A pore modeling method

Abstract

The present invention discloses a method for modeling a cellular cavity. The method comprises the following steps: using a three-dimensional motion detection system to scan a cellular cavity on a sample to be tested at different imaging depths from top to bottom to obtain a plurality of scanned images; performing distortion correction and brightness compensation on the acquired scanned images; calculating an optimal plane position and an optimal depth value for a pixel point in the scanned image, and obtaining an optimal coordinate of the current pixel point using the optimal plane position and the optimal depth value; combining the optimal coordinate with the color information of the pixel to generate three-dimensional point cloud data, generating a grid file using the three-dimensional point cloud data, and further reconstructing a three-dimensional model of the cellular cavity using the grid file; the present invention performs non-contact three-dimensional scanning and reconstruction of the cellular cavity based on a depth scanning method, and can conveniently, rapidly and robustly generate a high-precision three-dimensional cellular cavity model, and can ensure the modeling accuracy of the generated three-dimensional cellular cavity model.

Description

Technical Field

[0001] The invention belongs to the technical field of non-contact structure three-dimensional reconstruction and relates to a socket modeling method. Background Art

[0002] In recent years, with the rapid development of visual sensing and image processing technologies, visual measurement technology has matured and gradually become a relatively popular non-contact measurement method. Compared with contact measurement, machine vision offers non-contact, real-time, flexibility, and accuracy that traditional inspection methods cannot match.

[0003] Machine vision technology is widely used across various industries due to its non-contact, high-precision, and high-efficiency capabilities. Early machine vision measurement primarily relied on two-dimensional technology. While it could measure some dimensions of a cavity, it was unable to measure its most critical dimension, the depth. Three-dimensional measurement, however, addresses this limitation and can measure all dimensions of the cavity. Recovering 3D depth from 2D images has been a research hotspot. Compared to 2D images, 3D images are more capable of reconstructing the 3D shape of the cavity.

[0004] However, existing cellular 3D visual reconstruction technologies, such as laser scanning and structured light methods, have certain limitations. The laser scanning method has low precision, is easily affected by highly reflective surfaces, and loses surface 3D information. It is suitable for large-scene 3D creation. When the structured light method encounters a surface with a large curvature or a hole (steep slope) during projection, the transmitted stripes are not easy to be detected and the surface 3D information is also lost. The present invention overcomes the above shortcomings at the same time. Compared with the structured light method, the present invention can restore 3D reconstruction even when the steep slope angle reaches 80°. Compared with the laser scanning method, the present invention has lower requirements for surface reflectivity and higher reconstruction accuracy, and is suitable for microscopic 3D reconstruction. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for modeling the fossa, which performs non-contact three-dimensional scanning and reconstruction of the fossa based on depth scanning, can conveniently, quickly and robustly generate a high-precision fossa three-dimensional model, and can ensure the modeling accuracy of the generated fossa three-dimensional model.

[0006] The present invention is achieved through the following technical solutions:

[0007] A cellular modeling method, implemented using a three-dimensional motion detection system, comprises the following steps:

[0008] Step 1: Scan the pores on the sample to be tested at different imaging depths from top to bottom using a three-dimensional motion detection system to obtain a plurality of scanned images;

[0009] Step 2: performing distortion correction and brightness compensation on the scanned image acquired in step 1;

[0010] Step 3: Calculate the optimal plane position and optimal depth value for the pixel points in the scanned image, and use the optimal plane position and optimal depth value to obtain the optimal coordinates of the current pixel point;

[0011] Step 4: Combine the optimal coordinates with the color information of the pixels to generate three-dimensional point cloud data, use the three-dimensional point cloud data to generate a grid file, and further use the grid file to reconstruct the three-dimensional model of the socket.

[0012] In order to better implement the present invention, further, the step 3 includes:

[0013] Step 3.1, using the sum of the pixel values ​​of all pixels in the range area on the scanned image as an index, solving the optimal plane position, wherein the optimal plane position minimizes the sum of the absolute differences between the pixel values ​​of all pixels in the range area;

[0014] Step 3.2, find the focus of the scanned image passing through the optimal plane position, and calculate the optimal depth value through the focus;

[0015] Step 3.3: Based on the optimal plane position and the optimal depth value, the optimal coordinates of the pixel points are calculated using the intrinsic parameter matrix and the extrinsic parameter matrix of the three-dimensional motion detection system.

[0016] In order to better implement the present invention, further, the step of solving the optimal plane position in step 3 is specifically as follows:

[0017] Step 3.11: Select any scanned image as the reference image and the remaining scanned images as the transformed images. Select any pixel point on the reference image as the pixel center and establish an accumulation window with the pixel center as the reference.

[0018] Step 3.12: Establish a cost function based on the coordinates of the pixel center and the position description parameters of the plane where any transformed image is located;

[0019] Step 3.13, accumulating the absolute difference between the pixel values ​​of all pixels within the window range on the reference image and the transformed image through the cost function;

[0020] Step 3.14: Calculate the coordinates of the pixel center corresponding to the minimum value of the cost function and the position description parameters of the plane where the pixel center is located, and then calculate the optimal plane position.

[0021] In order to better implement the present invention, further, the cost function in step 3.12 is specifically:

[0022]

[0023] Where: C(x,y,∏k) represents the cost function; x represents the x-coordinate of the pixel center; y represents the y-coordinate of the pixel center; ∏k represents the position description parameter of the k-th transformed image; N represents the number of scanned images; k represents the plane where the k-th transformed image is located; i represents the width coordinate of the accumulation window; j represents the length coordinate of the accumulation window; W represents the range of the accumulation window; I ref Represents the pixel value of the reference image within the accumulation window; represents the gain ratio between the reference image and the kth transformed image; Represents the pixel value of the k-th transformed image within the accumulation window.

[0024] In order to better implement the present invention, further, the formula for solving the optimal plane position in step 3.14 is as follows:

[0025] ∏(x,y)=argminC(x,y,∏k);

[0026] Among them: ∏(x,y) represents the minimization function; argminC(x,y,∏k) represents the values ​​of x, y, and ∏k corresponding to the minimum value of the cost function C(x,y,∏k).

[0027] In order to better implement the present invention, further, the formula for calculating the optimal depth value in step 3.3 is:

[0028]

[0029] Where: z m (x,y) represents the optimal depth value of the pixel center; K ref represents the calibration matrix of the 3D motion detection system; n m Represents the normal of the plane where the pixel center is located; d m Indicates the optimal plane position where the pixel center is located.

[0030] In order to better implement the present invention, further, the brightness compensation specifically includes the following steps:

[0031] Step A1: normalize the grayscale range of the scanned image so that the grayscale range of the scanned image is normalized from 0-255 to 0-1;

[0032] Step A2: Calculate the histogram function of the normalized scanned image, and use the histogram function to complete image mapping of the scanned image at different grayscale levels;

[0033] Step A3: Expand the grayscale value range of the mapped image from 0-1 to 0-255.

[0034] In order to better implement the present invention, further, the histogram function is:

[0035]

[0036] Where: D represents the number of pixels in the scanned image; i represents the row coordinate of the pixel in the scanned image; j represents the column coordinate of the pixel in the scanned image; u(i, j) represents the value of the pixel with row coordinate i and column coordinate j; F(t) represents the histogram function; t represents the function's independent variable.

[0037] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0038] The present invention performs three-dimensional scanning of the fossa based on a depth scanning method, realizes non-contact high-precision three-dimensional reconstruction of the fossa, and can conveniently, quickly and robustly generate high-precision three-dimensional models of the fossa. It is applicable to different types of fossa, has strong versatility and practicality, and ensures the accuracy of the fossa three-dimensional modeling. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A schematic diagram of the process of the pore modeling method;

[0040] Figure 2 Schematic diagram of image mapping on each shooting plane;

[0041] Figure 3 Schematic diagram of several captured images;

[0042] Figure 4 Schematic diagram of three-dimensional modeling of the socket. DETAILED DESCRIPTION

[0043] The following detailed description is illustrative and is intended to provide further explanation of the present invention. Unless otherwise specified, all technical and scientific terms used in the present invention have the same meaning as commonly understood by those skilled in the art to which the present invention belongs.

[0044] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless otherwise expressly specified in the present invention, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0045] For the convenience of description, if the words "up", "down", "left" and "right" appear in the present invention, they only indicate that they are consistent with the up, down, left and right directions of the drawings themselves, and do not limit the structure. They are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or component referred to must have a specific orientation, be constructed and operate in a specific orientation. Therefore, they should not be understood as limiting the present invention.

[0046] Explanation of terms: The terms "install", "connect", "connect", "fix" and the like in the present invention should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral whole; it can be a mechanical connection, an electrical connection, a direct connection, or an indirect connection through an intermediate medium, an internal connection between two elements, or an interaction relationship between two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0047] Example 1:

[0048] A pore modeling method of this embodiment, such as Figure 1-Figure 4 As shown, the 3D motion detection system is used to realize the 3D motion detection. The 3D motion detection system includes a visual detection camera, which is used to photograph the pores on the sample to be tested. The following steps are included:

[0049] Step 1: Scan the pores on the sample to be tested at different imaging depths from top to bottom using a three-dimensional motion detection system to obtain a plurality of scanned images;

[0050] Step 2: performing distortion correction and brightness compensation on the scanned image acquired in step 1;

[0051] Step 3: Calculate the optimal plane position and optimal depth value for the pixel points in the scanned image, and use the optimal plane position and optimal depth value to obtain the optimal coordinates of the current pixel point;

[0052] Step 4: Combine the optimal coordinates with the color information of the pixels to generate three-dimensional point cloud data, use the three-dimensional point cloud data to generate a grid file, and further use the grid file to reconstruct the three-dimensional model of the socket.

[0053] The brightness compensation in step 2 specifically includes the following steps:

[0054] Step A1: normalize the grayscale range of the scanned image so that the grayscale range of the scanned image is normalized from 0-255 to 0-1;

[0055] Step A2: Calculate the histogram function of the normalized scanned image, and use the histogram function to complete image mapping of the scanned image at different grayscale levels;

[0056] Step A3: Expand the grayscale value range of the mapped image from 0-1 to 0-255.

[0057] The histogram function is:

[0058]

[0059] Where: D represents the number of pixels in the scanned image; i represents the row coordinate of the pixel in the scanned image; j represents the column coordinate of the pixel in the scanned image; u(i, j) represents the value of the pixel with row coordinate i and column coordinate j; F(t) represents the histogram function; and t represents the function variable.

[0060] The step 3 comprises:

[0061] Step 3.1, using the sum of the pixel values ​​of all pixels in the range area on the scanned image as an index, solving the optimal plane position, wherein the optimal plane position minimizes the sum of the absolute differences between the pixel values ​​of all pixels in the range area;

[0062] Step 3.2, find the focus of the scanned image passing through the optimal plane position, and calculate the optimal depth value through the focus;

[0063] Step 3.3: Based on the optimal plane position and the optimal depth value, the optimal coordinates of the pixel points are calculated using the intrinsic parameter matrix and the extrinsic parameter matrix of the three-dimensional motion detection system.

[0064] The steps for solving the optimal plane position in step 3 are specifically as follows:

[0065] Step 3.11: Select any scanned image as the reference image and the remaining scanned images as the transformed images. Select any pixel point on the reference image as the pixel center and establish an accumulation window with the pixel center as the reference.

[0066] Step 3.12: Establish a cost function based on the coordinates of the pixel center and the position description parameters of the plane where any transformed image is located;

[0067] Step 3.13, accumulating the absolute difference between the pixel values ​​of all pixels within the window range on the reference image and the transformed image through the cost function;

[0068] Step 3.14: Calculate the coordinates of the pixel center corresponding to the minimum value of the cost function and the position description parameters of the plane where the pixel center is located, and then calculate the optimal plane position.

[0069] The cost function in step 3.12 is specifically:

[0070]

[0071] Where: C(x,y,∏k) represents the cost function; x represents the x-coordinate of the pixel center; y represents the y-coordinate of the pixel center; ∏k represents the position description parameter of the k-th transformed image; N represents the number of scanned images; k represents the plane where the k-th transformed image is located; i represents the width coordinate of the accumulation window; j represents the length coordinate of the accumulation window; W represents the range of the accumulation window; I ref Represents the pixel value of the reference image within the accumulation window; represents the gain ratio between the reference image and the kth transformed image; Represents the pixel value of the k-th transformed image within the accumulation window.

[0072] The formula for solving the optimal plane position in step 3.14 is as follows:

[0073] ∏(x,y)=argminC(x,y,∏k);

[0074] Among them: argminC(x,y,∏k) is the value of x,y,∏k corresponding to the minimum value of C(x,y,∏k).

[0075] The formula for calculating the optimal depth value in step 3.3 is:

[0076]

[0077] Where: z m (x,y) represents the optimal depth value of the pixel center; K ref represents the calibration matrix of the 3D motion detection system; n m Represents the normal of the plane where the pixel center is located; d m Indicates the optimal plane position where the pixel center is located.

[0078] The rest of this embodiment is the same as that of embodiment 1, so it will not be described again.

[0079] Example 2:

[0080] A socket modeling method of this embodiment is improved on the basis of embodiment 1, and a three-dimensional motion detection system is built, and the lens of the visual detection camera in the three-dimensional motion detection system is adjusted to ensure that the sample to be tested is at the edge of the depth of field of the lens. The three-axis motion component of the three-dimensional motion detection system drives the visual detection camera to shoot and scan the sockets on the sample to be tested from top to bottom at different imaging depths to obtain a number of scanned images. In the process of shooting and scanning from top to bottom, the visual detection camera needs to ensure that the upper and lower surfaces of the sockets are shot and scanned, and the visual detection camera obtains the following images: Figure 3 Several scanned images are shown.

[0081] For each scanned image, brightness equalization strategy is used for brightness compensation. That is, the grayscale range of the scanned image is first normalized from 0-255 to 0-1, and a histogram function is established to realize the mapping of the scanned image. The histogram function is as follows:

[0082]

[0083] Where: D represents the number of pixels in the scanned image; i represents the row coordinate of the pixel in the scanned image; j represents the column coordinate of the pixel in the scanned image; u(i, j) represents the value of the pixel with row coordinate i and column coordinate j; F(t) represents the histogram function; and t represents the function variable.

[0084] Based on the histogram function, given the number of grayscale levels G, it is defined as:

[0085]

[0086] Where: G represents the number of gray levels; L represents the coefficient, L = 0, 1, ... G; F -1 (t) represents the reciprocal of F(t);

[0087] For any interval [x k , x k+1 ], construct a linear transformation T k (x), such that:

[0088]

[0089] Where: m k represents the slope, and

[0090] According to the above formula, the interval [x k , x k+1 ] linear transformation to [y k ,y k+1 ]. During the transformation process, if m k If it is too small, the histogram function will be compressed and the problem of large brightness differences caused by uneven lighting cannot be corrected; when m k When m is too large, it will produce a lot of noise on the original scanned image. Therefore, in order to obtain a good brightness compensation result, m k Should be kept within a fixed range, i.e. s min ≤m k ≤s max , where s min Indicates the minimum value of the empirical slope, s max Indicates the maximum empirical slope.

[0091] When the grayscale number G = 2, the dark image is mapped to the grayscale range of 0 to 128, and the bright image is mapped to the grayscale range of 129 to 255. When the grayscale number G = 3, the scanned image is mapped in three segments. Different grayscale numbers are selected for different materials of the sample to be tested.

[0092] For example, for a scanned image of a sample to be tested made of aluminum alloy, the grayscale gradation number G is selected to be 3; for example, for a scanned image of a sample to be tested made of composite material, the grayscale gradation number G is selected to be 5.

[0093] After completing the grayscale mapping of the scanned image, the grayscale value range of the mapped image is expanded from 0-1 to 0-255. This operation can correct the problem of excessive brightness differences in the scanned image caused by uneven lighting.

[0094] After completing the brightness compensation of the scanned image, the next step is to estimate the pore depth. Suppose the projection matrix of different visual inspection cameras is:

[0095]

[0096] Where: P k is the projection matrix of the visual detection camera; Q k is the calibration matrix of the visual inspection camera, used to eliminate distortion; is the transpose of the rotation matrix of the visual detection camera; C k is the translation matrix of the vision detection camera.

[0097] The reference plane is established with the origin of the coordinate system of the visual inspection camera, and the projection matrix of the visual inspection camera on the reference plane is P ref =Q ref [I 3×3 0], where P ref is the projection matrix of the reference plane, Q ref is the calibration matrix of the reference plane, I 3×3 is the identity matrix. Then the set of shooting planes at different depths from the reference plane can be defined as:

[0098]

[0099] Where: ∏k represents the kth shooting plane; n k represents the normalized normal vector of the kth shooting plane, n k =[0,0,1];d k represents the distance from the kth shooting plane to the reference plane; T represents the transposition operation.

[0100] The transformation of the image from the shooting plane to the reference plane belongs to planar mapping and can be described using the homography matrix:

[0101]

[0102] Where: Q k The calibration matrix of the visual inspection camera at the kth shooting plane; is the transpose of the rotation matrix of the visual detection camera in the kth shooting plane; C k is the translation matrix of the visual detection camera in the kth shooting plane; is the inverse of the calibration matrix of the reference plane.

[0103] The mapping relationship between the mapped pixel points on the reference plane and the pixel points on other shooting planes can be obtained through the homography matrix:

[0104]

[0105] Where: x0 represents the x-coordinate of the pixel point on the reference plane; y0 represents the y-coordinate of the pixel point on the reference plane; Indicates the x-coordinate of the pixel point mapped on the shooting plane; Indicates the y coordinate of the pixel point mapped on the shooting plane; Indicates the z-coordinate of the pixel mapped on the shooting plane.

[0106] The absolute difference between pixels on different planes is used to measure distance similarity. However, the pixel value at a pixel is usually sensitive. In order to reduce the sensitivity, it is necessary to combine the values ​​of the pixels in the range area around the current pixel for calculation.

[0107] Select any pixel point on the reference image as the pixel center, establish an accumulation window with the pixel center as the reference, and establish the cost function based on the coordinates of the pixel center and the position description parameters of the plane where any transformed image is located:

[0108]

[0109] Where: C(x,y,∏k) represents the cost function; x represents the x-coordinate of the pixel center; y represents the y-coordinate of the pixel center; ∏k represents the position description parameter of the k-th transformed image; N represents the number of scanned images; k represents the plane where the k-th transformed image is located; i represents the width coordinate of the accumulation window; j represents the length coordinate of the accumulation window; W represents the range of the accumulation window; I ref Represents the pixel value of the reference image within the accumulation window; represents the gain ratio between the reference image and the kth transformed image; Represents the pixel value of the k-th transformed image within the accumulation window.

[0110] After obtaining the cost function of the pixel point, the depth value of each pixel point can be extracted. Before extracting the depth value, the optimal plane position d needs to be selected for each pixel point in the reference image. m , the minimization function about the optimal plane position is as follows:

[0111] ∏(x,y)=argminC(x,y,∏k);

[0112] Where: ∏(x,y) represents the minimization function; argminC(x,y,∏k) represents the value of x,y,∏k corresponding to the minimum value of the cost function C(x,y,∏k); (x,y) represents the coordinates of the pixel point.

[0113] The depth value is calculated by finding the focus of the light passing through the plane ∏k and the center of the pixel. After the depth value is calculated, the three-dimensional topographic coordinates (x, y, z) of the surface of the sample to be tested are calculated by combining the intrinsic parameter matrix and the extrinsic parameter matrix of the visual inspection camera. The calculation formula is as follows:

[0114]

[0115] in: T p Represents the external parameter translation matrix of the visual detection camera; R p Represents the intrinsic rotation matrix of the visual detection camera; z m (x,y) represents the optimal depth value of the pixel center; K ref represents the calibration matrix of the 3D motion detection system; n m Represents the normal of the plane where the pixel center is located; d m Indicates the optimal plane position where the pixel center is located.

[0116] The rest of this embodiment is the same as that of embodiment 1, so it will not be described again.

[0117] The above are only preferred embodiments of the present invention and do not limit the present invention in any form. Any simple modifications or equivalent changes made to the above embodiments based on the technical essence of the present invention fall within the scope of protection of the present invention.

Claims

1. A cellular modeling method, implemented using a three-dimensional motion detection system, characterized in that: The following steps are involved: Step 1: Scan the pores on the sample to be tested at different imaging depths from top to bottom using a three-dimensional motion detection system to obtain a plurality of scanned images; Step 2: performing distortion correction and brightness compensation on the scanned image acquired in step 1; Step 3: Calculate the optimal plane position and optimal depth value for the pixel points in the scanned image, and use the optimal plane position and optimal depth value to obtain the optimal coordinates of the current pixel point; specifically, the following steps are involved: Step 3.1, using the sum of the pixel values ​​of all pixels in the range area on the scanned image as an index, solving the optimal plane position, wherein the optimal plane position minimizes the sum of the absolute differences between the pixel values ​​of all pixels in the range area; Step 3.2, find the focus of the scanned image passing through the optimal plane position, and calculate the optimal depth value through the focus; Step 3.3, based on the optimal plane position and the optimal depth value, the optimal coordinates of the pixel points are calculated using the intrinsic parameter matrix and the extrinsic parameter matrix of the 3D motion detection system; Step 4: Combine the optimal coordinates with the color information of the pixels to generate three-dimensional point cloud data, use the three-dimensional point cloud data to generate a grid file, and further use the grid file to reconstruct the three-dimensional model of the socket.

2. A pore modeling method according to claim 1, characterized in that: The steps for solving the optimal plane position in step 3 are specifically as follows: Step 3.11: Select any scanned image as the reference image and the remaining scanned images as the transformed images. Select any pixel point on the reference image as the pixel center and establish an accumulation window with the pixel center as the reference. Step 3.12: Establish a cost function based on the coordinates of the pixel center and the position description parameters of the plane where any transformed image is located; Step 3.13, accumulating the absolute difference between the pixel values ​​of all pixels within the window range on the reference image and the transformed image through the cost function; Step 3.14: Calculate the coordinates of the pixel center corresponding to the minimum value of the cost function and the position description parameters of the plane where the pixel center is located, and then calculate the optimal plane position.

3. A pore modeling method according to claim 2, characterized in that: The cost function in step 3.12 is specifically: Where: C(x,y,∏k) represents the cost function; x represents the x-coordinate of the pixel center; y represents the y-coordinate of the pixel center; ∏k represents the position description parameter of the k-th transformed image; N represents the number of scanned images; k represents the plane where the k-th transformed image is located; i represents the width coordinate of the accumulation window; j represents the length coordinate of the accumulation window; W represents the range of the accumulation window; I ref Represents the pixel value of the reference image within the accumulation window; represents the gain ratio between the reference image and the kth transformed image; Represents the pixel value of the k-th transformed image within the accumulation window.

4. A pore modeling method according to claim 3, characterized in that: The formula for solving the optimal plane position in step 3.14 is as follows: ∏(x,y)=argminC(x,y,∏k); Among them: argminC(x,y,∏k) is the value of x,y,∏k corresponding to the minimum value of C(x,y,∏k).

5. A pore modeling method according to claim 4, characterized in that: The formula for calculating the optimal depth value in step 3.3 is: Where: z m (x,y) represents the optimal depth value of the pixel center; K ref represents the calibration matrix of the 3D motion detection system; n m Represents the normal of the plane where the pixel center is located; d m Indicates the optimal plane position where the pixel center is located.

6. A pore modeling method according to any one of claims 1 to 5, characterized in that: The brightness compensation specifically includes the following steps: Step A1: normalize the grayscale range of the scanned image so that the grayscale range of the scanned image is normalized from 0-255 to 0-1; Step A2: Calculate the histogram function of the normalized scanned image, and use the histogram function to complete image mapping of the scanned image at different grayscale levels; Step A3: Expand the grayscale value range of the mapped image from 0-1 to 0-255.

7. A pore modeling method according to claim 6, characterized in that: The histogram function is: Where: D represents the number of pixels in the scanned image; i represents the row coordinate of the pixel in the scanned image; j represents the column coordinate of the pixel in the scanned image; u(i, j) represents the value of the pixel with row coordinate i and column coordinate j; F(t) represents the histogram function; t represents the function's independent variable.

Images