I have a two part question related to Boeing's OSMnx Tutorial 8 - Street network centrality analysis. Firstly, I have a knowledge question regarding edge closeness centrality and then a code-based question regarding edge betweenness centrality. My purpose is to calculate edge closeness and betweenness centrality around stations in various locations.
1. Edge Closeness Centrality
The following code works well for me:
# edge closeness centrality: convert graph to a line graph so edges become nodes and vice versa
edge_centrality = nx.closeness_centrality(nx.line_graph(G))
# list of edge values for the original graph
ev = [edge_centrality[edge + (0,)] for edge in G.edges()]
# color scale converted to list of colors for graph edges
norm = colors.Normalize(vmin=min(ev)*0.8, vmax=max(ev))
cmap = cm.ScalarMappable(norm=norm, cmap=cm.inferno)
ec = [cmap.to_rgba(cl) for cl in ev]
Question: Can anyone explain why in the normalisation code the mininum edge value is multiplied by 0.8 and the the maximum value is set at the maximum edge value? I am not too familiar with the literature so any advice would be appreciated.
2. Edge Betweenness Centrality
I am trying to calculate edge betweenness centrality in a similar way to the above code for edge closeness centrality on the same graph in the example. I have tried this and get the following:
# edge betweenness centrality
edge_bcentrality = nx.edge_betweenness_centrality(G)
# list of edge values for the orginal graph
ev1 = [edge_bcentrality[edge + (0,)] for edge in G.edges()]
# color scale converted to list of colors for graph edges
norm = colors.Normalize(vmin=min(ev1)*0.8, vmax=max(ev1))
cmap = cm.ScalarMappable(norm=norm, cmap=cm.inferno)
ec = [cmap.to_rgba(cl) for cl in ev1]
# color the edges in the original graph with betweeness centralities in the line graph
fig, ax = ox.plot_graph(G, bgcolor='k', axis_off=True, node_size=0, node_color='w', node_edgecolor='gray', node_zorder=2,
edge_color=ec, edge_linewidth=1.5, edge_alpha=1)
---------------------------------------------------------------------------
KeyError Traceback (most recent call last)
<ipython-input-14-6ee1d322067c> in <module>()
1 # list of edge values for the orginal graph
----> 2 ev1 = [edge_bcentrality[edge + (0,)] for edge in G.edges()]
3
4 # color scale converted to list of colors for graph edges
5 norm = colors.Normalize(vmin=min(ev)*0.8, vmax=max(ev))
KeyError: (53090322, 53082634, 0)
If anyone advice on the the best way to calculate edge betweenness centrality I would be greatly appreciated, as I am still a novice. Also, it would be appreciated if someone could share the best way to proceed with normalisation.
Thank you for your time,
BC
I applied this code and it works for me. Hope it helps.
#calculate betweenness
betweenness = nx.edge_betweenness(G=G, normalized=False)
# iterate over edges
edges = []
for i in betweenness.items():
i = i[0] + (0,)
edges.append(i)
for i,j in zip(edges,betweenness.keys()):
betweenness[i] = betweenness[j]
del betweenness[j]
# color scale converted to list of colors for graph edges
norm = colors.Normalize(vmin=min(betweenness.values())*0.8, vmax=max(betweenness.values()))
cmap = cm.ScalarMappable(norm=norm, cmap=cm.viridis)
ec = [cmap.to_rgba(cl) for cl in betweenness.values()]
# color the edges in the original graph with betweeness centralities in the line graph
fig, ax = ox.plot_graph(G, bgcolor='w', axis_off=True, node_size=0, node_color='w', node_edgecolor='gray', node_zorder=2,
edge_color=ec, edge_linewidth=1.5, edge_alpha=1)
fig.show()
Related
I have gone through the code below and would like to know how can I count the outlier points and inlier points after using RANSAC? could you point to a good code how it can be done?
Second question, which feature matching algorithm is better: BFMatcher.knnMatch() with Test ratio or bf = cv.BFMatcher(cv.NORM_HAMMING, crossCheck=True) with shortest distance? any reference for this comparison?
**# BFMatcher with default params
bf = cv.BFMatcher()
matches = bf.knnMatch(des1, des2, k=2)
# Apply ratio test
good_matches = []
for m,n in matches:
if m.distance < 0.75*n.distance:
good_matches.append([m])
# Draw matches
img3=cv.drawMatchesKnn(img1,kp1,img2,kp2,good_matches,None,flags=cv.DrawMatchesFlags_NOT_DRAW_SINGLE_POINTS)
cv.imwrite('matches.jpg', img3)
# Select good matched keypoints
ref_matched_kpts = np.float32([kp1[m[0].queryIdx].pt for m in good_matches])
sensed_matched_kpts = np.float32([kp2[m[0].trainIdx].pt for m in good_matches])
# Compute homography
H, status = cv.findHomography(sensed_matched_kpts, ref_matched_kpts, cv.RANSAC,5.0)**
Count number of outliers and inliers
# number of detected outliers: len(status) - np.sum(status)
# number of detected inliers: np.sum(status)
# Inlier Ratio, number of inlier/number of matches: float(np.sum(status)) / float(len(status))
Feature Matching Algorithm
I would say that if you are using the sparse feature-based algorithm (SIFT or SURF), BFMatcher.knnMatch() with Test ratio is preferred. While the bf = cv.BFMatcher(cv.NORM_HAMMING, crossCheck=True) is used for binary-based algorithm (ORB, FAST, etc). My suggestion would be try both algorithms on your project to investigate which one is better.
I am trying to use osmnx to find distances between a origin point (lat/lon) and nearest infrastructure, such as railways, water or parks.
1) I get the entire graph from an area with network_type='walk'.
2) Get the needed infrastructure, e.g. railway for that same area.
3) Compose the two graphs into one.
4) Find the nearest node from origin point in the original graph.
5) Find the nearest node from the origin point in the infrastructure graph
6) Find the shortest route length between the two nodes.
If you run the example below, you will see that it is missing 20% of the data because it cannot find a route between the nodes. For infrastructure='way["leisure"~"park"]' or infrastructure='way["natural"~"wood"]' this is even worse, with 80-90% of nodes not being connected.
Minimal reproducible example:
import osmnx as ox
import networkx as nx
bbox = [55.5267243, 55.8467243, 12.4100724, 12.7300724]
g = ox.graph_from_bbox(bbox[0], bbox[1], bbox[2], bbox[3],
retain_all=True,
truncate_by_edge=True,
simplify=False,
network_type='walk')
points = [(55.6790884456018, 12.568493971506154),
(55.6790884456018, 12.568493971506154),
(55.6867418740291, 12.58232314016353),
(55.6867418740291, 12.58232314016353),
(55.6867418740291, 12.58232314016353),
(55.67119624894504, 12.587201455313153),
(55.677406927839506, 12.57651997656002),
(55.6856574907879, 12.590500429002823),
(55.6856574907879, 12.590500429002823),
(55.68465359365924, 12.585474365063224),
(55.68153666806675, 12.582594757267945),
(55.67796979175, 12.583111746311117),
(55.68767346629932, 12.610040871066179),
(55.6830855237578, 12.575431380892427),
(55.68746749645466, 12.589488615911913),
(55.67514254640597, 12.574308210656602),
(55.67812748568291, 12.568454119053886),
(55.67812748568291, 12.568454119053886),
(55.6701733527419, 12.58989203029166),
(55.677700136266616, 12.582800629527789)]
railway = ox.graph_from_bbox(bbox[0], bbox[1], bbox[2], bbox[3],
retain_all=True,
truncate_by_edge=True,
simplify=False,
network_type='walk',
infrastructure='way["railway"]')
g_rail = nx.compose(g, railway)
l_rail = []
for point in points:
nearest_node = ox.get_nearest_node(g, point)
rail_nn = ox.get_nearest_node(railway, point)
if nx.has_path(g_rail, nearest_node, rail_nn):
l_rail.append(nx.shortest_path_length(g_rail, nearest_node, rail_nn, weight='length'))
else:
l_rail.append(-1)
There are 2 things that caught my attention.
OSMNX documentation specifies ox.graph_from_bbox parameters be given in the order of north, south, east, west (https://osmnx.readthedocs.io/en/stable/osmnx.html). I mention this because when I tried to run your code, I was getting empty graphs.
The parameter 'retain_all = True' is the key as you may already know. When set to true, it retains all nodes in the graph, even if they are not connected to any of the other nodes in the graph. This happens primarily due to the incompleteness of OpenStreetMap which contains voluntarily contributed geographic information. I suggest you set 'retain_all = False' meaning your graph now contains only the connected nodes. In this way, you get a complete list without any -1.
I hope this helps.
g = ox.graph_from_bbox(bbox[1], bbox[0], bbox[3], bbox[2],
retain_all=False,
truncate_by_edge=True,
simplify=False,
network_type='walk')
railway = ox.graph_from_bbox(bbox[1], bbox[0], bbox[3], bbox[2],
retain_all=False,
truncate_by_edge=True,
simplify=False,
network_type='walk',
infrastructure='way["railway"]')
g_rail = nx.compose(g, railway)
l_rail = []
for point in points:
nearest_node = ox.get_nearest_node(g, point)
rail_nn = ox.get_nearest_node(railway, point)
if nx.has_path(g_rail, nearest_node, rail_nn):
l_rail.append(nx.shortest_path_length(g_rail, nearest_node, rail_nn, weight='length'))
else:
l_rail.append(-1)
print(l_rail)
Out[60]:
[7182.002999999995,
7182.002999999995,
5060.562000000002,
5060.562000000002,
5060.562000000002,
6380.099999999999,
7127.429999999996,
4707.014000000001,
4707.014000000001,
5324.400000000003,
6153.250000000002,
6821.213000000002,
8336.863999999998,
6471.305,
4509.258000000001,
5673.294999999996,
6964.213999999994,
6964.213999999994,
6213.673,
6860.350000000001]
I am trying to find a simple algorithm to find the correspondence between two sets of 2D points (registration). One set contains the template of an object I'd like to find and the second set mostly contains points that belong to the object of interest, but it can be noisy (missing points as well as additional points that do not belong to the object). Both sets contain roughly 40 points in 2D. The second set is a homography of the first set (translation, rotation and perspective transform).
I am interested in finding an algorithm for registration in order to get the point-correspondence. I will be using this information to find the transform between the two sets (all of this in OpenCV).
Can anyone suggest an algorithm, library or small bit of code that could do the job? As I'm dealing with small sets, it does not have to be super optimized. Currently, my approach is a RANSAC-like algorithm:
Choose 4 random points from set 1 and from set 2.
Compute transform matrix H (using openCV getPerspective())
Warp 1st set of points using H and test how they aligned to the 2nd set of points
Repeat 1-3 N times and choose best transform according to some metric (e.g. sum of squares).
Any ideas? Thanks for your input.
With python you can use Open3D librarry, wich is very easy to install in Anaconda. To your purpose ICP should work fine, so we'll use the classical ICP, wich minimizes point-to-point distances between closest points in every iteration. Here is the code to register 2 clouds:
import numpy as np
import open3d as o3d
# Parameters:
initial_T = np.identity(4) # Initial transformation for ICP
distance = 0.1 # The threshold distance used for searching correspondences
(closest points between clouds). I'm setting it to 10 cm.
# Read your point clouds:
source = o3d.io.read_point_cloud("point_cloud_1.xyz")
target = o3d.io.read_point_cloud("point_cloud_0.xyz")
# Define the type of registration:
type = o3d.pipelines.registration.TransformationEstimationPointToPoint(False)
# "False" means rigid transformation, scale = 1
# Define the number of iterations (I'll use 100):
iterations = o3d.pipelines.registration.ICPConvergenceCriteria(max_iteration = 100)
# Do the registration:
result = o3d.pipelines.registration.registration_icp(source, target, distance, initial_T, type, iterations)
result is a class with 4 things: the transformation T(4x4), 2 metrict (rmse and fitness) and the set of correspondences.
To acess the transformation:
I used it a lot with 3D clouds obteined from Terrestrial Laser Scanners (TLS) and from robots (Velodiny LIDAR).
With MATLAB:
We'll use the point-to-point ICP again, because your data is 2D. Here is a minimum example with two point clouds random generated inside a triangle shape:
% Triangle vértices:
V1 = [-20, 0; -10, 10; 0, 0];
V2 = [-10, 0; 0, 10; 10, 0];
% Create clouds and show pair:
points = 5000
N1 = criar_nuvem_triangulo(V1,points);
N2 = criar_nuvem_triangulo(V2,points);
pcshowpair(N1,N2)
% Registrate pair N1->N2 and show:
[T,N1_tranformed,RMSE]=pcregistericp(N1,N2,'Metric','pointToPoint','MaxIterations',100);
pcshowpair(N1_tranformed,N2)
"criar_nuvem_triangulo" is a function to generate random point clouds inside a triangle:
function [cloud] = criar_nuvem_triangulo(V,N)
% Function wich creates 2D point clouds in triangle format using random
% points
% Parameters: V = Triangle vertices (3x2 Matrix)| N = Number of points
t = sqrt(rand(N, 1));
s = rand(N, 1);
P = (1 - t) * V(1, :) + bsxfun(#times, ((1 - s) * V(2, :) + s * V(3, :)), t);
points = [P,zeros(N,1)];
cloud = pointCloud(points)
end
results:
You may just use cv::findHomography. It is a RANSAC-based approach around cv::getPerspectiveTransform.
auto H = cv::findHomography(srcPoints, dstPoints, CV_RANSAC,3);
Where 3 is the reprojection threshold.
One traditional approach to solve your problem is by using point-set registration method when you don't have matching pair information. Point set registration is similar to method you are talking about.You can find matlab implementation here.
Thanks
I am trying to extract Rotation matrix and Translation vector from the essential matrix.
<pre><code>
SVD svd(E,SVD::MODIFY_A);
Mat svd_u = svd.u;
Mat svd_vt = svd.vt;
Mat svd_w = svd.w;
Matx33d W(0,-1,0,
1,0,0,
0,0,1);
Mat_<double> R = svd_u * Mat(W).t() * svd_vt; //or svd_u * Mat(W) * svd_vt;
Mat_<double> t = svd_u.col(2); //or -svd_u.col(2)
</code></pre>
However, when I am using R and T (e.g. to obtain rectified images), the result does not seem to be right(black images or some obviously wrong outputs), even so I used different combination of possible R and T.
I suspected to E. According to the text books, my calculation is right if we have:
E = U*diag(1, 1, 0)*Vt
In my case svd.w which is supposed to be diag(1, 1, 0) [at least in term of a scale], is not so. Here is an example of my output:
svd.w = [21.47903827647813; 20.28555196246256; 5.167099204708699e-010]
Also, two of the eigenvalues of E should be equal and the third one should be zero. In the same case the result is:
eigenvalues of E = 0.0000 + 0.0000i, 0.3143 +20.8610i, 0.3143 -20.8610i
As you see, two of them are complex conjugates.
Now, the questions are:
Is the decomposition of E and calculation of R and T done in a right way?
If the calculation is right, why the internal rules of essential matrix are not satisfied by the results?
If everything about E, R, and T is fine, why the rectified images obtained by them are not correct?
I get E from fundamental matrix, which I suppose to be right. I draw epipolar lines on both the left and right images and they all pass through the related points (for all the 16 points used to calculate the fundamental matrix).
Any help would be appreciated.
Thanks!
I see two issues.
First, discounting the negligible value of the third diagonal term, your E is about 6% off the ideal one: err_percent = (21.48 - 20.29) / 20.29 * 100 . Sounds small, but translated in terms of pixel error it may be an altogether larger amount.
So I'd start by replacing E with the ideal one after SVD decomposition: Er = U * diag(1,1,0) * Vt.
Second, the textbook decomposition admits 4 solutions, only one of which is physically plausible (i.e. with 3D points in front of the camera). You may be hitting one of non-physical ones. See http://en.wikipedia.org/wiki/Essential_matrix#Determining_R_and_t_from_E .
I want to understand how to compute big-O for a dense versus sparse graph.
"Algorithms in a nutshell" says that for sparse graph, O(E) is O(V) and for dense graph O(E) is closer to O(V^2). Does anyone know how is that derived?
Assuming the graph is simple - at the worst case every node can be connected to all |V|-1 other nodes, resulting in [in not directed graph:] |E| = (|V|-1) + (|V| -2) + ... + 1 <= |V| * (|V| -1) = O(|V|^2). And in directed graph: |E| = |V| * (|V|-1) = O(|V|^2).
A good example for a dense graph is a clique - which have all the edges.
For sparsed graph - we assume the number of edges connected to each vertex is bounded by a constant. Let this constant be k. Thus: |E| <= k* |V|, and we get |E| = O(|V|)
A good example for a sparsed graph is the internet, where every URL is a node and every link is an edge.
Note that if the graph is not simple, you cannot bound |E| with any function of |V|.
It's not derived, it's a definition. In a fully connected (directed) graph with self-loops, the number of edges |E| = |V|² so the definition of a dense graph is reasonable. The definition of a sparse graph is one where O(|E|) = O(|V|), so there's a constant maximum number of edges per vertex.
Note that if the number of edges is much smaller, e.g. O(lg |V|), then it's still O(|V|) as well. One could imagine a "semi-sparse" class of graphs with |E| = O(|V| lg |V|) or something like that, but I personally have never encountered such a class in practice.