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figure-5.py
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import numpy as np
def rasterize(V):
"""
Polygon rasterization (scanlines).
Given an ordered set of vertices V describing a polygon,
return all the (integer) points inside the polygon.
See http://alienryderflex.com/polygon_fill/
Parameters:
-----------
V : (n,2) shaped numpy array
Polygon vertices
"""
n = len(V)
X, Y = V[:, 0], V[:, 1]
ymin = int(np.ceil(Y.min()))
ymax = int(np.floor(Y.max()))
#ymin = int(np.round(Y.min()))
#ymax = int(np.round(Y.max()))
P = []
for y in range(ymin, ymax+1):
segments = []
for i in range(n):
index1, index2 = (i-1) % n, i
y1, y2 = Y[index1], Y[index2]
x1, x2 = X[index1], X[index2]
if y1 > y2:
y1, y2 = y2, y1
x1, x2 = x2, x1
elif y1 == y2:
continue
if (y1 <= y < y2) or (y == ymax and y1 < y <= y2):
segments.append((y-y1) * (x2-x1) / (y2-y1) + x1)
segments.sort()
for i in range(0, (2*(len(segments)//2)), 2):
x1 = int(np.ceil(segments[i]))
x2 = int(np.floor(segments[i+1]))
# x1 = int(np.round(segments[i]))
# x2 = int(np.round(segments[i+1]))
P.extend([[x, y] for x in range(x1, x2+1)])
if not len(P):
return V
return np.array(P)
def centroid(vertices):
A = 0
Cx = 0
Cy = 0
for i in range(len(vertices)-1):
s = vertices[i, 0] * vertices[i+1, 1] - vertices[i+1, 0] * vertices[i, 1]
A += s
Cx += (vertices[i, 0] + vertices[i+1, 0]) * s
Cy += (vertices[i, 1] + vertices[i+1, 1]) * s
Cx /= 3*A
Cy /= 3*A
return np.array([[Cx, Cy]])
def weighted_centroid(vertices):
V = rasterize(vertices)
return V.sum(axis=0)/float(len(V))
def plot_shape(ax, seed, radius, letter):
np.random.seed(seed)
n = 12
T = np.linspace(0, 2*np.pi, n, endpoint=True)
R = radius * np.random.uniform(0.75, 1.25, n)
R[-1] = R[0]
V = np.zeros((n,2))
V[:,0] = R*np.cos(T)
V[:,1] = R*np.sin(T)
V[:,0] -= V[:,0].min()
V[:,1] -= V[:,1].min()
P = rasterize(V)
C = centroid(V)
wC = weighted_centroid(V[:-1])
Pi = np.round(P).astype(int)
shape = Pi[:,1].max()+1, Pi[:,0].max()+1
D = np.zeros(shape)
D[Pi[:,1], Pi[:,0]] = 1
ax.imshow(.15*D, extent=[0, D.shape[1], 0, D.shape[0]], origin="lower",
interpolation="nearest", vmin=0, vmax=1, cmap=plt.cm.gray_r, )
ax.plot(V[:,0], V[:,1], color="k", linewidth=2)
ax.scatter(V[:,0], V[:,1], s=25, edgecolor="k", facecolor="w", zorder=10)
ax.scatter(C[:,0], C[:,1], s=50, marker="o", linewidth=1,
edgecolor="k", facecolor="w", alpha=1, zorder=10)
ax.scatter(wC[0], wC[1], s=100, marker="x", linewidth=2,
edgecolor="None", facecolor="k", zorder=20)
ax.set_xlim(-0.5,shape[1]+0.5)
ax.set_xticks(np.arange(shape[1]+1))
ax.set_xticklabels([])
ax.xaxis.set_ticks_position('none')
ax.set_ylim(-0.5,shape[0]+0.5)
ax.set_yticks(np.arange(shape[0]+1))
ax.set_yticklabels([])
ax.yaxis.set_ticks_position('none')
ax.text(.05,.95, letter, fontsize="xx-large", weight="bold",
ha="center", va="center", transform=ax.transAxes)
ax.grid(color="0.6", linestyle='solid')
import matplotlib.pyplot as plt
fig = plt.figure(figsize=(12,6))
ax = plt.subplot(2,3,1, frameon=False)
plot_shape(ax, 1, 3, "A")
ax = plt.subplot(2,3,2, frameon=False)
plot_shape(ax, 1, 5, "B")
ax = plt.subplot(2,3,3, frameon=False)
plot_shape(ax, 1, 10, "C")
ax = plt.subplot(2,1,2)
def difference(seed, radius):
np.random.seed(seed)
n = 16
T = np.linspace(0, 2*np.pi, n, endpoint=True)
R = radius * np.random.uniform(0.5, 1.5, n)
R[-1] = R[0]
V = np.zeros((n,2))
V[:,0] = R*np.cos(T)
V[:,1] = R*np.sin(T)
V[:,0] -= V[:,0].min()
V[:,1] -= V[:,1].min()
P = rasterize(V)
C = centroid(V)
wC = weighted_centroid(V[:-1])
return np.sqrt(np.sum((C-wC)**2))
P = np.zeros((100,50))
for i,radius in enumerate(np.linspace(2,50,P.shape[0])):
for seed in range(P.shape[1]):
P[i,seed] = difference(seed, radius)
X = np.linspace(2,50,P.shape[0])
D = np.mean(P,axis=1)
S = np.std(P,axis=1)
ax.plot(X,D)
ax.fill_between(X, D+S, D-S, alpha=.15)
ax.scatter( [X[6],X[17],X[37]], [D[6],D[17],D[37]], s=25,
facecolor="w", edgecolor="k", zorder=30)
dy = 0.01
ax.text(X[6], D[6]-dy, "A", fontsize="large",
ha="center", va="top", transform=ax.transData)
ax.text(X[17], D[17]-dy, "B", fontsize="large",
ha="center", va="top", transform=ax.transData)
ax.text(X[37], D[37]-dy, "C", fontsize="large",
ha="center", va="top", transform=ax.transData)
ax.set_xticks([10,20,30,40,50])
ax.set_xticklabels(["10×10","20×20","30×30","40×40","50×50"])
ax.set_xlabel("Size (pixels²)")
ax.set_ylabel("Precision (pixels)")
plt.tight_layout()
plt.savefig("figures/figure-5.pdf")
plt.show()