A two-dimensional profile shape characterization method based on the current Fourier transform

Abstract

This invention discloses a two-dimensional contour morphology representation method based on current-round Fourier transform, belonging to the field of geometric morphometrics. It solves the technical problems of contour rotation and misalignment caused by elliptical Fourier descriptors. The key point of its technical solution is to complete the reconstruction and normalization of contour shape through current-round Fourier descriptors, achieving contour shape description effect comparable to elliptical Fourier analysis and contour normalization (alignment) effect comparable to the Pu-style method. It can also avoid the graphic pseudo-rotation problem caused by standardization during the extraction of elliptical Fourier descriptors, without the need for prior standardization of contours using other methods, and quantitatively describes contour features based on current-round Fourier descriptors.

Description

Technical Field

[0001] This application relates to the field of geometric morphometrics, and in particular to a two-dimensional contour morphology representation method based on current Fourier transform. Background Technology

[0002] Morphometrics is a discipline that quantitatively describes, analyzes, and interprets the morphology and changes of organisms or their organs. Among these methods, coordinate-based morphometrics is currently the mainstream approach, its advantage being the preservation of complete geometric morphological information throughout the data collection, analysis, and visualization process. In the morphological differences reflected by these coordinate data, the overall proportion (size) effect often plays a dominant role; therefore, normalization (including scaling, translation, and rotation) must be performed to eliminate the influence of non-shape differences before further analysis. For the morphological reconstruction and analysis of two-dimensional contours, elliptic Fourier analysis is one of the most commonly used morphometric methods, extracting features from the two-dimensional contour through Fourier series. However, when using elliptic Fourier analysis to numerically normalize near-circular two-dimensional contours, it may lead to contour rotation and misalignment. Current solutions involve generalized Procrustes alignment of contours of the same type before elliptic Fourier analysis, a computationally intensive process. Therefore, a concise and efficient method for representing the morphology of two-dimensional contours is urgently needed. Summary of the Invention

[0003] This application provides a two-dimensional contour morphology representation method based on the current Fourier transform. Its technical purpose is to complete the reconstruction and normalization of contour shape through the current Fourier descriptor, while achieving contour shape description effect comparable to elliptical Fourier analysis and contour normalization (alignment) effect comparable to the Pu-style method, and finally quantitatively representing contour features based on the current Fourier descriptor.

[0004] The above-mentioned technical objective of this application is achieved through the following technical solution:

[0005] A two-dimensional contour morphology representation method based on current-round Fourier transform includes:

[0006] S1: Standardize the clockwise direction of the analyzed sample contour to obtain discrete sampling points of the sample contour;

[0007] S2: Project the discrete sampling points on the sample contour onto the complex plane to obtain the complex coordinates of the discrete sampling points;

[0008] S3: Perform an m-order discrete-time Fourier series expansion on the complex coordinates of the discrete sampling points, and then perform an m-order discrete-time inverse Fourier transform to finally obtain the reconstructed and fitted contour curve of the sample contour.

[0009] S4: Perform numerical standardization processing on the reconstructed fitted contour curve by translation and rotation to obtain the final reconstructed fitted contour curve.

[0010] S5: Output the magnitude and phase angle of each Fourier series of the final reconstructed fitted contour curve as feature parameters.

[0011] S6: Represent the two-dimensional contour morphology based on the feature parameters;

[0012] In step S2, the complex coordinates of the discrete sampling points are represented as follows:

[0013] s[n] = x[n] + iy[n];

[0014] Where s[n] represents the coordinates of the nth discrete sampling point on the sample contour in the complex plane, and x[n] and y[n] represent the abscissa and ordinate of the nth discrete sampling point in the complex coordinate system, respectively;

[0015] In step S4, the numerical standardization process for translation includes:

[0016] S411: Calculate the centroid of the reconstructed fitted contour curve. The location of this centroid is represented as:

[0017]

[0018] Where S0 represents the centroid position, which is the 0th order Fourier harmonic spectrum, that is, the center position of the first epicycle circle O1.

[0019] S412: Subtract the centroid position S0 from the coordinates of each point on the reconstructed fitted contour curve to obtain the translated reconstructed fitted contour curve.

[0020] Before performing numerical standardization for rotation, the modulus of the first pair of Fourier series is calculated and expressed as follows:

[0021]

[0022] Among them, R1 and R -1 This represents the modulus of the 1st and -1st order Fourier series, that is, the first pair of epicycle circles O1 and O2. -1 The radius;

[0023] Then in R1 and R -1 Choose the larger modulus value, represented as: R main =max{R1,R -1}; where main represents the index of the Fourier series;

[0024] In step S4, the numerical standardization process for rotation includes:

[0025] S431: For the first pair of Fourier series, the larger value of the modulus R main The phase angle of the corresponding Fourier series is calculated and expressed as:

[0026]

[0027] Among them, a main and b main These represent the real and imaginary parts of the Fourier series corresponding to the larger modulus in the first pair of Fourier series, respectively.

[0028] S432: Based on the phase angle The rotation matrix is ​​calculated and represented as follows:

[0029]

[0030] S433: According to the rotation matrix The coordinates of each point on the reconstructed fitting contour curve are rotated to obtain the rotated reconstructed fitting contour curve.

[0031] The beneficial effects of this application are as follows: The two-dimensional contour morphology representation method based on the current Fourier transform described in this application completes the reconstruction and normalization of the contour shape through the current Fourier descriptor, achieving a contour shape description effect comparable to elliptical Fourier analysis and a contour normalization (alignment) effect comparable to the Pu-style method. It can also avoid the graphic pseudo-rotation problem caused by standardization during the extraction of elliptical Fourier descriptors, without the need for other methods to standardize the contour in advance, and finally quantitatively describe the contour features based on the current Fourier descriptor. Attached Figure Description

[0032] Figure 1 This is a flowchart of the two-dimensional contour morphology representation method based on the current round Fourier transform described in this application;

[0033] Figure 2 This is a schematic diagram of the corpus callosum profile characterized by current-cycle Fourier analysis at harmonic number m=1. The solid line represents the original profile, and the dashed lines represent the two current-cycle circles O1 and O2 corresponding to the first harmonic. -1 The fitted curve generated by superposition has an ellipse shape;

[0034] Figure 3 The diagram shows the outline of the human corpus callosum as characterized by current Fourier analysis at different harmonic numbers m, where m = 1 for a, m = 5 for b, m = 10 for c, and m = 15 for d.

[0035] Figure 4 This is a schematic diagram of the morphological characterization and numerical standardization results of a set of heart-shaped contour lines using elliptical Fourier analysis, where the solid lines are the fitted curves and the dashed lines are the first-order parent ellipse.

[0036] Figure 5 This is a schematic diagram showing the morphological characterization and numerical standardization results of a set of heart-shaped contours using this round of Fourier analysis.

[0037] Figure 6 Representative outline diagrams of nine groups of leaf samples (n=145): a is cherry tree leaf (n=18), b is dogwood leaf (n=14), c is rubber tree leaf (n=16), d is hickory leaf (n=16), e is mulberry leaf (n=17), f is red maple leaf (n=19), g is red oak leaf (n=11), h is sugar maple leaf (n=18), and i is white oak leaf (n=16).

[0038] Figure 7 To utilize this round of Fourier analysis (harmonic number m = 15) to... Figure 6 The diagram shows the morphological characterization results of 145 leaf sample contours: a is the contour with the lowest fitting error (pecan leaf), and b is the contour with the highest fitting error (red oak leaf). The solid line is the original contour, and the dashed line is the fitted contour.

[0039] Figure 8 To utilize the Fourier descriptor pairs in this round Figure 7 The following are distribution diagrams of the corresponding values ​​of the first three canonical discriminant functions obtained by stepwise discriminant analysis of the fitting results shown: a is the convex hull diagram of the 1st and 2nd canonical discriminant functions, and b is the convex hull diagram of the 1st and 3rd canonical discriminant functions. Detailed Implementation

[0040] The technical solution of this application will be described in detail below with reference to the accompanying drawings.

[0041] like Figure 1 As shown, the two-dimensional contour morphology representation method based on current-round Fourier transform described in this application includes:

[0042] S1: Mark the analyzed sample contours with a uniform clockwise direction to obtain discrete sampling points of the sample contours.

[0043] Specifically, in this embodiment, the detailed steps for uniformly marking the clockwise direction of the analyzed sample contour are as follows:

[0044] (1.1) The two-dimensional closed contour is extracted from the binary black and white image of the leaf scan using the eight-neighbor tracking method;

[0045] (1.2) The bottom of the petiole is taken as the starting point of the two-dimensional closed profile. The MATLAB function CSCVN is used to perform smooth interpolation on the two-dimensional closed profile, and 500 points are resampled uniformly on each two-dimensional closed profile.

[0046] Figure 6 The diagram shows representative contours of nine leaf samples used in this embodiment. All contours start at the tip of the petiole and are sampled in a clockwise / counterclockwise direction.

[0047] S2: Project the discrete sampling points on the sample contour onto the complex plane to obtain the complex coordinates of the discrete sampling points.

[0048] Specifically, the complex coordinates of the discrete sampling points are represented as follows:

[0049] s[n] = x[n] + iy[n];

[0050] Where s[n] represents the coordinates of the nth discrete sampling point on the sample contour in the complex plane, and x[n] and y[n] represent the x-coordinate and y-coordinate of the nth point in the complex coordinate system, respectively.

[0051] S3: Perform an m-order discrete-time Fourier series expansion on the complex coordinates of the discrete sampling points, and then perform an m-order discrete-time inverse Fourier transform to finally obtain the reconstructed and fitted contour curve of the sample contour.

[0052] In this embodiment of the application, the harmonic number m = 30 is selected.

[0053] The specific process of the k-th order discrete-time Fourier series expansion of s[n] is as follows:

[0054]

[0055] Among them, S k S represents the k-th order Fourier harmonic spectrum, where k is an integer from -m to m, and N represents the total number of discrete sampling points; k When expressed in complex form, the real part is a. k The imaginary part is b k .

[0056] For S k Performing an m-th order discrete-time inverse Fourier transform, we obtain:

[0057]

[0058] Among them, s t [n] represents the reconstructed and fitted contour curve of the original sample contour obtained after inverse Fourier transform, x t [n]、y t [n] represents s respectivelyt [n] represents the x-coordinate and y-coordinate in a complex coordinate system.

[0059] The k-th order inverse Fourier transform represents a radius... The current round O k There is a vector S on it that points from the center of the circle to the circumference. k S k by The angular velocity of the rotation is such that its initial phase angle is... The vectors S obtained from the m-th order harmonic series k Connecting them sequentially forms the reconstructed fitting contour curve. The geometric meaning of the reconstructed fitting contour curve is the trajectory formed by the superposition of 2m circular primary rings, that is: O k In O -(k-1) On the circumference angular velocity of revolution (O) k The center of the circle is at S -(k-1) (the location indicated), and with The angular velocity of rotation, while O -k In O k On the circumference angular velocity of movement (O) -k The center of the circle is at S k (the location indicated), and with The angular velocity of the rotation (clockwise rotation has a positive angular velocity).

[0060] Based on the example of the above method: Figure 2 This is a schematic diagram of the corpus callosum profile characterized by current-cycle Fourier analysis at harmonic number m=1. The solid line represents the original profile, and the dashed lines represent the two current-cycle circles O1 and O2 corresponding to the first harmonic. -1 The fitted curve generated by superposition is ellipse in shape. Figure 3 This is a schematic diagram of the human corpus callosum outline characterized by current-cycle Fourier analysis at different harmonic numbers m, where m = 1 for a, m = 5 for b, m = 10 for c, and m = 15 for d. For comparison, we have... Figure 4 This diagram illustrates the morphological characterization and numerical standardization results of a set of heart-shaped contour lines using elliptical Fourier analysis. The solid lines represent the fitted curves, and the dashed lines represent the first-order parent ellipse.

[0061] S4: Perform numerical standardization on the reconstructed fitted contour curve by any or any combination of translation, scaling and rotation to obtain the final reconstructed fitted contour curve.

[0062] Specifically, the reconstructed fitted contour curve is shifted so that its centroid is located at the origin, thereby eliminating positional differences between samples. The numerical standardization process for this shift includes:

[0063] S411: Calculate the centroid of the reconstructed fitted contour curve. The location of this centroid is represented as:

[0064]

[0065] Where S0 represents the centroid position, which is the 0th order Fourier harmonic spectrum, that is, the center position of the first epicycle circle O1.

[0066] S412: Subtract the centroid position S0 from the coordinates of each point on the reconstructed fitted contour curve to obtain the translated reconstructed fitted contour curve. This shifts the centroid of each contour curve to the origin, thereby eliminating the difference in contour position.

[0067] The processing steps in this application prior to numerical normalization for scaling and rotation include:

[0068] (1) First, calculate the modulus of the first pair of Fourier series, which are expressed as follows:

[0069]

[0070] Among them, R1 and R -1 This represents the modulus of the 1st and -1st order Fourier series, that is, the first pair of epicycle circles O1 and O2. -1 The radius;

[0071] (2) Then in S1 and R -1 Choose the larger modulus value, represented as: R main =max{R1,R -1}; where main represents the index of the Fourier series.

[0072] Specifically, the numerical standardization process for scaling includes:

[0073] S421: Divide the coordinates of each point on the reconstructed fitted contour curve by R. main The scaled reconstructed fitted contour curve is obtained.

[0074] In this embodiment of the application, since leaf size is also an important criterion in plant species classification based on leaf morphology, and the original data of leaf scanning retains the true information of the relative size of each leaf, no scaling processing was used when performing the Fourier numerical standardization in this case; only translation and rotation processing were performed.

[0075] Specifically, the numerical standardization process for rotation includes:

[0076] S431: For the first pair of Fourier series, the larger value of the modulus R main The phase angle of the corresponding Fourier series is calculated and expressed as:

[0077]

[0078] Among them, a main and b main These represent the real and imaginary parts of the Fourier series corresponding to the larger modulus in the first pair of Fourier series, respectively.

[0079] S432: Based on the phase angle The rotation matrix is ​​calculated and represented as follows:

[0080]

[0081] S433: According to the rotation matrix The coordinates of each point on the reconstructed fitting contour curve are rotated to obtain the rotated reconstructed fitting contour curve.

[0082] After the three normalization processes of translation, scaling, and rotation, we obtain the Fourier series of various orders, where the k-th order Fourier series is expressed as:

[0083]

[0084] In this embodiment, since size is an important criterion for leaf classification, no scaling is performed during standardization. Figure 7 This demonstrates the use of current-cycle Fourier analysis with a harmonic number of 30. Figure 6 The 145 leaf outlines described in the paper were reconstructed and normalized to a partial fitting result, in which... Figure 7 In the diagram, 'a' represents the contour with the highest fitting accuracy. Figure 7 In the diagram, b represents the contour with the lowest fitting accuracy.

[0085] S5: Output the magnitude and phase angle of each Fourier series of the final reconstructed fitted contour curve as feature parameters.

[0086] Specifically, step S5 includes:

[0087] S51: Calculate the modulus of each Fourier series in the final reconstructed fitted contour curve, expressed as:

[0088]

[0089] Among them, S′ k S′ represents the k-th order Fourier harmonic spectrum of the final reconstructed fitted profile curve; k When expressed in complex form, a′ k b′ is the real part. k R' is the imaginary part. k S′ k The model;

[0090] S52: The phase angles of each Fourier series of the final reconstructed fitted contour curve are calculated and expressed as follows:

[0091]

[0092] S6: Represent the two-dimensional contour morphology based on the feature parameters. Figure 5 This is a schematic diagram illustrating the morphological characterization and numerical standardization results of a set of heart-shaped contour lines using this round of Fourier analysis. It can be seen that compared to... Figure 4 The use of elliptical Fourier analysis to characterize and standardize the heart-shaped contours resulted in inconsistent orientations among the contours. However, this round of Fourier analysis is not affected by whether the contour shape is close to a circle or symmetrical, and the morphological characterization and standardization results are relatively accurate.

[0093] In this embodiment, the feature parameters extracted in step S5 are subjected to stepwise discriminant analysis, and the nine plant species are classified according to their contour morphology. The results are as follows: Figure 8 As shown. Leave-one-out cross-validation was used, and the results showed that the classification prediction accuracy of the original and cross-validation of the 145 leaf samples in this embodiment was 100% and 98.6%, respectively, indicating that this application can successfully characterize the two-dimensional contour morphology and extract its features.

[0094] The above are exemplary embodiments of this application, and the scope of protection of this application is defined by the claims and their equivalents.

Claims

1. A two-dimensional contour morphology representation method based on current-round Fourier transform, characterized in that, include: S1: Standardize the clockwise direction of the analyzed sample contour to obtain discrete sampling points of the sample contour; S2: Project the discrete sampling points on the sample contour onto the complex plane to obtain the complex coordinates of the discrete sampling points; S3: Perform an m-order discrete-time Fourier series expansion on the complex coordinates of the discrete sampling points, and then perform an m-order discrete-time inverse Fourier transform to finally obtain the reconstructed and fitted contour curve of the sample contour. S4: Perform numerical standardization processing on the reconstructed fitted contour curve by translation and rotation to obtain the final reconstructed fitted contour curve. S5: Output the magnitude and phase angle of each Fourier series of the final reconstructed fitted contour curve as feature parameters. S6: Represent the two-dimensional contour morphology based on the feature parameters; In step S2, the complex coordinates of the discrete sampling points are represented as follows: s[n] = x[n] + iy[n]; Where s[n] represents the coordinates of the nth discrete sampling point on the sample contour in the complex plane, and x[n] and y[n] represent the abscissa and ordinate of the nth discrete sampling point in the complex coordinate system, respectively; In step S4, the numerical standardization process for translation includes: S411: Calculate the centroid of the reconstructed fitted contour curve. The location of this centroid is represented as: Where S0 represents the centroid position, which is the 0th order Fourier harmonic spectrum, that is, the center position of the first epicycle circle O1. S412: Subtract the centroid position S0 from the coordinates of each point on the reconstructed fitted contour curve to obtain the translated reconstructed fitted contour curve. Before performing numerical standardization for rotation, the modulus of the first pair of Fourier series is calculated and expressed as follows: Among them, R1 and R -1 This represents the modulus of the 1st and -1st order Fourier series, that is, the first pair of epicycle circles O1 and O2. -1 radius; Then in R1 and R -1 Choose the larger modulus value, represented as: R main =max{R1,R -1 }; where main represents the index of the Fourier series; In step S4, the numerical standardization process for rotation includes: S431: For the first pair of Fourier series, the larger value of the modulus R main The phase angle of the corresponding Fourier series is calculated and expressed as: Among them, a main and b main These represent the real and imaginary parts of the Fourier series corresponding to the larger modulus in the first pair of Fourier series, respectively. S432: Based on the phase angle The rotation matrix is ​​calculated and represented as follows: S433: According to the rotation matrix The coordinates of each point on the reconstructed fitting contour curve are rotated to obtain the rotated reconstructed fitting contour curve.

2. The method as described in claim 1, characterized in that, In step S4, the method for numerically standardizing the reconstructed fitted contour curve further includes scaling, and the numerical standardization process for scaling includes: S421: Divide the coordinates of each point on the reconstructed fitted contour curve by R. main The scaled reconstructed fitted contour curve is obtained.

3. The method as described in claim 2, characterized in that, Step S5 includes: S51: Calculate the modulus of each Fourier series in the final reconstructed fitted contour curve, expressed as: Among them, S′ k S′ represents the k-th order Fourier harmonic spectrum of the final reconstructed fitted profile curve; k When expressed in complex form, a′ k b′ is the real part. k R' is the imaginary part. k S′ k The model; S52: The phase angles of each Fourier series of the final reconstructed fitted contour curve are calculated and expressed as follows:

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