρ) in peripheral regionmax-r0) The gray value of each pixel point; the concave area and the equal mapping circle area are m multiplied by r0The gray value of each pixel point is calculated by all pixel points contained in an inner concave area and an equal mapping circle area of the digital image I to be processed, and mx (rho) in the peripheral areamax-r0) The gray value of each pixel point is calculated by all pixel points contained in the peripheral area inside and outside the digital image I to be processed.
Compared with the traditional technology, the invention has the following advantages: the method can ensure that the mapping near the equal mapping circle conforms to the mapping characteristic of logarithmic polar coordinate transformation, and can ensure that the mapping of the peripheral area conforms to the mapping characteristic of logarithmic polar coordinate transformation; meanwhile, the set radius adjusting coefficient can conveniently realize the sizes of the image sizes after different conversions.
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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 is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of an optimization method for log-polar transformation based on digital image processing according to the present invention;
FIG. 2 is a schematic diagram of a rectangular coordinate system of a digital image to be processed;
FIG. 3 is an exemplary diagram of an image after log-polar transformation according to an embodiment of the present invention;
FIG. 4 is an exemplary diagram of rotation and scale invariance of the log-polar transformation provided by embodiments of the present invention;
fig. 5 is a graph showing the result of log-polar transformation under different adjustment coefficients according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1, it is a flowchart of an optimization method based on log-polar transformation of digital image processing according to the present invention; the optimization method of the logarithm polar coordinate transformation based on the digital image processing comprises the following steps:
step 1, acquiring a digital image I to be processed, wherein the size of the digital image I to be processed is X multiplied by Y, coordinates of any pixel point in the digital image I to be processed are recorded as (X, Y), the gray value of the pixel point with the coordinates of (X, Y) is I (X, Y), X is more than or equal to 0 and less than X, Y is more than or equal to 0 and less than Y, and X, Y are integers which are more than 0 respectively.
Respectively calculating the coordinates (x) of the central point of the digital image I to be processed0,y0) And the maximum radius r of the digital image I to be processedmaxThe expressions are respectively:
Figure BDA0001182638090000031
establishing a digital image rectangular coordinate system to be processed, and referring to fig. 2, wherein the digital image rectangular coordinate system to be processed is a schematic diagram; in the rectangular coordinate system of the digital image to be processed shown in fig. 2, the horizontal axis is x, and is positive to the right; the vertical axis is y, positive downwards; the central point O of the rectangular coordinate system of the digital image to be processed is the central point coordinate (x) of the digital image I to be processed0,y0) (ii) a Selecting any pixel point P (i, j) in the rectangular coordinate system of the digital image to be processed,
Figure BDA0001182638090000032
the included angle between the pixel point P (i, j) and the horizontal axis x is phi, and the pixel point P (i, j) and the digital image to be processedThe distance of the central point O of the rectangular coordinate system is r, and r is an integer greater than or equal to 0.
In fig. 2, the inner solid circle of the digital image rectangular coordinate system to be processed is the equal mapping circle at the angular sampling rate m of the digital image rectangular coordinate system to be processed, and the area within the equal mapping circle is the concave area and is represented by the shaded part; the areas outside the mapping circles are peripheral areas and are represented by unshaded lines; and respectively determining the radius of an equal mapping circle in a digital image rectangular coordinate system to be processed as r0The diameter of the equal mapping circle in the digital image rectangular coordinate system to be processed is c, r0＝m/2π，c＝2×r0M represents the angular sampling rate of the digital image rectangular coordinate system to be processed; the number of pixels actually covered by the circumference of the equal mapping circle is equal to the number of sampling points of the circumference of the equal mapping circle; in this embodiment, the angular sampling rate m of the rectangular coordinate system of the digital image to be processed is 64, and the radius r of the mapping circle is equal to0Is 10, the diameter of the iso-map circle is 20.
Illustratively, the size of the digital image to be processed is 320 × 320, the coordinates of the center point of the digital image to be processed I are (160 ), and the maximum radius of the digital image to be processed I is 226; x and Y are natural numbers greater than 0, respectively.
In a digital image I to be processed, taking a central point O of a rectangular coordinate system of the digital image I to be processed as a circle center, taking the distance from any pixel point in the digital image I to be processed to the central point O of the rectangular coordinate system of the digital image I to be processed as a radius, and taking d as a set radius step increment to obtain N concentric circles; n, d are integers greater than 0, and N is not greater than the maximum radius r of the digital image I to be processedmax(ii) a In this example, d is 1.
Step 2, performing log-polar coordinate transformation on the digital image rectangular coordinate system to be processed to obtain an image J after log-polar coordinate transformation, as shown in fig. 3, where fig. 3 is an exemplary diagram of an image after log-polar coordinate transformation according to an embodiment of the present invention.
The horizontal direction of the image J after the log-polar coordinate transformation is a plurality of angle quantization values obtained by respectively sampling the circumferences of N concentric circles at equal intervals by using the angle sampling rate m of a digital image rectangular coordinate system to be processed, and any one of the angle quantization values is represented by k; the vertical direction of the image J after the log-polar coordinate transformation is a distance quantization value obtained after log-polar coordinate transformation is carried out on the distance between each pixel point in the digital image rectangular coordinate system to be processed and the central point O of the digital image rectangular coordinate system to be processed, and any one distance quantization value is represented by rho; the shaded area in fig. 3 is the area mapped by the pit in fig. 2, and the double-shaded area in fig. 3 is the position mapped by the pixel point P (i, j) in fig. 2.
Therefore, any pixel point in the image J after log-polar transformation is denoted as J (k, ρ), and its expression is:
J(k,ρ)＝LPT{I(x,y)；(x0,y0)}
the LPT { } represents that any pixel point gray value I (x, y) in the digital image rectangular coordinate system to be processed is subjected to logarithmic polar coordinate transformation by using a central point O of the digital image rectangular coordinate system to be processed, (x) represents that the gray value I (x, y) is subjected to logarithmic polar coordinate transformation by using a central point O of the digital image rectangular coordinate system to be processed0,y0) Representing the coordinates of the central point of the digital image I to be processed.
For the region outside the mapping circle in the digital image rectangular coordinate system to be processed, the mapping from the digital image rectangular coordinate system to be processed to the image J after log-polar coordinate transformation is under-sampled, namely, multiple points in the region outside the mapping circle in the digital image rectangular coordinate system to be processed are mapped to the same point in the image J after log-polar coordinate transformation, and along with the increase of the distance from the pixel point in the digital image rectangular coordinate system to be processed to the central point O, the number of the pixel points mapped to the same pixel point in the image J after log-polar coordinate transformation is also increased, so that the region outside the mapping circle in the digital image rectangular coordinate system to be processed is marked as a peripheral region; illustratively, after the peripheral region in fig. 2 is subjected to log-polar transformation, the corresponding region is the non-shaded portion in fig. 3, and the point P (i, j) in the peripheral region in fig. 2 is mapped to the cross-hatched region in fig. 3.
Recording the radius of any concentric circle in the digital image I to be processed as r, and obtaining the mapping distance rho of the radius r of any concentric circle in the digital image I to be processed in the image J after the logarithmic polar coordinate transformation, wherein the expression is as follows:
Figure BDA0001182638090000051
wherein r represents the radius of any concentric circle in the digital image I to be processed, ρ represents the mapping distance of the radius r of any concentric circle in the digital image I to be processed in the image J after the log-polar coordinate transformation, M represents the gain coefficient of the mapping distance in the log-polar coordinate transformation, a represents the radius correction coefficient of the mapping distance in the log-polar coordinate transformation, b represents the offset coefficient of the mapping distance in the log-polar coordinate transformation,
Figure BDA0001182638090000052
represents rounding down; any concentric circle with the radius of r in the digital image I to be processed is mapped into the rho-th row in the image J after logarithmic polar coordinate transformation.
Derivation of formula (1) yields:
the radius of the digital image I to be processed is r0Is mapped to the rho-th circle in the log-polar transformed image J0Line, then let the radius of the digital image I to be processed be r0The circle of +1 is mapped to the ρ -th circle in the image J after the number-polar coordinate conversion0+1 line; the mapping distance rho between the radius r of any concentric circle in the digital image I to be processed and the radius r of any concentric circle in the digital image I to be processed in the image J after the logarithmic polar coordinate transformation must satisfy:
Figure BDA0001182638090000054
combined c 2r0Formula (2) and formula (3) to yield: m ═ r0+a＝0.5c+a(4)
Selecting N ' concentric circles from the N concentric circles, wherein the radius range of the N ' concentric circles is not more than the diameter of a mapping circle in the digital image I to be processed, and the radiuses of the N ' concentric circles are r in sequenced,rd+1,...,rmax，0<N'<N，rmax-rdC denotes the diameter of the mapping circle in the digital image I to be processed, μ denotes a set radius adjustment coefficient, 0<μ≤1；rdRepresents the radius of the smallest concentric circle of the N' concentric circles, and the expression is as follows:
Figure BDA0001182638090000055
wherein r is0Representing the radius of a mapping circle in a digital image I to be processed, and a representing a radius correction coefficient of the mapping distance in logarithmic polar coordinate transformation; the number of mapping distances for respectively mapping N' concentric circles in the digital image I to be processed into the image J after logarithmic polar coordinate transformation is controlled by a set radius adjusting coefficient mu; the smaller the set radius adjustment coefficient μ is, the smaller the number of mapping distances by which N' concentric circles are mapped to the image J after log-polar transformation is, and the lower the degree of compression is.
Maximum radius r of digital image I to be processedmaxThe mapping distance in the log-polar-transformed image J is taken as the farthest mapping distance ρ in the log-polar-transformed image JmaxDue to the furthest mapping distance ρmaxThe maximum number of lines in the image J after the log-polar transformation is also included, so the size of the set radius adjustment coefficient μ also determines the maximum size in the vertical direction in the image J after the log-polar transformation; referring to fig. 4, an exemplary diagram of rotation and scale invariance of a log-polar transformation is provided for an embodiment of the present invention.
Table 1 shows the maximum number of lines ρ in the image J after log-polar transformation in the case where the set radius adjustment coefficient μ differs in valuemaxThe value of (1); and the diameter c of the iso-map circle of the digital image I to be processed in table 1 is 24, the most significant of the digital image I to be processedLarge radius rmaxAt 400, the digital image I to be processed has an angular sampling rate m of 64.
μ ρmax
1.000000 64
0.994923 64
0.696994 80
0.487496 100
0.100000 255
TABLE 1
Referring to fig. 5, a result diagram of log-polar transformation under different adjustment coefficients provided by the embodiment of the present invention is shown; further determining the size range of the image J in the vertical direction after the log-polar coordinate transformation according to the value of the set radius adjusting coefficient mu; generally, the size of the image J after log-polar transformation is sufficiently large when the set radius adjustment coefficient μ is 0.1, and in the case where the compression rate is pursued, it is more desirable that the image J after log-polar transformation has a small image size; in this embodiment, the minimum size of the image J after log-polar transformation in the vertical direction is 64, and the maximum size is 255.
Determining the maximum radius r of a digital image I to be processedmaxRadius r of the smallest concentric circle of the N' concentric circlesdSatisfies the following conditions: r isd＝rmax-μc (6)
Wherein c represents the diameter of an equal mapping circle in a rectangular coordinate system of the digital image to be processed; further, according to equations (5) and (6), a radius correction coefficient a of the mapping distance in the log-polar transformation is calculated and obtained, and the expression is:
Figure BDA0001182638090000071
according to the formula (4) and the formula (7), calculating a gain coefficient M of the mapping distance in the logarithmic polar coordinate transformation, wherein the expression is as follows:
Figure BDA0001182638090000072
and further calculating to obtain a bias coefficient b of the mapping distance in the logarithmic polar coordinate transformation, wherein the expression is as follows:
b＝M ln M-0.5c(9)
then calculating to obtain the farthest mapping distance rho of the image J after logarithmic polar coordinate transformationmaxThe expression is as follows:
Figure BDA0001182638090000073
and then respectively determining the horizontal size of the image J after log-polar coordinate transformation as the angular sampling rate m of a digital image rectangular coordinate system to be processed, and the vertical size of the image J after log-polar coordinate transformation as the farthest mapping distance rho of the image J after log-polar coordinate transformationmax。
Illustratively, in the embodiment of the present invention, the radius adjustment coefficient μ is set to 0.435949, and the farthest mapping distance ρ of the image J after log-polar transformation is setmaxThe horizontal size of the log-polar-transformed image J is 64, and the vertical size of the log-polar-transformed image J is 64.
Step 3, calculating to obtain the corresponding logarithm poles of the concave area and the equal mapping circle in the digital image rectangular coordinate system to be processedConcave region and m × r within equal mapping circle region of image J after coordinate transformation0Gray value of individual pixel point, r0And the radius of an equal mapping circle in a rectangular coordinate system of the digital image to be processed is represented.
Specifically, in the log-polar transformation, a plurality of sampling points may fall into the same pixel position in the rectangular coordinate system of the digital image to be processed on the circumference of each concentric circle in the concave area in the image J after the log-polar transformation. Therefore, in order to obtain the mapping image of the concave area and the equal mapping circle in the rectangular coordinate system of the digital image to be processed in the image J after the logarithm polar coordinate transformation, the invention adopts the method that k is more than or equal to 0 and less than or equal to m-1 and p is more than or equal to 1 and less than or equal to r in the image J after the logarithm polar coordinate transformation0And (4) inversely calculating the gray value of the corresponding pixel point in the digital image I to be processed according to the coordinates of all the pixel points in the range.
It is set that the concave area and the equal mapping circle area in the image J after the log-polar coordinate transformation include m × r0The concave area and the equal mapping circle area in the image J after the log-polar coordinate transformation are mapping areas of the concave area and the equal mapping circle area in a digital image rectangular coordinate system to be processed in the image J after the log-polar coordinate transformation; wherein the t-th pixel coordinate is marked as (k)t,ρt)，ktRepresenting the angle quantization value, p, of the t-th pixeltExpressing the distance quantization value of the t-th pixel point, wherein the t-th pixel point corresponds to the d-th pixel point in the digital image rectangular coordinate system to be processed,
t∈{1,2,…,m×r0d ∈ {1,2, …, X × Y }; respectively setting the angle quantization value k of the t-th pixel pointtCorresponding to the actual angle value phi of the d-th pixel pointdSetting the coordinate (k) of the t-th pixel pointt,ρt) Coordinate (x) corresponding to the d-th pixel pointd,yd) The expressions are respectively:
0≤kt≤m-1，0≤ρt≤r0，
Figure BDA0001182638090000081
J(kt,ρt)＝I(xd,yd)
wherein r is0The radius of an equal mapping circle in a digital image rectangular coordinate system to be processed is represented, and m represents the angular sampling rate of the digital image rectangular coordinate system to be processed; (x)0,y0) Representing the coordinates of the central point of the digital image I to be processed; t th pixel coordinate (k)t,ρt) Gray value J (k) oft,ρt) Coordinate (x) of the d-th pixel point in the rectangular coordinate system of the digital image to be processedd,yd) Gray value of (x)d,yd) Are equal.
Let t take 1 to mxr respectively0Further, the concave area in the image J after log-polar coordinate transformation and the 1 st pixel point to the m × r in the equal mapping circle area are obtained respectively0Each pixel point is respectively corresponding to m multiplied by r in a digital image rectangular coordinate system to be processed0Actual angle value, and gray value from 1 st pixel point in concave area and equal mapping circle in digital image rectangular coordinate system to be processed to m multiplied by r0The gray values of the pixel points are further respectively and correspondingly obtained to obtain the concave area of the image J and the m multiplied by r in the equal mapping circle area after the conversion of the log-polar coordinates corresponding to the concave area and the equal mapping circle in the rectangular coordinate system of the digital image to be processed0The gray value of each pixel point.
Step 4, calculating to obtain mx (rho) in the peripheral area in the image J after the logarithm polar coordinate corresponding to the peripheral area in the digital image rectangular coordinate system to be processed is transformedmax-r0) The gray value of each pixel point; and obtaining a final image after log-polar coordinate transformation, wherein the final image after log-polar coordinate transformation comprises a concave area and an m multiplied by r in an equal mapping circle area0Gray value of each pixel point, and mx (ρ) in peripheral regionmax-r0) The gray value of each pixel point; the concave area and the equal mapping circle area are m multiplied by r0The gray value of each pixel point is within the digital image I to be processedAll pixel points contained in the concave area and the equal mapping circle area are obtained through calculation, and mx (rho) in the peripheral areamax-r0) The gray value of each pixel point is calculated by all pixel points contained in the peripheral area inside and outside the digital image I to be processed.
Specifically, in the log-polar coordinate transformation, sampling points of the same angle range of the circumferences of a plurality of concentric circles in a peripheral area outside an equal mapping circle in a digital image rectangular coordinate system to be processed fall into the same pixel point position in an image J after log-polar coordinate transformation; therefore, in order to obtain a mapping region image of the peripheral region of the digital image rectangular coordinate system to be processed in the image J after log-polar coordinate transformation, pixel points in the peripheral region of the digital image rectangular coordinate system to be processed are subjected to log-polar coordinate transformation respectively, pixel gray values of a plurality of digital image rectangular coordinate systems to be processed, which fall into the same pixel point of the image J after log-polar coordinate transformation, are accumulated and counted, and finally, the pixels to be filled in the image J after log-polar coordinate transformation are filled with the average values of the accumulation and counting respectively; the pixel coordinate to be filled in the image J after the log-polar coordinate transformation is a pixel point coordinate contained in a difference image between the image J after the log-polar coordinate transformation and a concave area and an equal mapping circle area in the image J after the log-polar coordinate transformation.
The substep of step 4 is:
(4a) respectively establishing a gray scale accumulation two-dimensional array R with the same size as the image J after log-polar coordinate transformation and a counting accumulation two-dimensional array C with the same size as the image J after log-polar coordinate transformation; then, H gray scale accumulation sums contained in the gray scale accumulation two-dimensional array R are respectively set, and the jth gray scale accumulation sum is recorded as
Figure BDA0001182638090000091
j∈{1,2,…,H}，
Figure BDA0001182638090000092
The ith pixel point of the digital image I to be processed in the digital image rectangular coordinate system to be processed corresponds to the ith pixel point in the gray scale accumulation two-dimensional array RThe angular quantization values for the j gray-scale accumulated sums,
Figure BDA0001182638090000093
and the ith pixel point of the digital image I to be processed in the digital image rectangular coordinate system to be processed corresponds to the distance quantization value of the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R.
Setting the count accumulation two-dimensional array C to include
Figure BDA0001182638090000094
A count of accumulated sums, a
Figure BDA0001182638090000095
A count is added and recorded as
Figure BDA0001182638090000096
And respectively recording the initial gray scale accumulation sum of the gray scale accumulation two-dimensional array R asRecording the initial count accumulation sum of the count accumulation two-dimensional array C as
Figure BDA0001182638090000098
And is
Figure BDA0001182638090000099
And
Figure BDA00011826380900000910
all have a value of 0; the total number of the gray scale accumulations contained in the gray scale accumulation two-dimensional array R is equal to the total number of the count accumulation sums contained in the count accumulation two-dimensional array C, and each gray scale accumulation sum corresponds to one count accumulation sum;
Figure BDA00011826380900000911
H<m×ρmax。
setting the digital image I to be processed in the rectangular coordinate system of the digital image to be processed to be X multiplied by YThe digital image I to be processed contains X multiplied by Y pixel points, I belongs to {1,2, …, X multiplied by Y }, I represents the ith pixel point, and the coordinate of the ith pixel point is marked as (X)i,yi) The gray value of the ith pixel point is I (x)i,yi) The initial value of i is 1; x, Y are each integers greater than 0.
(4b) According to the coordinates (x) of the central point of the digital image I to be processed0,y0) And calculating to obtain the radius r of the ith pixel pointiThe expression is as follows:
Figure BDA0001182638090000101
(4c) judging the radius r of the ith pixel pointiIs larger than the radius r of an equal mapping circle in a digital image rectangular coordinate system to be processed0Whether the result is true or not; if not, adding 1 to i, and returning to the substep (4 b); if yes, calculating to obtain a distance quantization value of the ith pixel point corresponding to the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R
Figure BDA0001182638090000102
The expression is as follows:
Figure BDA0001182638090000105
(4d) respectively calculating the coordinates (x) of the ith pixel point according to the following formulai,yi) Included angle phi between the horizontal axis of the rectangular coordinate system of the digital image to be processed and the horizontal axis of the rectangular coordinate system of the digital image to be processediAnd the ith pixel point corresponds to the angle quantization value of the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R
Figure BDA0001182638090000106
The expressions are respectively:
φi＝tan-1((yi-y0)/(xi-x0))
wherein,
Figure BDA0001182638090000108
indicating rounding down, tan indicates the tangent operation, and the superscript-1 indicates the inversion operation.
(4e) Respectively converting the gray value I (x) of the ith pixel pointi,yi) And the j-th gray scale accumulated sum
Figure BDA0001182638090000109
As a sum ofA sum of gray scale
Figure BDA00011826380900001011
Will be first
Figure BDA00011826380900001012
A count of accumulated sums
Figure BDA00011826380900001013
Plus 1 as
Figure BDA00011826380900001014
A count of accumulated sums
Figure BDA00011826380900001015
And respectively order
Figure BDA00011826380900001016
And
Figure BDA00011826380900001017
adding 1;
Figure BDA00011826380900001018
and
Figure BDA00011826380900001019
are respectively 1, 1 st
Figure BDA00011826380900001020
A sum of gray scale
Figure BDA00011826380900001021
Corresponds to the first
Figure BDA00011826380900001022
A count of accumulated sums
Figure BDA00011826380900001023
Figure BDA00011826380900001024
(4f) Adding 1 to I, and returning to the substep (4b) until the X multiplied by Y pixel points in the digital image I to be processed in the digital image rectangular coordinate system to be processed are traversed, and further obtaining H gray scale accumulation sums in the gray scale accumulation two-dimensional array R and the counting accumulation two-dimensional array C
Figure BDA00011826380900001025
The counts are added up and respectively
Figure BDA0001182638090000111
And
Figure BDA0001182638090000112
reset to 1.
(4g) Judgment of
Figure BDA0001182638090000113
A count of accumulated sums
Figure BDA0001182638090000114
Whether greater than 0 is true; if so, calculate
Figure BDA0001182638090000115
Individual gray scale accumulations and corresponding image gray scale values
Figure BDA0001182638090000116
If not, go to substep (4 h); wherein, the firstA count of accumulated sums
Figure BDA0001182638090000118
Corresponds to the first
Figure BDA0001182638090000119
And (4) gray scale accumulation sum.
The first mentioned
Figure BDA00011826380900001110
Individual gray scale accumulations and corresponding image gray scale values
Figure BDA00011826380900001111
The expression is as follows:
Figure BDA00011826380900001112
wherein,
Figure BDA00011826380900001113
indicating a rounding down.
(4h) Order to
Figure BDA00011826380900001114
Adding 1, returning to the substep (4g) until the H gray scale accumulation sums in the gray scale accumulation two-dimensional array R and the count accumulation two-dimensional array C are traversed
Figure BDA00011826380900001115
The 1 st gray scale accumulation sum obtained at the moment is added to the corresponding image gray scale value
Figure BDA00011826380900001116
To the H-th gray scale accumulation and corresponding image gray scale value
Figure BDA00011826380900001117
As mx (rho) in the peripheral region inside the image J after the transformation of the log-polar coordinates corresponding to the peripheral region in the rectangular coordinate system of the digital image to be processedmax-r0) Gray value of each pixel, H ═ mx (ρ)max-r0)，H<m×ρmax。
And obtaining a final image after log-polar coordinate transformation, wherein the final image after log-polar coordinate transformation comprises a concave area and an m multiplied by r in an equal mapping circle area0Gray value of each pixel point, and mx (ρ) in peripheral regionmax-r0) The gray value of each pixel point; the concave area and the equal mapping circle area are m multiplied by r0The gray value of each pixel point is calculated by all pixel points contained in an inner concave area and an equal mapping circle area of the digital image I to be processed, and mx (rho) in the peripheral areamax-r0) The gray value of each pixel point is calculated by all pixel points contained in the peripheral area inside and outside the digital image I to be processed.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present invention, and all the changes or substitutions should be covered within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the appended claims.

Claims (5)

1. A log-polar transformation optimization method based on digital image processing is characterized by comprising the following steps:
step 1, obtaining a digital image I to be processed, wherein the size of the digital image I to be processed is X multiplied by Y, X, YAre each an integer greater than 0; calculating the maximum radius r of the digital image I to be processedmax；
Establishing a digital image rectangular coordinate system to be processed, and determining an equal mapping circle in the digital image rectangular coordinate system to be processed; the region inside the mapping circle in the digital image rectangular coordinate system to be processed is a concave region, and the region outside the mapping circle is a peripheral region;
step 2, carrying out logarithmic polar coordinate transformation on a digital image rectangular coordinate system to be processed to obtain an image J after the logarithmic polar coordinate transformation, and respectively determining the horizontal size of the image J after the logarithmic polar coordinate transformation as an angular sampling rate m and the vertical size of the image J after the logarithmic polar coordinate transformation as the farthest mapping distance rho of the image J after the logarithmic polar coordinate transformationmax；
Wherein the maximum radius r of the digital image I to be processedmaxThe mapping distance in the log-polar-transformed image J is the farthest mapping distance ρ in the log-polar-transformed image JmaxM represents the angular sampling rate of the digital image rectangular coordinate system to be processed;
step 3, calculating to obtain m multiplied by r in the concave area of the image J and the equal mapping circle area after the concave area and the equal mapping circle corresponding to the concave area and the equal mapping circle in the digital image rectangular coordinate system to be processed are transformed by the log-polar coordinates0Gray value of individual pixel point, r0Representing the radius of an equal mapping circle in a digital image rectangular coordinate system to be processed;
step 4, calculating to obtain mx (rho) in the peripheral area in the image J after the logarithm polar coordinate corresponding to the peripheral area in the digital image rectangular coordinate system to be processed is transformedmax-r0) The gray value of each pixel point; and obtaining a final image after log-polar coordinate transformation, wherein the final image after log-polar coordinate transformation comprises a concave area and an m multiplied by r in an equal mapping circle area0Gray value of each pixel point, and mx (ρ) in peripheral regionmax-r0) The gray value of each pixel point; the concave area and the equal mapping circle area are m multiplied by r0The gray value of each pixel point is calculated by all pixel points contained in the concave area and the equal mapping circle area of the digital image I to be processedMx (ρ) in peripheral regionmax-r0) The gray value of each pixel point is calculated by all pixel points contained in the peripheral area inside and outside the digital image I to be processed;
wherein the process of the step 2 is as follows:
the horizontal direction of the image J after log-polar coordinate transformation is a plurality of angle quantization values obtained by respectively sampling the circumferences of N concentric circles at equal intervals by using the angle sampling rate m of a digital image rectangular coordinate system to be processed, and any one of the angle quantization values is represented by k; the vertical direction of the image J after the log-polar coordinate transformation is a distance quantization value obtained after log-polar coordinate transformation is carried out on the distance between each pixel point in the digital image rectangular coordinate system to be processed and the central point O of the digital image rectangular coordinate system to be processed, and any one distance quantization value is represented by rho;
wherein, the obtaining process of the N concentric circles is as follows: in a digital image I to be processed, taking a central point O of a rectangular coordinate system of the digital image I to be processed as a circle center, taking the distance from any pixel point in the digital image I to be processed to the central point O of the rectangular coordinate system of the digital image I to be processed as a radius, and taking d as a set radius step increment to obtain N concentric circles; n, d are integers greater than 0, and N is not greater than the maximum radius r of the digital image I to be processedmax；
Recording any pixel point in the image J after the logarithm polar coordinate transformation as J (k, rho), wherein the expression is as follows:
J(k,ρ)＝LPT{I(x,y)；(x0,y0)}
the LPT { } represents that any pixel point gray value I (x, y) in the digital image rectangular coordinate system to be processed is subjected to logarithmic polar coordinate transformation by using a central point O of the digital image rectangular coordinate system to be processed, (x) represents that the gray value I (x, y) is subjected to logarithmic polar coordinate transformation by using a central point O of the digital image rectangular coordinate system to be processed0,y0) Representing the coordinates of the central point of the digital image I to be processed;
recording the radius of any concentric circle in the digital image I to be processed as r, and obtaining the mapping distance rho of the radius r of any concentric circle in the digital image I to be processed in the image J after the logarithmic polar coordinate transformation, wherein the expression is as follows:
Figure FDA0002276852360000021
wherein,
Figure FDA0002276852360000022
represents rounding-down, r represents the radius of any concentric circle in the digital image I to be processed, p represents the mapping distance of the radius r of any concentric circle in the digital image I to be processed in the image J after logarithmic polar coordinate transformation, M represents the gain coefficient of the mapping distance in the logarithmic polar coordinate transformation,
Figure FDA0002276852360000023
a denotes a radius correction coefficient of the mapping distance in the log-polar transformation,b represents a bias coefficient of the mapping distance in logarithmic polar coordinate transformation, and b is M ln M-0.5 c; μ denotes a set radius adjustment coefficient, 0<μ≤1；
Maximum radius r of digital image I to be processedmaxThe mapping distance in the log-polar-transformed image J is taken as the farthest mapping distance ρ in the log-polar-transformed image JmaxAnd then respectively determining the horizontal size of the image J after log-polar coordinate transformation as the angular sampling rate m of a digital image rectangular coordinate system to be processed, and the vertical size of the image J after log-polar coordinate transformation as the farthest mapping distance rho of the image J after log-polar coordinate transformationmax。
2. The method for optimizing log-polar transformation based on digital image processing as claimed in claim 1, wherein in step 1, the digital image I to be processed further comprises:
recording the coordinate of any pixel point in the digital image I to be processed as (x, y), wherein the gray value of the pixel point with the coordinate of (x, y) is I(x,y)，0≤x<X,0≤y<Y, X, Y are each integers greater than 0; and respectively calculating the coordinates (x) of the central point of the digital image I to be processed0,y0) And the maximum radius r of the digital image I to be processedmaxThe expressions are respectively:
x0＝X/2；y0＝Y/2；
Figure FDA0002276852360000032
the digital image rectangular coordinate system to be processed further comprises: in a digital image rectangular coordinate system to be processed, a horizontal axis is x, and a right direction is positive; the vertical axis is y, positive downwards; the central point O of the rectangular coordinate system of the digital image to be processed is the central point coordinate (x) of the digital image I to be processed0,y0) (ii) a Selecting any pixel point P (i, j) in the rectangular coordinate system of the digital image to be processed,the included angle between the pixel point P (i, j) and the horizontal axis x is phi, the distance between the pixel point P (i, j) and the central point O of the digital image rectangular coordinate system to be processed is r, and r is an integer greater than or equal to 0;
the radius of the equal mapping circle in the digital image rectangular coordinate system to be processed is r0Diameter c, r0＝m/2π，c＝2×r0And m represents the angular sampling rate of the digital image rectangular coordinate system to be processed.
3. The method for optimizing log-polar transformation based on digital image processing according to claim 1, wherein a represents a radius correction coefficient of the mapping distance in log-polar transformation, and is calculated by:
selecting N ' concentric circles from the N concentric circles, wherein the radius range of the N ' concentric circles is not more than the diameter of a mapping circle in the digital image I to be processed, and the radiuses of the N ' concentric circles are r in sequenced,rd+1,...,rmax，0<N'<N，rmax-rdC denotes the diameter of the mapping circle in the digital image I to be processed, μ denotes a set radius adjustment coefficient, 0<μ≤1；rdRepresents the radius of the smallest concentric circle of the N' concentric circles, and the expression is as follows:
Figure FDA0002276852360000041
wherein r is0Represents the radius of a mapping circle in the digital image I to be processed;
determining the maximum radius r of a digital image I to be processedmaxRadius r of the smallest concentric circle of the N' concentric circlesdSatisfies the following conditions: r isd＝rmax- μ c, c represents the diameter of the equal mapping circle in the rectangular coordinate system of the digital image to be processed, and then the radius correction coefficient a of the mapping distance in the log-polar coordinate transformation is calculated,
Figure FDA0002276852360000042
4. the method for optimizing log-polar transformation based on digital image processing according to claim 1, wherein the procedure of step 3 is:
it is set that the concave area and the equal mapping circle area in the image J after the log-polar coordinate transformation include m × r0The concave area and the equal mapping circle area in the image J after the log-polar coordinate transformation are mapping areas of the concave area and the equal mapping circle area in a digital image rectangular coordinate system to be processed in the image J after the log-polar coordinate transformation; wherein the t-th pixel coordinate is marked as (k)t,ρt)，ktRepresenting the angle quantization value, p, of the t-th pixeltExpressing the distance quantization value of the t-th pixel point, wherein the t-th pixel point corresponds to the d-th pixel point in the digital image rectangular coordinate system to be processed,
t∈{1,2,…,m×r0d ∈ {1,2, …, X × Y }; respectively setting the angle quantization value k of the t-th pixel pointtActual of the d-th pixel pointAngle value phidSetting the coordinate (k) of the t-th pixel pointt,ρt) Coordinate (x) corresponding to the d-th pixel pointd,yd) The expressions are respectively:
0≤kt≤m-1，0≤ρt≤r0，
Figure FDA0002276852360000051
Figure FDA0002276852360000052
J(kt,ρt)＝I(xd,yd)
wherein r is0The radius of an equal mapping circle in a digital image rectangular coordinate system to be processed is represented, and m represents the angular sampling rate of the digital image rectangular coordinate system to be processed; (x)0,y0) Representing the coordinates of the central point of the digital image I to be processed; t th pixel coordinate (k)t,ρt) Gray value J (k) oft,ρt) Coordinate (x) of the d-th pixel point in the rectangular coordinate system of the digital image to be processedd,yd) Gray value of (x)d,yd) Equal;
let t take 1 to mxr respectively0Further, the concave area in the image J after log-polar coordinate transformation and the 1 st pixel point to the m × r in the equal mapping circle area are obtained respectively0Each pixel point is respectively corresponding to m multiplied by r in a digital image rectangular coordinate system to be processed0Actual angle value, and gray value from 1 st pixel point in concave area and equal mapping circle in digital image rectangular coordinate system to be processed to m multiplied by r0The gray values of the pixel points are further respectively and correspondingly obtained to obtain the concave area of the image J and the m multiplied by r in the equal mapping circle area after the conversion of the log-polar coordinates corresponding to the concave area and the equal mapping circle in the rectangular coordinate system of the digital image to be processed0The gray value of each pixel point.
5. The method for optimizing log-polar transformation based on digital image processing according to claim 1, wherein the substep of step 4 is:
(4a) respectively establishing a gray scale accumulation two-dimensional array R with the same size as the image J after log-polar coordinate transformation and a counting accumulation two-dimensional array C with the same size as the image J after log-polar coordinate transformation; then, H gray scale accumulation sums contained in the gray scale accumulation two-dimensional array R are respectively set, and the jth gray scale accumulation sum is recorded as
Figure FDA0002276852360000056
Figure FDA0002276852360000054
The ith pixel point of the digital image I to be processed in the digital image rectangular coordinate system to be processed corresponds to the angle quantization value of the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R,
Figure FDA0002276852360000055
representing the distance quantization value of the ith pixel point of the digital image I to be processed in the digital image rectangular coordinate system to be processed, corresponding to the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R;
setting the count accumulation two-dimensional array C to include
Figure FDA0002276852360000061
A count of accumulated sums, a
Figure FDA0002276852360000062
A count is added and recorded as
Figure FDA0002276852360000063
And respectively recording the initial gray scale accumulation sum of the gray scale accumulation two-dimensional array R as
Figure FDA0002276852360000064
Recording the initial count accumulation sum of the count accumulation two-dimensional array C as
Figure FDA0002276852360000065
And isAnd
Figure FDA0002276852360000067
all have a value of 0; the total number of the gray scale accumulations contained in the gray scale accumulation two-dimensional array R is equal to the total number of the count accumulation sums contained in the count accumulation two-dimensional array C, and each gray scale accumulation sum corresponds to one count accumulation sum;
Figure FDA0002276852360000068
H<m×ρmax；
setting the size of a digital image I to be processed to be X multiplied by Y in a rectangular coordinate system of the digital image to be processed, wherein the digital image I to be processed comprises X multiplied by Y pixel points, making I belong to {1,2, …, X multiplied by Y }, and I represents the ith pixel point, and marking the coordinate of the ith pixel point as (X) Yi,yi) The gray value of the ith pixel point is I (x)i,yi) The initial value of i is 1; x, Y are each integers greater than 0;
(4b) according to the coordinates (x) of the central point of the digital image I to be processed0,y0) And calculating to obtain the radius r of the ith pixel pointiThe expression is as follows:
Figure FDA0002276852360000069
(4c) judging the radius r of the ith pixel pointiIs larger than the radius r of an equal mapping circle in a digital image rectangular coordinate system to be processed0Whether the result is true or not; if not, adding 1 to i, and returning to the substep (4 b); if yes, calculating to obtain the jth gray scale accumulation in the gray scale accumulation two-dimensional array R corresponding to the ith pixel pointSummed distance quantization value
Figure FDA00022768523600000610
The expression is as follows:
Figure FDA00022768523600000611
Figure FDA00022768523600000612
(4d) respectively calculating the coordinates (x) of the ith pixel point according to the following formulai,yi) Included angle phi between the horizontal axis of the rectangular coordinate system of the digital image to be processed and the horizontal axis of the rectangular coordinate system of the digital image to be processediAnd the ith pixel point corresponds to the angle quantization value of the jth gray scale accumulation sum in the gray scale accumulation two-dimensional array R
Figure FDA0002276852360000071
The expressions are respectively:
φi＝tan-1((yi-y0)/(xi-x0))
Figure FDA0002276852360000072
wherein,
Figure FDA0002276852360000073
indicating rounding down, tan indicating the operation of finding the tangent value, and superscript-1 indicating the operation of finding the inverse;
(4e) respectively converting the gray value I (x) of the ith pixel pointi,yi) And the j-th gray scale accumulated sum
Figure FDA0002276852360000074
As a sum ofFirst, theA sum of gray scale
Figure FDA0002276852360000076
Will be first
Figure FDA0002276852360000077
A count of accumulated sums
Figure FDA0002276852360000078
Plus 1 asA count of accumulated sums
Figure FDA00022768523600000710
And respectively order
Figure FDA00022768523600000711
And
Figure FDA00022768523600000712
adding 1;
Figure FDA00022768523600000713
and
Figure FDA00022768523600000714
are respectively 1, 1 st
Figure FDA00022768523600000715
A sum of gray scaleCorresponds to the first
Figure FDA00022768523600000717
A count of accumulated sums
Figure FDA00022768523600000719
(4f) Adding 1 to I, and returning to the substep (4b) until the X multiplied by Y pixel points in the digital image I to be processed in the digital image rectangular coordinate system to be processed are traversed, and further obtaining H gray scale accumulation sums in the gray scale accumulation two-dimensional array R and the counting accumulation two-dimensional array CThe counts are added up and respectively
Figure FDA00022768523600000721
And
Figure FDA00022768523600000722
resetting to 1;
(4g) judgment of
Figure FDA00022768523600000723
A count of accumulated sums
Figure FDA00022768523600000724
Whether greater than 0 is true; if so, calculate
Figure FDA00022768523600000725
Individual gray scale accumulations and corresponding image gray scale values
Figure FDA00022768523600000726
If not, go to substep (4 h); wherein, the first
Figure FDA00022768523600000727
A count of accumulated sums
Figure FDA00022768523600000728
Corresponds to the first
Figure FDA00022768523600000729
A gray scale cumulative sum;
the first mentioned
Figure FDA00022768523600000730
Individual gray scale accumulations and corresponding image gray scale values
Figure FDA00022768523600000731
The expression is as follows:
wherein,
Figure FDA00022768523600000733
represents rounding down;
(4h) order to
Figure FDA00022768523600000734
Adding 1, returning to the substep (4g) until the H gray scale accumulation sums in the gray scale accumulation two-dimensional array R and the count accumulation two-dimensional array C are traversed
Figure FDA00022768523600000735
The 1 st gray scale accumulation sum obtained at the moment is added to the corresponding image gray scale value
Figure FDA0002276852360000081
To the H-th gray scale accumulation and corresponding image gray scale valueAs the external in the rectangular coordinate system of the digital image to be processedMx (ρ) in the peripheral region inside image J after log-polar transformation corresponding to the peripheral regionmax-r0) Gray value of each pixel, H ═ mx (ρ)max-r0)，H<m×ρmax。
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