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均匀三次B样条曲线.续 [复制链接]

ミルク

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Rank: 4

发表于 2012-8-1 13:06:37 |显示全部楼层

懒得码字了，直接上代码+测试。
注：附上了完整的TCAX下的测试工程，要执行，请把tcCurve.py放到TCAX的util目录下。

实现

tcCurve.py (10.02 KB, 下载次数: 1748)

###########################################################################################################################
###########################################################################################################################
###################### Uniform Cubic B-Spline Python Implementation, Object-Oriented Version With Const Speed Utility
class UCBSpline:
# pre-calculate coefficients
def __init__(self, p):
self.p = p # tuple or list of control points (x, y)
n = len(p) - 1 # control points p0 ~ pn, i.e., p[0] ~ p[n]
m = 3 + n + 1 # knots vector u0 ~um, i.e., u[0] ~ u[m]
self.m = m
step = 1 / (m - 2 * 3)
self.u = 3 * [0] + [_u * step for _u in range(m - 2 * 3 + 1)] + [1] * 3 # knots vector
num = n + 1 - 3 # number of curve segments
self.num = num
self.a1 = []
self.a2 = []
self.a3 = []
self.a4 = []
self.b1 = []
self.b2 = []
self.b3 = []
self.b4 = []
for i in range(num):
self.a1.append((-p[i][0] + 3 * p[i + 1][0] - 3 * p[i + 2][0] + p[i + 3][0]) / 6.0)
self.a2.append((3 * p[i][0] - 6 * p[i + 1][0] + 3 * p[i + 2][0]) / 6.0)
self.a3.append((-p[i][0] + p[i + 2][0]) / 2.0)
self.a4.append((p[i][0] + 4 * p[i + 1][0] + p[i + 2][0]) / 6.0)
self.b1.append((-p[i][1] + 3 * p[i + 1][1] - 3 * p[i + 2][1] + p[i + 3][1]) / 6.0)
self.b2.append((3 * p[i][1] - 6 * p[i + 1][1] + 3 * p[i + 2][1]) / 6.0)
self.b3.append((-p[i][1] + p[i + 2][1]) / 2.0)
self.b4.append((p[i][1] + 4 * p[i + 1][1] + p[i + 2][1]) / 6.0)
self.SIMPSON = 100 # number of sub-intervals of composite simpson integration, must be even, usually 100 is good enough
self.NEWTON = 0.0001 # epsilon, effective digits of the root calculated by Newton's method
# calculate the value at t
def __call__(self, t):
if t < 0 or t >= 1:
return None
else: # find the curve segment
for i in range(3, self.m - 3):
if self.u[i] <= t and t < self.u[i + 1]:
break
idx = i - 3 # get coefficients
a1 = self.a1[idx]
a2 = self.a2[idx]
a3 = self.a3[idx]
a4 = self.a4[idx]
b1 = self.b1[idx]
b2 = self.b2[idx]
b3 = self.b3[idx]
b4 = self.b4[idx]
t = (t - self.u[i]) / (self.u[i + 1] - self.u[i]) # map to the current curve segment
return (a1 * t * t * t + a2 * t * t + a3 * t + a4,
b1 * t * t * t + b2 * t * t + b3 * t + b4, 255) # 255 is the point's alpha value, to meet the TCAX's point structure requirement ((x, y, a), (x, y, a), ...)
# total arc-length
def length(self):
from math import sqrt
sqr = lambda x: x * x
l = 0
for i in range(self.num):
a1 = self.a1[i]
a2 = self.a2[i]
a3 = self.a3[i]
a4 = self.a4[i]
b1 = self.b1[i]
b2 = self.b2[i]
b3 = self.b3[i]
b4 = self.b4[i]
f_grand = lambda t: sqrt(sqr(3 * a1 * t * t + 2 * a2 * t + a3) + sqr(3 * b1 * t * t + 2 * b2 * t + b3)) # integrand of UCB
l += CompositeSimpson(f_grand, 0, 1, self.SIMPSON)
return l
# find t2 according to a specific arc-length s and a starting point t1
def t2OfLength(self, t1, s):
if t1 < 0 or t1 >= 1:
return None
else: # find the curve segment
for i in range(3, self.m - 3):
if self.u[i] <= t1 and t1 < self.u[i + 1]:
break
idx = i - 3 # get coefficients
a1 = self.a1[idx]
a2 = self.a2[idx]
a3 = self.a3[idx]
a4 = self.a4[idx]
b1 = self.b1[idx]
b2 = self.b2[idx]
b3 = self.b3[idx]
b4 = self.b4[idx]
t1 = (t1 - self.u[i]) / (self.u[i + 1] - self.u[i]) # map to the current curve segment
# Assumption: t1 and t are in the same piece of curve
from math import sqrt
sqr = lambda x: x * x
f_grand = lambda t: sqrt(sqr(3 * a1 * t * t + 2 * a2 * t + a3) + sqr(3 * b1 * t * t + 2 * b2 * t + b3)) # integrand of UCB
s1 = CompositeSimpson(f_grand, t1, 1, self.SIMPSON) # length from the starting point t1 to the end of the current curve segment
if s1 >= s: # t1 and t are in the same segment of curve
f_grand_d = lambda t: ((3 * a1 * t * t + 2 * a2 * t + a3) * (6 * a1 * t + 2 * a2) + (3 * b1 * t * t + 2 * b2 * t + b3) * (6 * b1 * t + 2 * b2 * t)) / f_grand(t) # derivative of f_grand
# using Newton's method to find the root of function f
f = lambda t: CompositeSimpson(f_grand, t1, t, self.SIMPSON) - s
fd = lambda t: CompositeSimpsonDerivative(f_grand, f_grand_d, t1, t, self.SIMPSON) # derivative of f
return NewtonMethod(f, fd, t1, self.NEWTON) * (self.u[i + 1] - self.u[i]) + self.u[i] # find the t and map to the whole curve
# otherwise, t is in next segment(s)
else:
if idx + 1 == self.num: # no more curve segment
return 1
for i in range(idx + 1, self.num):
a1 = self.a1[i]
a2 = self.a2[i]
a3 = self.a3[i]
a4 = self.a4[i]
b1 = self.b1[i]
b2 = self.b2[i]
b3 = self.b3[i]
b4 = self.b4[i]
f_grand = lambda t: sqrt(sqr(3 * a1 * t * t + 2 * a2 * t + a3) + sqr(3 * b1 * t * t + 2 * b2 * t + b3)) # integrand of UCB
s0 = s1
s1 += CompositeSimpson(f_grand, 0, 1, self.SIMPSON)
if s1 >= s: # we find at which segment the t lies
break
if i == self.num: # t is beyond the whole curve
return 1
f_grand_d = lambda t: ((3 * a1 * t * t + 2 * a2 * t + a3) * (6 * a1 * t + 2 * a2) + (3 * b1 * t * t + 2 * b2 * t + b3) * (6 * b1 * t + 2 * b2 * t)) / f_grand(t) # derivative of f_grand
# using Newton's method to find the root of function f
f = lambda t: CompositeSimpson(f_grand, 0, t, self.SIMPSON) - (s - s0)
fd = lambda t: CompositeSimpsonDerivative(f_grand, f_grand_d, 0, t, self.SIMPSON) # derivative of f
return NewtonMethod(f, fd, 0, self.NEWTON) * (self.u[i + 3 + 1] - self.u[i + 3]) + self.u[i + 3] # find the t and map to the whole curve, note, here i is actually idx, should + 3 to be real i
# f=function, a=initial value, b=end value, n=number of intervals of size h, n must be even
def CompositeSimpson(f, a, b, n):
h = (b - a) / n
S = f(a)
for i in range(1, n, 2):
x = a + h * i
S += 4 * f(x)
for i in range(2, n - 1, 2):
x = a + h * i
S += 2 * f(x)
S += f(b)
return h * S / 3
# f=function, fd=derivative of f, t1=initial value (constant), t=end value (variable), n=number of intervals of size h, n must be even
def CompositeSimpsonDerivative(f, fd, t1, t, n):
h = (t - t1) / n
hd = 1 / n
S = f(t1)
for i in range(1, n, 2):
x = t1 + h * i
S += 4 * f(x)
for i in range(2, n - 1, 2):
x = t1 + h * i
S += 2 * f(x)
S += f(t)
part1 = hd * S / 3
# part2
S = 0
for i in range(1, n, 2):
x = t1 + h * i
xd = hd * i
S += 4 * fd(x) * xd
for i in range(2, n - 1, 2):
x = t1 + h * i
xd = hd * i
S += 2 * fd(x) * xd
S += fd(t)
part2 = h * S / 3
return part1 + part2
# f is the function whose root needs to be found, fd is the derivative of function f, x0 is the starting point, e is an epsilon, i.e., effective digits
def NewtonMethod(f, fd, x0, e):
x = x0
while True:
x_new = x - f(x) / fd(x)
if abs(x_new - x) < e:
return x_new
x = x_new
###########################################################################################################################
###########################################################################################################################
###################### Uniform Cubic B-Spline Python Implementation
def UniformCubicBSpline(p, t):
n = len(p) - 1 # control points p0 ~ pn, i.e., p[0] ~ p[n]
m = 3 + n + 1 # knots vector u0 ~um, i.e., u[0] ~ u[m]
step = 1 / (m - 2 * 3)
u = 3 * [0] + [_u * step for _u in range(m - 2 * 3 + 1)] + [1] * 3
if t < 0 or t >= 1:
return None
else:
for i in range(3, m - 3):
if u[i] <= t and t < u[i + 1]:
break
return UniformCubicBSpline_Calc(p[i - 3], p[i - 2], p[i - 1], p[i], (t - u[i]) / (u[i + 1] - u[i])) # can use either calc or calc2
def UniformCubicBSpline_Calc(p1, p2, p3, p4, t):
a1 = (-p1[0] + 3 * p2[0] - 3 * p3[0] + p4[0]) / 6.0
a2 = (3 * p1[0] - 6 * p2[0] + 3 * p3[0]) / 6.0
a3 = (-p1[0] + p3[0]) / 2.0
a4 = (p1[0] + 4 * p2[0] + p3[0]) / 6.0
b1 = (-p1[1] + 3 * p2[1] - 3 * p3[1] + p4[1]) / 6.0
b2 = (3 * p1[1] - 6 * p2[1] + 3 * p3[1]) / 6.0
b3 = (-p1[1] + p3[1]) / 2.0
b4 = (p1[1] + 4 * p2[1] + p3[1]) / 6.0
return (a1 * t * t * t + a2 * t * t + a3 * t + a4,
b1 * t * t * t + b2 * t * t + b3 * t + b4, 255) # 255 is the point's alpha value, to meet the TCAX's point structure requirement ((x, y, a), (x, y, a), ...)
def UniformCubicBSpline_Calc2(p1, p2, p3, p4, t):
A1 = (1 - t) * (1 - t) * (1 - t) / 6.0
A2 = (3 * t * t * t - 6 * t * t + 4) / 6.0
A3 = (-3 * t * t * t + 3 * t * t + 3 * t + 1) / 6.0
A4 = t * t * t / 6.0
return (A1 * p1[0] + A2 * p2[0] + A3 * p3[0] + A4 * p4[0],
A1 * p1[1] + A2 * p2[1] + A3 * p3[1] + A4 * p4[1], 255) # 255 is the point's alpha value, to meet the TCAX's point structure requirement ((x, y, a), (x, y, a), ...)

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测试用例

test_curves.rar (1.48 KB, 下载次数: 7000)

from tcaxPy import *
from util.cairo import *
from util.tcCurve import *
def tcaxPy_User():
surface = ImageSurface(FORMAT_ARGB32, GetVal(val_ResolutionX), GetVal(val_ResolutionY))
ctx = Context(surface)
# 控制点列表1
P = [(200, 500), (300, 100), (700, 100), (400, 500), (600, 700), (800, 400), (700, 200), (900, 300), (660, 400)]
# 控制点列表2
# P = [(640, 0)]
# for i in range(10):
# P.extend([(240 + randint(-40, 40), 200 + randint(-50, 50)), (1040 + randint(-40, 40), 200 + randint(-50, 50)), (640, 0)])
# P.extend([(340 + randint(-60, 60), 600 + randint(-50, 50)), (940 + randint(-60, 60), 600 + randint(-50, 50)), (640, 0)])
# 控制点列表3
# P = [(0, 100), (100, 0), (200, 0), (300, 100), (400, 200), (500, 200), (600, 100), (400, 400), (700, 50), (800, 200)]
# 控制点列表4
# P = [(300, 100), (400, 200), (500, 200), (600, 100)]
# 控制点列表5
# P = [(400, 100), (400, 500), (500, 500), (500, 100)]
# 绘制控制点
ctx.set_source_rgb(0, 0, 0)
num = len(P)
for i in range(num):
ctx.arc(P[i][0], P[i][1], 2, 0, 2 * pi)
ctx.fill()
# 计算曲线
ucb = UCBSpline(P)
L = ucb.length() # 曲线总长度
STEP_N = int(L / 10 + 0.5) # 取样点数
step_size = 1 / STEP_N
print('控制点: {0} 曲线总长度: {1:.02f} 所取点数: {2}'.format(len(P), L, STEP_N))
t1 = 0
points = []
for i in range(STEP_N):
# 普通1, 使用UniformCubicBSpline函数
#points.append(UniformCubicBSpline(P, i * step_size))
# 普通2
#points.append(ucb(i * step_size))
# 匀速版本
points.append(ucb(t1))
t1 = ucb.t2OfLength(t1, step_size * L)
# 绘制独立点
ctx.set_source_rgb(0, 1, 0)
num = len(points)
for i in range(num):
ctx.arc(points[i][0], points[i][1], 2, 0, 2 * pi)
ctx.fill()
# 绘制连续曲线
# ctx.move_to(points[0][0], points[0][1])
# for i in range(1, num):
# ctx.line_to(points[i][0], points[i][1])
# ctx.stroke()
# save the result
surface.write_to_png(abspath('1.png'))
PIX = surface.get_pix()
PIX = PixBlur(PIX, 10)
SavePix(abspath('2.png'), PIX)
#sys.exit()

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若干效果截图

计算效率初步测试

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ミルク发表于12楼查看完整内容

根据这篇论文，重新理解了下UCBS，并简化了代码（更好理解）。

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BurySakura

Super Moderator

{\clip\t(\clip)}∀.I.R.nesSub

Rank: 6 Rank: 6

发表于 2012-8-1 17:33:50 |显示全部楼层

本帖最后由 BurySakura 于 2012-8-1 17:43 编辑

效率相对我那个匀速贝塞尔真心好。
但是这个怎么控制曲线开始，结束的位置？
还有点的导数在哪？

curve.rar (18.44 KB, 下载次数: 1856)

psp.希望牛奶提供的大概思路，我看我那曲线是否能改下提高效率。

ミルク

正式会员

Rank: 4

发表于 2012-8-1 17:44:27 |显示全部楼层

BurySakura 发表于 2012-8-1 17:33
效率相对我那个匀速贝塞尔真心好。
但是这个怎么控制开始，结束点的位置？
还有点的导数在哪？

开始点和结束点好像只起一个引导作用。可以重叠两个点为开始点，效果就稍微接近一点。

P = [(200, 500), (200, 500), (300, 100), (700, 100), (400, 500), (600, 700), (800, 400), (700, 200), (900, 300), (660, 400), (660, 400)]

复制代码

导数应该是 f_grand_d, fd, CompositeSimpsonDerivative

ミルク

正式会员

Rank: 4

发表于 2012-8-1 17:51:51 |显示全部楼层

补充：对于有n（n比较大）个控制点的情况，直接使用贝塞尔曲线，会造成曲线阶数很高，从而导致效率下降（阶数过高的贝塞尔曲线形状也不容易控制）。
可以用Composite Bezier Spline（复合贝塞尔样条曲线），降低阶数以提高效率。

p.s. 如果你的代码对于4, 5个控制点的情况效率不错，就说明优化的价值（提升的空间）不高了