A digital-based in-depth design system for old residential areas

Abstract

The present invention belongs to the field of three-dimensional modeling, and discloses a digital-based in-depth design system for old residential areas, including a drone, a height calculation module, and a drone control module; the drone includes a first shooting device and a second shooting device on the same horizontal plane; the first shooting device is used to obtain a color image of a building; the second shooting device is used to obtain modeling data of the building; the height calculation module is used to calculate the next scanning height according to the color image obtained at the previous scanning height; the drone control module is used to control the drone to fly around the building at the next scanning height, so that the first shooting device and the second shooting device can respectively obtain the color image and modeling data of the building at the next scanning height. The present invention can shorten the length of the total flight path as much as possible while ensuring the accuracy of the modeling data obtained at two adjacent scanning heights when they are fused.

Description

Technical Field

[0001] The present invention relates to the field of three-dimensional modeling, and in particular to a digital-based in-depth design system for old residential areas. Background Art

[0002] When renovating old residential communities, it is usually necessary to redesign the exterior of the buildings. However, since old residential communities usually only have two-dimensional floor plans and no three-dimensional models of the buildings in the community, this brings difficulties to the design. It is usually necessary to re-model the buildings in the old communities to obtain the corresponding three-dimensional models, and then redesign the buildings based on the three-dimensional models.

[0003] In the prior art, three-dimensional modeling is usually carried out using a three-dimensional scanning device. However, the effective working distance of a three-dimensional scanning device is usually relatively small, and a single scan cannot obtain a complete three-dimensional model of a building. Therefore, when it is necessary to model a building in an old community, it is necessary to use a drone carrying a three-dimensional scanning device to scan around the building at different heights, and then merge the modeling data (such as point cloud data) obtained by scanning multiple areas to obtain a complete three-dimensional model of the building.

[0004] The existing scanning process is usually: first plan the flight route, and then the pilot controls the drone to fly according to the flight route. When planning the flight route, the size between two adjacent scanning heights is usually the same, which is smaller than the maximum scanning length of the scanning device in the longitudinal direction, so that the modeling data obtained by scanning the building between two adjacent heights has a certain overlap, so that in the subsequent fusion process, the modeling data of adjacent scanning heights can be accurately spliced ​​and fused based on the modeling data of the overlapping parts.

[0005] However, this scanning method does not take into account the complexity of the details of the building's facade. If the overlapping area is too large, the total flight path will be too long, affecting the efficiency of scanning the building. If the overlapping area is too small, the more detailed and complex areas of the building's facade may not be accurately integrated, making the established three-dimensional model inaccurate, thus affecting subsequent design. Summary of the invention

[0006] The purpose of the present invention is to disclose a digital-based in-depth design system for old residential communities, and to solve the problem of how to reasonably determine the difference between two adjacent scanning heights when using drones to perform three-dimensional modeling of buildings in old residential communities, so as to shorten the length of the total flight path as much as possible while ensuring the accuracy of the modeling data obtained from the two adjacent scanning heights when they are fused.

[0007] In order to achieve the above object, the present invention provides the following technical solutions:

[0008] The present invention provides a digital-based in-depth design system for old residential areas, including a drone, a height calculation module and a drone control module;

[0009] The drone includes a first shooting device and a second shooting device on the same horizontal plane; the first shooting device is used to obtain a color image of a building; the second shooting device is used to obtain modeling data of the building;

[0010] The height calculation module is used to calculate the next scanning height based on the color image obtained from the previous scanning height;

[0011] The drone control module is used to control the drone to fly around the building at the next scanning height, so that the first shooting device and the second shooting device can respectively obtain the color image and modeling data of the building at the next scanning height.

[0012] Optionally, the drone control module is also used to control the drone to land at the take-off point when the next scanning height exceeds the height of the building.

[0013] Optionally, a data fusion module is also included;

[0014] The data fusion module is used to fuse the modeling data obtained from different scanning heights to obtain complete modeling data.

[0015] Optionally, a pre-processing module is also included;

[0016] The preprocessing module is used to preprocess the complete modeling data to obtain preprocessed modeling data.

[0017] Optionally, a modeling module is also included;

[0018] The modeling module is used to model the preprocessed modeling data to obtain a three-dimensional model of the building.

[0019] Optionally, a design module is also included;

[0020] The design module is used to design based on the three-dimensional model of the building and obtain the design plan of the building.

[0021] Optionally, the vertical field of view of the first camera device is 1.5 times the vertical field of view of the second camera device.

[0022] Optionally, a horizontal field of view angle of the first camera device is the same as a horizontal field of view angle of the second camera device.

[0023] Optionally, control the drone to fly around the building at the next scanning altitude, including:

[0024] When flying, the same horizontal distance from the building is maintained, the scanning height is maintained at the same, and the flying speed is maintained at the same. After each fixed length of flight, the drone hovers so that the first camera can obtain a color image of the building while hovering and the second camera can obtain modeling data of the building while hovering.

[0025] Optionally, at the same scanning height, there is an overlap between the modeling data obtained from two adjacent hoverings.

[0026] Beneficial effects:

[0027] Compared with the prior art, the present invention calculates the next scanning height based on the color image obtained at the previous scanning height. Therefore, the difference between two adjacent scanning heights can adaptively change with the complexity of the details of the building's facade. When the complexity of the details of the building's facade is low, the difference between the two adjacent scanning heights is made as large as possible, thereby reducing the length of the total flight path of the drone and improving the scanning efficiency. When the complexity of the details of the building's facade is high, the difference between the two adjacent scanning heights is reduced to ensure that there is a large overlapping area between the modeling data obtained at adjacent heights, so as to facilitate the subsequent data fusion to obtain sufficiently accurate fusion results. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for describing the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying creative work.

[0029] Figure 1 A schematic diagram of the digital-based in-depth design system for old residential areas of the present invention.

[0030] Figure 2 Another schematic diagram of the digital-based in-depth design system for old residential areas of the present invention. DETAILED DESCRIPTION

[0031] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope parameters of the protection of the present invention.

[0032] like Figure 1In one embodiment shown, the present invention provides a digital-based in-depth design system for old residential areas, including a drone, a height calculation module, and a drone control module;

[0033] The drone includes a first shooting device and a second shooting device on the same horizontal plane; the first shooting device is used to obtain a color image of a building; the second shooting device is used to obtain modeling data of the building;

[0034] The height calculation module is used to calculate the next scanning height based on the color image obtained from the previous scanning height;

[0035] The drone control module is used to control the drone to fly around the building at the next scanning height, so that the first shooting device and the second shooting device can respectively obtain the color image and modeling data of the building at the next scanning height.

[0036] Compared with the prior art, the present invention calculates the next scanning height based on the color image obtained at the previous scanning height. Therefore, the difference between two adjacent scanning heights can adaptively change with the complexity of the details of the building's facade. When the complexity of the details of the building's facade is low, the difference between the two adjacent scanning heights is made as large as possible, thereby reducing the length of the total flight path of the drone and improving the scanning efficiency. When the complexity of the details of the building's facade is high, the difference between the two adjacent scanning heights is reduced to ensure that there is a large overlapping area between the modeling data obtained at adjacent heights, so as to facilitate the subsequent data fusion to obtain sufficiently accurate fusion results.

[0037] Optionally, the first camera and the second camera have the same shooting angle, and the bottom of the first camera and the bottom of the second camera are in the same horizontal plane. The first camera and the second camera both acquire data along the horizontal direction.

[0038] Optionally, the second shooting device is a depth camera.

[0039] Optionally, the drone control module is also used to control the drone to land at the take-off point when the next scanning height exceeds the height of the building.

[0040] Specifically, for example, when the next scanning height is 50 meters and the height of the building is 45 meters, the drone is controlled to land at the take-off point. Because at this time, if the drone is shooting at a height of 50 meters, the second shooting device it carries will not be able to capture new features of the building's facade because it has already been shot at the previous scanning height.

[0041] Optionally, the modeling data is point cloud data. By acquiring the point cloud data of the building, three-dimensional reconstruction of the building can be achieved.

[0042] Optional, such as Figure 2 As shown, it also includes a data fusion module;

[0043] The data fusion module is used to fuse the modeling data obtained from different scanning heights to obtain complete modeling data.

[0044] Specifically, when the modeling data is point cloud data, the modeling data obtained at two adjacent scanning heights are spliced ​​to obtain complete modeling data.

[0045] Optionally, a pre-processing module is also included;

[0046] The preprocessing module is used to preprocess the complete modeling data to obtain preprocessed modeling data.

[0047] Specifically, the process of preprocessing the modeling data includes removing noise from the modeling data.

[0048] Optionally, a modeling module is also included;

[0049] The modeling module is used to model the preprocessed modeling data to obtain a three-dimensional model of the building.

[0050] Specifically, the process of modeling the modeling data includes:

[0051] The preprocessed modeling data is modeled using algorithms such as surface reconstruction method, model fitting method and voxel method to obtain a three-dimensional model of the building.

[0052] Optionally, a design module is also included;

[0053] The design module is used to design based on the three-dimensional model of the building and obtain the design plan of the building.

[0054] Specifically, during the design, the designed 3D model of the building part can be directly added to the 3D model of the building to realize the redesign of the building. The 3D model of the building part includes windows, balconies, walls, etc.

[0055] Optionally, the vertical field of view of the first camera device is 1.5 times the vertical field of view of the second camera device.

[0056] Specifically, the field of view angle is also called the field of view in optical engineering. The size of the field of view angle determines the field of view of the optical instrument. The vertical field of view angle refers to the size of the field of view angle of the shooting device in the vertical direction.

[0057] By setting the vertical field of view of the first camera device to 1.5 times the vertical field of view of the second camera device, the first camera device can capture the area of ​​the building facade that the second camera device needs to scan at the next scanning height at the previous scanning height. The next scanning height can be calculated based on the complexity of the details of the overlapping part between the area of ​​the facade that the second camera device needs to capture at two adjacent scanning heights, so as to avoid maintaining the same difference between two adjacent scanning heights, resulting in low scanning efficiency or affecting the final modeling accuracy.

[0058] When the field of view angle is 1.5 times, in the color image obtained by the first camera device, the actual length of the image in the vertical direction is twice the actual length of the point cloud data obtained by the second camera device in the vertical direction. In this way, only the upper quarter of the area in the color image needs to be calculated to obtain the next scanning height.

[0059] Optionally, calculating the next scanning height based on the color image obtained at the previous scanning height includes:

[0060] Calculate the next scan height using the following formula:

[0061] scahei z+1 =scahei z +adphei z

[0062] scahei z+1 and Scahei z denote the z+1th and zth scanning heights, adphei z represents the height increase calculated based on the color image captured at the z-th scanning height;

[0063] adphei z The calculation process includes:

[0064] The first step is to calculate the height increase of each color image captured at the z-th scanning height:

[0065]

[0066] adphei(calpho z ) represents the height increase calculated based on the color image calphoz captured at the z-th scanning height, graydif z Represents the complex coefficient of grayscale, calpix z Represents the image calpho z The set of pixels in grayvaluei Represents calpix z The grayscale value of pixel i in numcalpix z Represents calpix z The total number of pixels in grayvalue z,max Represents calpix z The maximum grayscale value of all pixels contained, epxnum z Indicates calpho z The total number of pixels that meet the recognition requirements, b1 and b2 represent the first weight and the second weight respectively; basehei represents the maximum value of the height increase, calpho z The number of rows in the color image captured at the z-th scanning height is in the interval The image is composed of pixels in ; nrow represents the total number of rows in the color image, and the nrowth row is the last row in the color image from bottom to top. Indicates rounding down;

[0067] In the second step, the average of all height increases calculated in the first step is taken as adphei z .

[0068] Specifically, the next scanning height is obtained by adding the height increase calculated based on the color image obtained from the previous scanning height to the previous scanning height. Therefore, the increase in scanning height is closely related to the complexity of the building's facade, so that the adaptive change of the difference between the scanning heights when two adjacent scans at different heights are achieved, which can shorten the total flight mileage of the drone as much as possible and improve the scanning efficiency while ensuring the accuracy of the obtained modeling data after fusion.

[0069] When calculating the height increase, the present invention obtains it by comprehensive calculation from the complexity of the grayscale and the complexity of the contour of the color image (that is, the total number of pixel points that meet the recognition requirements). When the complexity of the grayscale is smaller and the complexity of the contour is smaller, it means that at the next scanning height, the complexity of the building's facade is lower. Correspondingly, the greater the height increase, thereby reducing the number of circles made by the drone during aerial photography and improving the scanning efficiency. Conversely, the smaller the height increase, the modeling accuracy of the modeling data obtained by the scan is guaranteed.

[0070] Optionally, the first weight and the second weight are 0.4 and 0.6 respectively.

[0071] Optionally, the maximum value of the height increase is calculated as:

[0072]

[0073] bsD represents the safe shooting distance of the UAV, and θ represents the vertical field of view angle of the second shooting device.

[0074] Specifically, the safe shooting distance of the drone is 6 meters.

[0075] Optionally, the process of determining whether a pixel point meets the recognition requirements includes:

[0076] For calpix z For pixel i in , calculate the recognition coefficient of pixel i;

[0077] If the recognition coefficient is greater than the set recognition coefficient threshold, the pixel i meets the recognition requirements.

[0078] Specifically, whether the recognition requirements are met is judged based on the recognition coefficient. The larger the recognition coefficient, the greater the probability of meeting the recognition requirements, thereby identifying the pixels that can represent the contour. The more pixels are identified, the more complex the building's facade is at the next scanning height.

[0079] Optionally, the calculation process of the recognition coefficient of pixel i includes:

[0080] Calculate the first recognition value:

[0081] idecoef i，1 =grayvalue(x-1,y-1)+2grayvalue(x,y-1)+grayvalue(x+1,y-1)-

[0082] grayvalue(x-1,y+1)-2grayvalue(x,y+1)-grayvalue(x+1,y+1)

[0083] idecoef i，1 represents the first identification value of pixel i, grayvalue(x-1,y-1), grayvalue(x,y-1), grayvalue(x+1,y-1), grayvalue(x-1,y+1), grayvalue(x,y+1), and grayvalue(x+1,y+1) represent the grayscale values ​​of the pixels at coordinates (x-1,y-1), (x,y-1), (x+1,y-1), (x-1,y+1), (x,y+1), and (x+1,y+1), respectively, and (x,y) represents the coordinates of pixel i;

[0084] Calculate the second identification value:

[0085] idecoef i，2=grayvalue(x+1,y+1)+2grayvalue(x+1,y)+grayvalue(x+1,y-1)-

[0086] grayvalue(x-1,y+1)-2grayvalue(x-1,y)-grayvalue(x-1,y-1)

[0087] idecoef i，2 represents the second identification value of pixel i, grayvalue(x+1,y) and grayvalue(x-1,y) represent the grayscale values ​​of the pixel at coordinates (x+1,y) and (x-1,y) respectively;

[0088] Calculate the recognition coefficient:

[0089]

[0090] idecoeft i Represents the recognition coefficient of pixel i.

[0091] Specifically, the recognition coefficient is calculated based on the pixel points in the 8-neighborhood of the pixel point i. Moreover, by calculating the first recognition value and the second recognition value respectively, the possibility of the pixel point belonging to the image contour in the vertical direction and the horizontal direction is expressed. The larger the recognition coefficient, the greater the possibility that the pixel point i belongs to the pixel point of the image contour.

[0092] Optionally, the recognition coefficient threshold is set to 0.9.

[0093] Optionally, get calpho z The process of calculating the grayscale value of a pixel in includes:

[0094] About Calpho z Perform grayscale processing to obtain a grayscale image G;

[0095] For calpho z The grayscale value of the pixel with coordinates (x, y) in the grayscale image G is the pixel value of the pixel with coordinates (x, y) in the grayscale image G.

[0096] Optional, for calpho z Perform grayscale processing to obtain a grayscale image G, including:

[0097] About Calpho z Perform grayscale processing to obtain a processed image;

[0098] The processed image is subjected to competitive filtering to obtain a grayscale image G.

[0099] Specifically, traditional filtering algorithms are generally based on a single filtering algorithm for filtering. However, such an algorithm does not take the type of noise into consideration when filtering, and a single filtering algorithm can only filter one type of pixel. If the filtering results of two different types of filtering algorithms are directly weighted and fused, more details in the image will be lost.

[0100] Optionally, competitive filtering is performed on the processed image to obtain a grayscale image G, including:

[0101] For the pixel point p in the processed image,

[0102] Using a linear filtering algorithm and a nonlinear filtering algorithm to filter p respectively, obtaining a linear filtering grayscale value and a nonlinear filtering grayscale value;

[0103] Based on the linear filtering gray value and the nonlinear filtering gray value, respectively calculate the linear filtering effective coefficient and the nonlinear filtering effective coefficient of the pixel point p;

[0104] If the effective coefficient of the linear filter is greater than the effective coefficient of the nonlinear filter, the linear filter grayscale value is used as the grayscale value of the grayscale point p in the grayscale image G; otherwise, the nonlinear filter grayscale value is used as the grayscale value of the grayscale point p in the grayscale image G.

[0105] Specifically, the present invention uses competitive filtering processing to compare the size relationship between the effective coefficients of linear filtering and the effective coefficients of nonlinear filtering, thereby retaining filtering results with good filtering effects, thereby being able to retain more details while filtering.

[0106] Optionally, the calculation process of the linear filter effective coefficient and the nonlinear filter effective coefficient includes:

[0107] The linear filter effective coefficient is calculated using the following formula:

[0108]

[0109] liefltcoef p Represents the effective coefficient of linear filtering of pixel p, liegray p Represents the linear filtered gray value of pixel p, grayvalue j represents the gray value of pixel j, neip represents the set of pixels in a 5×5 square area centered on p; num gray represents the total number of pixels with gray value gray in neip, β1 and β2 represent the pixel quantity weight and pixel gray distribution weight respectively;

[0110] The nonlinear filter effective coefficient is calculated using the following formula:

[0111]

[0112] nliefltcoef p Represents the effective coefficient of nonlinear filtering of pixel p, nliegray p represents the nonlinear filtering gray value of pixel p, and dli represents the result obtained by normalizing the variables in the brackets.

[0113] Specifically, when calculating the effective filtering coefficient, the present invention, in addition to calculating the difference between the filtering result and the original image, also calculates the distribution of the grayscale values ​​of the pixels in the square area after filtering. The greater the difference between the filtering result and the original image, the more complex the distribution of the grayscale values ​​of the pixels in the square area, which means that the filtering effect is better and more details are retained.

[0114] Optionally, the pixel quantity weight and the pixel grayscale distribution weight are 0.5 and 0.5 respectively.

[0115] Optionally, a horizontal field of view angle of the first camera device is the same as a horizontal field of view angle of the second camera device.

[0116] Optionally, control the drone to fly around the building at the next scanning altitude, including:

[0117] When flying, the same horizontal distance from the building is maintained, the scanning height is maintained at the same, and the flying speed is maintained at the same. After each fixed length of flight, the drone hovers so that the first camera can obtain a color image of the building while hovering and the second camera can obtain modeling data of the building while hovering.

[0118] Specifically, the above horizontal distance is the safe shooting distance of the drone.

[0119] Optionally, at the same scanning height, there is an overlap between the modeling data obtained from two adjacent hoverings.

[0120] Specifically, there are also overlapping parts in the horizontal direction, so that subsequent data fusion can be achieved.

[0121] In the several embodiments provided by the present invention, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are only schematic, for example, the division of units is only a logical function division, and there may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of devices or units, which can be electrical, mechanical or other forms.

[0122] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0123] In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0124] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope parameters of the technical solutions of the embodiments of the present invention.

Claims

1. The digital-based in-depth design system for old residential areas is characterized by: It includes a drone, a height calculation module and a drone control module; The drone includes a first shooting device and a second shooting device on the same horizontal plane; the first shooting device is used to obtain a color image of a building; the second shooting device is used to obtain modeling data of the building; The height calculation module is used to calculate the next scanning height based on the color image obtained from the previous scanning height; The drone control module is used to control the drone to fly around the building at the next scanning height, so that the first shooting device and the second shooting device can respectively obtain the color image and modeling data of the building at the next scanning height; Calculate the next scanning height based on the color image obtained from the previous scanning height, including: Calculate the next scan height using the following formula: scaly z+1 =scahei z +adphei z scahei z+1 and Scahei z denote the z+1th and zth scanning heights, adphei z represents the height increase calculated based on the color image captured at the z-th scanning height; adphei z The calculation process includes: The first step is to calculate the height increase of each color image captured at the z-th scanning height: adphei(calpho z ) represents the height increase calculated based on the color image calphoz captured at the z-th scanning height, graydif z Represents the complex coefficient of grayscale, calpix z Represents the image calpho z The set of pixels in grayvalue i Represents calpix z The grayscale value of pixel i in numcalpix z Represents calpix z The total number of pixels in grayvalue z,max Represents calpix z The maximum grayscale value of all pixels contained, epxnum z Indicates calpho z The total number of pixels that meet the recognition requirements, b1 and b2 represent the first weight and the second weight respectively; basehei represents the maximum value of the height increase, calpho z The number of rows in the color image captured at the z-th scanning height is in the interval The image is composed of pixels in ; nrow represents the total number of rows in the color image, and the nrowth row is the last row in the color image from bottom to top. Indicates rounding down; In the second step, the average of all height increases calculated in the first step is taken as adphei z .

2. The digital-based in-depth design system for old residential areas according to claim 1 is characterized in that: The drone control module is also used to control the drone to land at the take-off point when the next scanning height exceeds the height of the building.

3. The digital-based in-depth design system for old residential areas according to claim 1 is characterized in that: It also includes a data fusion module; The data fusion module is used to fuse the modeling data obtained from different scanning heights to obtain complete modeling data.

4. The digital-based in-depth design system for old residential areas according to claim 3 is characterized in that: It also includes a pre-processing module; The preprocessing module is used to preprocess the complete modeling data to obtain preprocessed modeling data.

5. The digital-based in-depth design system for old residential areas according to claim 4 is characterized in that: Also includes modeling modules; The modeling module is used to model the preprocessed modeling data to obtain a three-dimensional model of the building.

6. The digital-based in-depth design system for old residential areas according to claim 5 is characterized in that: Also includes design modules; The design module is used to design based on the three-dimensional model of the building and obtain the design plan of the building.

7. The digital-based old residential area in-depth design system according to claim 1 is characterized in that: The vertical field of view of the first camera device is 1.5 times the vertical field of view of the second camera device.

8. The digital-based in-depth design system for old residential areas according to claim 1 is characterized in that: The horizontal field of view angle of the first camera device is the same as the horizontal field of view angle of the second camera device.

9. The digital-based in-depth design system for old residential areas according to claim 1 is characterized in that: Control the drone to fly around the building at the next scanning height, including: When flying, the same horizontal distance from the building is maintained, the scanning height is maintained at the same, and the flying speed is maintained at the same. After each fixed length of flight, the drone hovers so that the first camera can obtain a color image of the building while hovering and the second camera can obtain modeling data of the building while hovering.

10. The digital-based in-depth design system for old residential areas according to claim 9 is characterized in that: At the same scanning height, there is overlap between the modeling data obtained from two adjacent hoverings.

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