CN116309034A - An Optimal Stitching Line Acquisition Method for Ultra-Large File Remote Sensing Images - Google Patents

Abstract

The invention provides an optimal splicing line acquisition method of an ultra-large file remote sensing image, which comprises the following steps: acquiring two ultra-large file remote sensing images with geographic coordinate references, performing equal resolution downsampling treatment on the overlapped area of the two ultra-large file remote sensing images, and establishing an undirected graph for pixel points after the resolution is reduced; searching the minimum cut of the undirected graph by using a graph cut algorithm, and obtaining a rough spelling line; generating a buffer based on the rough stitching line; obtaining a banded overlapping region of two scenery images obtained by mapping the buffer region in the original oversized file remote sensing image; establishing an undirected graph for pixel points in the banded overlapping region; and obtaining the optimal splicing line of the banded overlapping region by applying a graph cut algorithm. According to the invention, through the resolution sampling of the ultra-large file remote sensing image, the number of pixel points to be traversed in the determining process of the stitching line is reduced, so that the number of nodes and edges to be calculated in a graph cutting algorithm is met, and the nodes and edges are ensured to be global optimal solutions, thereby obtaining the optimal stitching line of the ultra-large file remote sensing image.

Description

An Optimal Stitching Line Acquisition Method for Ultra-Large File Remote Sensing Images

technical field

The invention relates to the field of mosaic of remote sensing images, in particular to a method for obtaining an optimal stitching line of remote sensing images of super-large files.

Background technique

Image stitching technology is the technology of stitching several overlapping images (which may be obtained at different times and by different sensors) into a seamless panorama or high-resolution image. In the application of remote sensing images, in order to obtain a larger field of view, better unified processing analysis and interpretation of remote sensing image information, it is necessary to stitch two or more remote sensing images with overlapping areas into an image map. Big data remote sensing images mainly refer to satellite remote sensing images. With the improvement of the resolution of satellite remote sensing images, the amount of image data is gradually increasing. Graph Cut algorithm (Graph Cut) is a very useful and popular energy optimization algorithm, which is widely used in the field of computer vision for image segmentation, stereo vision, image matting, etc. Due to the energy optimization characteristics of the graph cut algorithm, it can be applied to find the optimal stitching line between images by assigning values to the image edges, so that the spectral difference between the pixels on both sides of the image stitching line is the smallest. However, the current method of finding the optimal stitching line between two images by establishing an undirected graph and applying the graph cut algorithm is limited to the maximum number of nodes and edges that the graph cut algorithm can accommodate, which greatly limits its application to remote sensing images. Versatility. For example, if the number of pixels in the overlapping area of two large data remote sensing images is 5000*5000, and the maximum capacity that the graph cut algorithm can calculate is 1000*1000, the graph cut algorithm may not be applicable. In addition, the simple direct application of the graph cut algorithm will also cause many invalid points to be added to the calculation, greatly increasing its calculation time.

The traditional image mosaic optimization algorithm is only suitable for small data images. When it is applied to super large data remote sensing images, it usually cannot be calculated due to the large amount of data. Stitching between augmented remote sensing images.

Contents of the invention

Aiming at the deficiencies of the prior art, the present invention proposes a method for obtaining an optimal splicing line of a remote sensing image of a super-large file. This method is aimed at the problem that the calculation cannot be performed or the time is too long due to the large amount of data in the process of finding the mosaic line of the remote sensing image of the two scenes. The nested application of the graph cut algorithm finds the global optimal mosaic of the two remote sensing images. Wire. It should be pointed out that the special feature of the present invention lies in the nested search for stitching lines, which is different from the general search for stitching lines at one time or the processing of finding stitching lines in blocks. After the rough splicing line, make a buffer zone, look for finer splicing lines in the buffer zone, and finally splice and uniformly color based on the splicing lines to obtain a seamlessly spliced big data remote sensing image.

To achieve the above object, the specific technical solutions of the present invention are as follows:

A method for obtaining an optimal splicing line of an ultra-large file remote sensing image, the method comprising the following steps:

Step 1: Obtain the remote sensing image of the super-large file with geographic coordinate reference of the two scenes, and perform equal-resolution down-sampling processing on the overlapping area of the remote sensing image of the two super-large file to obtain a rough splicing line;

Described step one is specifically realized through the following sub-steps:

(1.1) Calculate the geographical overlapping area of the remote sensing images of the two super-large files;

(1.2) Calculate the number of pixels in the geographical overlapping area of the remote sensing images of the two super-large files and the resampling resolution of the image stitching;

(1.3) Perform equal-resolution down-sampling processing on the overlapping area of the remote sensing images of the two super-large files;

(1.4) Establish an undirected graph for the pixel points in the overlapped area of the remote sensing images of the two super-large files after the downsampling process;

(1.5) apply the graph cut algorithm to find the minimum cut of the undirected graph obtained in step (1.4), and obtain a rough splicing line after reducing the resolution;

Step 2: Generate a buffer zone based on the rough splicing line obtained in step 1. The buffer zone forms a band-shaped overlapping area in the original two-scene super-large file remote sensing image, and obtains the global optimal stitching line in the band-shaped overlapping area;

The second step is specifically implemented through the following sub-steps:

(2.1) Generate a buffer zone based on the rough stitching line obtained in step 1;

(2.2) Obtain the band-shaped overlapping area of the two scene super-large-file remote-sensing images formed in the original super-large-file remote-sensing image by the buffer zone that step (2.1) generates;

(2.3) Establish an undirected graph for the pixel points in the band-shaped overlapping area;

(2.4) Apply the graph cut algorithm to find the minimum cut of the undirected graph obtained in step (2.3), and obtain the optimal stitching line of the strip-shaped overlapping area, which is the global optimal stitching line.

Further, the step (1.3) includes:

Select the sampling factor n, where n>1, apply the bilinear interpolation method to down-sample the remote sensing images A and B of the two super-large files in step 1, and obtain two reduced images NA and B that are reduced in the same proportion as A and B. NB, as an image looking for rough stitching lines.

Further, the step (1.4) includes:

Determine the scope of the overlapping area on the reduced images NA, NB, and based on the pixels in the overlapping area, establish an undirected graph G 1 (V, E) with weight information on each side, the undirected graph G ₁ (V, E) The directed graph G ₁ (V, E) contains nodes V{C(x ₁ , y ₁ ), C(x ₁ , y ₂ ), ..., C(x ₁ , y _t ), ..., C(x _k , y _t )} and edge E{R(L), S(L), B(L)}; where, C(x ₁ , y ₁ ) represents the spectral value at position (x ₁ , y ₁ ), C( x ₁ , y ₂ ) represents the spectral value at (x ₁ , y ₂ ), C(x ₁ , y _t ) represents the spectral value at (x ₁ , y _t ), C(x _k , y _t ) Represents the spectral value at (x _k , y _t ); R(L) represents the edge between the same row of internal nodes, S(L) represents the edge between different rows of internal nodes, B(L) represents the most the outer edge;

The assignment rules of each node and edge in the undirected graph G ₁ (V, E) are as follows:

Edges between the same row:

R(x, y)＝|C _NA (x ₁ , y ₁ )-C _NB (x ₁ , y ₁ )|+|C _NA (x ₂ , y ₁ )-C _NB (x ₂ , y ₁ )| (1) Edges between different rows:

S(x, y)＝|C _NA (x ₁ , y ₁ )-C _NB (x ₁ , y ₁ )|+|C _NA (x ₁ , y ₂ )-C _NB (x ₁ , y ₂ )| (2)

The outermost edge of the frame:

In the formula, the value of the node is the spectral value of the corresponding pixel of the original super-large file remote sensing image: C _NA (x ₁ , y ₁ ) is the spectral value of the image NA at the position (x 1 , y ₁ ), C _NA (x ₁ , y 1 ) ₂ , y ₁ ) is the spectral value of image NA at position (x ₂ , y ₁ ), C _NA (x ₁ , y ₂ ) is the spectral value of image NA at position (x ₁ , y ₂ ), C _NA (x _p , y _q ) is the spectral value of image NA at position (x _p , y _q ); C _NB (x ₁ , y ₁ ) is the spectral value of image NB at position (x ₁ , y ₁ ), C _NB (x ₂ , y ₁ ) is the spectral value of image NB at position (x ₂ , y ₁ ), C _NB (x ₁ , y ₂ ) is the spectrum of image NB at position (x ₁ , y ₂ ) value, C _NB (x _p , y _q ) is the spectral value of image NB at position (x _p , y _q ).

Further, the step (2.1) includes:

After obtaining the rough splicing line, select the buffer radius as r, make buffers on both sides along the normal direction of the rough splicing line, and upsample the buffer to the size of the original image to obtain an image P .

Further, the step (2.2) includes:

Mapping the positions of the buffers in the image P to the original images A and B one by one, obtaining a strip-shaped overlapping area on the images A and B, and finding an optimal splicing line in the strip-shaped overlapping area;

Further, the step (2.3) includes:

An undirected graph G ₂ (V, E) is established based on the pixels in the band-shaped overlapping area, wherein the assignment rules of each node and edge are the same as the assignment rules of each node and edge in step (2.4). The beneficial effects of the present invention are:

(1) The present invention uses the nested application graph cut algorithm to find the global optimal splicing line of the two remote sensing images, avoiding the traditional splicing line search method that cannot be calculated or cannot be calculated due to the large amount of data in the splicing line search process. The problem of time-consuming is too long. Compared with the traditional method of finding stitching lines, the number of nodes for image calculation is greatly reduced, which in turn reduces the amount of calculation and improves the computational efficiency of finding stitching lines.

(2) The present invention is suitable for searching splice lines of various sensors, and has universal applicability to different sensors.

Description of drawings

Fig. 1 is the block flow diagram of the optimal splicing line acquisition method of super-large file remote sensing image of the present invention;

Fig. 2 is a schematic diagram after down-sampling of the overlapping area of two scene images in an embodiment of the present invention;

Fig. 3 is a schematic diagram of establishing multiple buffer zones based on obtained rough stitching lines in one embodiment of the present invention;

Fig. 4 (a) is the position in the buffer zone after the upsampling of the roughly spliced line obtained in the overlapping area in one of the embodiments of the present invention;

Fig. 4 (b) is the position of the globally optimal splicing line obtained in the buffer zone in the strip-shaped overlapping region in one of the embodiments of the present invention;

Figure 5(a) is a schematic panorama of the results after splicing and color uniformity in one of the embodiments of the present invention;

Fig. 5(b) is a schematic diagram after selecting the splicing result in Fig. 5(a) and partially zooming in.

Detailed ways

The present invention will be described in detail below according to the accompanying drawings and preferred embodiments, and the purpose and effect of the present invention will become clearer. The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described here are only used to explain the present invention, not to limit the present invention.

As shown in Figure 1, it is a flow chart of the optimal splicing line acquisition method of the super-large file remote sensing image of the present invention, the method comprises the following steps:

Step 1: Obtain the super-large file remote sensing image (Landsat8) with geographic coordinate reference of the two scenes, and perform equal-resolution downsampling on the overlapping area of the two images to obtain the first rough splicing line.

Step 1 is specifically implemented through the following sub-steps:

1.1. Calculate the geographical overlapping area of the remote sensing images of the two super-large files;

1.2. Calculate the number of pixels in the geographic overlapping area of the two images, and obtain 9,048,576 pixels in the overlapping area; calculate the resampling resolution of the image stitching;

1.3. Equal resolution downsampling is performed on the overlapping area of the two scene images. Specific steps are as follows:

The sampling factor is selected as 10, that is, the 10*10 pixel block on the original image is used as a pixel point in the reduced image. Apply bilinear interpolation to the original images A and B for downsampling, and obtain two reduced images NA and NB that are reduced to the same ratio as the original images A and B, so that the pixels in the geographically overlapping area of the reduced images The number of points is reduced to 62697, and the overlapping area is used as the range to find the rough stitching line.

As shown in FIG. 2 , it is a schematic diagram after down-sampling the overlapping regions of the images of the two scenes in this embodiment. In the figure, the black area is the overlapping area, the position in the downsampled image NA.

1.4. Create an undirected graph for the pixel points in the overlapping area of the two scene images after the downsampling process. Specific steps are as follows:

Determine the scope of the overlapping area on the reduced image, and based on the pixels in the overlapping area, establish an undirected graph G ₁ (V, E) with weight information on each side, which contains the node V{C(x ₁ , y ₁ ), C(x ₁ , y ₂ ), ..., C(x ₁ , y _t ), ..., C(x _k , y _t )} and the edge E{R(L), S(L) , B(L)}; where, C(x ₁ , y ₁ ) represents the spectral value at position (x ₁ , y ₁ ), and C(x ₁ , y ₂ ) represents the value at (x ₁ , y ₂ ) , C(x ₁ , y _t ) represents the spectral value at (x ₁ , y _t ), C(x _k , y _t ) represents the spectral value at (x _k , y _t ); R( L) represents the edge between the same row of internal nodes, S(L) represents the edge between different rows of internal nodes, and B(L) represents the outermost edge of the graph frame, which is used for the graph cut algorithm.

The assignment rules of each node and edge in the undirected graph G ₁ (V, E) are as follows:

Edges between the same row:

R(x, y)＝|C _NA (x ₁ , y ₁ )-C _NB (x ₁ , y ₁ )|+|C _NA (x ₂ , y ₁ )-C _NB (x ₂ , y ₁ )| (1)

Edges between different rows:

S(x, y)＝|C _NA (x ₁ , y ₁ )-C _NB (x ₁ , y ₁ )|+|C _NA (x ₁ , y ₂ )-C _NB (x ₁ , y ₂ )| (2)

The outermost edge of the frame:

In the formula, the value of the node is the spectral value of the corresponding pixel of the original image: C _NA (x1, y1) corresponds to the spectral value of the image NA at the position (x ₁ , y ₁ ), C _NA (x ₂ , y ₁ ) Corresponding to the spectral value of image NA at position (x ₂ , y ₁ ), C _NA (x ₁ , y ₂ ) corresponds to the spectral value of image NA at position (x ₁ , y ₂ ), C _NA (x _p , y _q ) is the spectral value of image NA at position (x _p , y _q ); C _NB (x ₁ , y ₁ ) corresponds to the spectral value of image NB at position (x ₁ , y ₁ ), C _NB (x ₂ , y ₁ ) corresponds to the spectral value of image NB at position (x ₂ , y ₁ ), C _NB (x ₁ , y ₂ ) corresponds to the spectral value of image NB at position (x ₁ , y ₂ ), C _NB ( x _p , y _q ) is the spectral value of the image NB at position (x _p , y _q ).

1.5. Apply the graph cut algorithm to find the minimum cut of the undirected graph obtained in step 1.4, and obtain a rough splicing line with reduced resolution. Specific steps are as follows:

All edges in the undirected graph are assigned a non-negative weight W _e , called the cost. The graph cut algorithm can disconnect the edges in an undirected graph to form a set of two disconnected edges. Such a set of edges is called a "cut". If a "cut" has the smallest sum of all costs of its edges, then this is called a "minimum cut". In this embodiment, the graph cut algorithm is specifically graph cut. The minimum cut in the undirected graph can be obtained by applying the maximum flow/minimum cut algorithm in the graph cut algorithm. This cut divides the image into two parts, then this dividing line is the energy optimal dividing line for shrinking the images NA and NB, which is Rough stitching lines.

Step 2: Generate a buffer zone based on the rough stitching line obtained in step 1. The buffer zone forms a band-like overlapping area in the original two-scene images, and obtains the global optimal stitching line in the band-like overlapping area.

Step 2 is specifically implemented through the following sub-steps:

2.1. Generate a buffer based on the rough stitching line obtained in step 2. Specific steps are as follows:

After obtaining the rough splicing line of the reduced image, select the radius of the buffer zone r to be 17, and make a buffer along the normal direction of the splicing line to both sides, and obtain a strip buffer in the reduced image NA. The element value is assigned as 1, and the upsampling is performed to expand to the original image size, and the image P is obtained.

As shown in FIG. 3 , it is a schematic diagram of the buffer zone established in the reduced image according to the obtained rough splicing line. The buffer area in the figure is represented by the middle gray strip. The area on the upper left of the buffer is the area where the image NA should be selected as the pixel value, and the area on the lower right of the buffer is the area where the image NB should be selected as the pixel value. area.

2.2. Obtain the band-shaped overlapping area of the two scene images formed by the buffer generated in step 2.1 in the original remote sensing image of the super-large file. Specific steps are as follows:

Map the positions of buffers in image P to original images A and B one by one, and obtain the range of striped buffers on images A and B, considering that there are some buffer positions in image p after upsampling (at the beginning and end of splicing line ) pixels beyond the overlapping area of the original image, in this embodiment, after the image p is mapped to the original image, corresponding searches are performed to find points on the edge that do not belong to the overlapping area of the original image and then deleted to obtain the band-shaped overlapping area of the two images . Find the optimal stitching line within the band overlapping region.

2.3. Create an undirected graph for the pixel points in the band overlapping area;

Establish an undirected graph G ₂ (V,E) based on the pixels in the band-shaped overlapping area obtained in step 2.2, where the assignment rules of each node and edge are the same as those of each node and edge in step 2.4.

2.4. Use the graph cut algorithm to find the minimum cut of the undirected graph G ₂ (V,E) obtained in step 2.3. For specific steps, refer to step 2.5 to obtain the optimal stitching line of the band-shaped overlapping area, which is the global optimal stitching line.

Figure 4(a) shows the corresponding position of the rough stitching line in the buffer zone; Figure 4(b) shows the corresponding position of the global optimal stitching line found in the buffer zone in the band-shaped overlapping area. It can be seen in the figure that the rough stitching line obtained by the first search is rougher than the global optimal stitching line obtained by the second search. In this embodiment, the rough stitching line obtained for the first time is divided into 10*10 pixels on the original image, and the global optimal stitching line obtained in the second time is divided into each pixel on the original image. segmentation.

Based on the global optimal stitching line obtained in step 2, the original two large data remote sensing images are uniformly colored and stitched to obtain a seamlessly stitched big data remote sensing image.

Figure 5(a) shows the gray scale panorama after splicing and uniform coloring of two large data remote sensing images; Figure 5(b) is a schematic diagram of the partially enlarged selected splicing results. It can be seen from the figure that there is no obvious color difference at the splicing point of the two images, and the splicing point basically does not pass through the prominent objects.

In summary, judging from the results of the two splicing line searches, large data remote sensing images can be searched for splicing lines through the optimal splicing line acquisition method for super-large file remote sensing images of the present invention. Different from the traditional one-time graph-cut algorithm to find splicing lines, the present invention greatly reduces the number of graph nodes in the graph-cut algorithm through nested application of the graph-cut algorithm. It should be pointed out that the present invention is suitable for finding splice lines of various sensors, and has universal applicability to different sensors.

Those of ordinary skill in the art can understand that the above description is only a preferred example of the invention, and is not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, for those skilled in the art, it can still be understood. The technical solutions described in the foregoing examples are modified, or some of the technical features are equivalently replaced. All modifications, equivalent replacements, etc. within the spirit and principles of the invention shall be included in the scope of protection of the invention.

Claims (6)

1. The optimal spelling line acquisition method of the ultra-large file remote sensing image is characterized by comprising the following steps of:

step one: acquiring two ultra-large file remote sensing images with geographic coordinate references, and performing equal resolution downsampling on the overlapped area of the two ultra-large file remote sensing images to acquire a rough splicing line;

the first step is realized by the following substeps:

(1.1) calculating the geographic overlapping area of the remote sensing images of the two-scene oversized file;

(1.2) calculating the quantity of pixels in the geographic overlapping area of the remote sensing images of the two-scene oversized file and the resampling resolution of image stitching;

(1.3) performing equal resolution downsampling treatment on the overlapping area of the remote sensing images of the two-scene oversized file;

(1.4) establishing an undirected graph for pixel points of an overlapping area of the remote sensing images of the two-scene oversized file after the downsampling treatment;

(1.5) searching the minimum cut of the undirected graph obtained in the step (1.4) by using a graph cut algorithm to obtain a rough split joint line after resolution reduction;

step two: generating a buffer area based on the rough spelling line obtained in the first step, wherein the buffer area forms a banded overlapping area in the remote sensing image of the original two-scene oversized file, and the global optimal spelling line is obtained in the banded overlapping area;

the second step is specifically realized by the following substeps:

(2.1) generating a buffer based on the rough stitching line obtained in step one;

(2.2) obtaining a banded overlapping region of the two-scene oversized file remote sensing image formed in the original oversized file remote sensing image by the buffer area generated in the step (2.1);

(2.3) establishing an undirected graph for pixel points in the banded overlapping region;

and (2.4) searching the minimum cut of the undirected graph obtained in the step (2.3) by using a graph cut algorithm, and obtaining the optimal splicing line of the banded overlapping region, namely the global optimal splicing line.

2. The method for obtaining an optimal stitching line of a remote sensing image of an oversized document according to claim 1, wherein the step (1.3) includes:

sampling coefficients n, n >1 are selected, and the two-scene oversized file remote sensing image A, B in the first step is subjected to downsampling by a bilinear interpolation method to obtain two reduced images NA and NB which are reduced by the same proportion compared with A, B, and the two reduced images NA and NB are used as images for searching rough spelling lines.

3. The method for obtaining an optimal stitching line of a remote sensing image of an oversized document according to claim 2, wherein the step (1.4) includes:

determining the range of the overlapping area on the contracted images NA and NB, and establishing an undirected graph G with weight information on each side based on the pixel points in the overlapping area ₁ (V, E) the undirected graph G ₁ (V, E) includes node V { C (x) ₁ ，y ₁ )，C(x ₁ ，y ₂ )，…，C(x ₁ ，y _t )，…，C(x _k ，y _t ) -and edges E { R (L), S (L), B (L) }; wherein C (x) ₁ ，y ₁ ) Is expressed in the position (x ₁ ，y ₁ ) Spectral values at C (x ₁ ，y ₂ ) Represented at (x ₁ ，y ₂ ) Spectral values at C (x ₁ ，y _t ) Represented at (x ₁ ，y _t ) Spectral values at C (x _k ，y _t ) Represented at (x _k ，y _t ) Spectral values at; r (L) represents edges between the same rows of internal nodes, S (L) represents edges between different rows of internal nodes, and B (L) represents the outermost edge of the frame;

undirected graph G ₁ The assignment rules for each node and edge in (V, E) are as follows:

edges between the same rows:

R(x，y)＝|C _NA (x ₁ ，y ₁ )-C _NB (x ₁ ，y ₁ )|+|C _NA (x ₂ ，y ₁ )-C _NB (x ₂ ，y ₁ )| (1)

edges between different rows:

S(x，y)＝|C _NA (x ₁ ，y ₁ )-C _NB (x ₁ ，y ₁ )|+|C _NA (x ₁ ，y ₂ )-C _NB (x ₁ ，y ₂ )| (2)

the outermost edges of the frame:

in the formula, the value of the node is the spectrum value of the pixel point corresponding to the remote sensing image of the original oversized file: c (C) _NA (x ₁ ，y ₁ ) For image NA at position (x ₁ ，y ₁ ) Spectral value at C _NA (x ₂ ，y ₁ ) For image NA at position (x ₂ ，y ₁ ) Spectral value at C _NA (x ₁ ，y ₂ ) For image NA at position (x ₁ ，y ₂ ) Spectral value at C _NA (x _p ，y _q ) For image NA at position (x _p ，y _q ) Spectral values at; c (C) _NB (x ₁ ，y ₁ ) For image NB in position (x ₁ ，y ₁ ) Spectral value at C _NB (x ₂ ，y ₁ ) For image NB in position (x ₂ ，y ₁ ) Spectral value at C _NB (x ₁ ，y ₂ ) For image NB in position (x ₁ ，y ₂ ) Spectral value at C _NB (x _p ，y _q ) For image NB in position (x _p ，y _q ) Spectral values at.

4. The method for obtaining an optimal stitching line for a remote sensing image of an oversized document according to claim 3, wherein the step (2.1) comprises:

after the rough splicing line is obtained, selecting a buffer area with the radius r, taking buffer areas towards two sides along the normal direction of the rough splicing line, and up-sampling and expanding the buffer areas to the original image size to obtain an image P.

5. The method for obtaining an optimal stitching line for a remote sensing image of an oversized document according to claim 4, wherein the step (2.2) includes:

the positions of the buffers in the image P are mapped onto the original image A, B one by one, a banded overlapping region is obtained on the image A, B, and an optimal stitching line is found in the banded overlapping region.

6. The method for obtaining an optimal stitching line for a remote sensing image of an oversized document according to claim 5, wherein the step (2.3) includes:

establishing an undirected graph G based on pixel points in the banded overlapping region ₂ (V, E) wherein the assignment rules for each node and edge are the same as the assignment rules for each node and edge in step (2.4).

Images