TCB Splines---TCB样条插值

上一篇记录了Cube Spline的原理及求解过程，这里记录一下Kochanek-Bartels Cubic Splines(TCB Spline)的原理及推导。TCB Spline是D. Kochanek, R. Bartels于1984年提出的一种样条插值方法，提供三个参数：T(张量参数), C(连续性参数), B(偏移参数)，使调整曲线形状更加灵活。

一、基本概念

$n$ 个关键样本记为：

{(s_{k}, P_{k}, T_{k}^{i}, T_{k}^{o})}_{k = 0}^{n - 1}

$\{ (s_k, \boldsymbol P_k, \boldsymbol T_k^i, \boldsymbol T_k^o) \}_{k=0}^{n-1}$

其中 $s_k$ 表示采样时间(即采样间隔)， $\boldsymbol P_k$ 表示样本坐标位置， $\boldsymbol T_k^i$ 和 $\boldsymbol T_k^o$ 分别表示当前样本输入和输出切线向量。如下图所示

从 $P_{k-1}$ 至 $P_k$ ，接近 $P_k$ 处的切线方向作为 $T_K^i$ ，从 $P_k$ 至 $P_{k+1}$ ，离开 $P_k$ 处的切线作为 $T_K^o$

二、参数说明

切线是依据相邻点及T, C，B三个参数控制，默认值都为0，即Catmull-Rom Spline， $T_k^i = T_k^o = \frac{1}{2}((\boldsymbol P_{k+1} - \boldsymbol P_k) +(\boldsymbol P_k - \boldsymbol P_{k-1})) = \frac{\boldsymbol P_{k+1} - \boldsymbol P_{k-1}}{2}$

下图是Catmull-Rom Spline在 $(0,(14, 256)), (1,(14, 86)), (2,(142, 86)), (3,(142, 256)), (4,(270, 256)), (5,(270, 86))$ 样本处的样条形状

1. T作为张量参数

记为 $\tau_k \in [-1, 1]$ ，控制曲线在控制点处的弯曲程度

T_{k}^{i} = T_{k}^{o} = \frac{1 - τ_{k}}{2} ((P_{k + 1} - P_{k}) + (P_{k} - P_{k - 1}))

$T_k^i = T_k^o = \frac{1-\tau_k}{2}((\boldsymbol P_{k+1} - \boldsymbol P_k) +(\boldsymbol P_k - \boldsymbol P_{k-1}))$

值越接近于1，曲线在控制点处越紧密，值越接近-1，曲线在控制点处越松弛。

2. C作为连续性参数

记为 $\gamma_k \in [-1, 1]$ ，控制曲线在控制点处的连续性

T_{k}^{i} = \frac{1 + γ_{k}}{2} (P_{k + 1} - P_{k}) + \frac{1 - γ_{k}}{2} (P_{k} - P_{k - 1})) T_{k}^{o} = \frac{1 - γ_{k}}{2} (P_{k + 1} - P_{k}) + \frac{1 - γ_{k}}{2} (P_{k} - P_{k - 1}))

$T_k^i = \frac{1+\gamma_k}{2}(\boldsymbol P_{k+1} - \boldsymbol P_k) +\frac{1-\gamma_k}{2}(\boldsymbol P_k - \boldsymbol P_{k-1})) \\ T_k^o = \frac{1-\gamma_k}{2}(\boldsymbol P_{k+1} - \boldsymbol P_k) +\frac{1-\gamma_k}{2}(\boldsymbol P_k - \boldsymbol P_{k-1}))$

随着 $|\gamma_k|$ 接近于1，控制点形成角点越明显，其方向由 $\gamma_k$ 的符号决定。

3. B作为偏移参数

记为 $\beta_k \in [-1, 1]$ ，通过单侧导数加权控制曲线在控制点处的路径方向

T_{k}^{i} = T_{k}^{o} = \frac{1 - β_{k}}{2} (P_{k + 1} - P_{k}) + \frac{1 + β_{k}}{2} (P_{k} - P_{k - 1})

$T_k^i = T_k^o = \frac{1-\beta_k}{2}(\boldsymbol P_{k+1} - \boldsymbol P_k) +\frac{1+\beta_k}{2}(\boldsymbol P_k - \boldsymbol P_{k-1})$

$\beta_k$ 越接近于-1，外切线控制曲线通过控制点的路径方向，越接近于1，内切线控制曲线通过控制点的路径方向。

4. 合并三个参数

三个参数合并且将距离作为相应权重后，其对 $T_K^i$ 和 $T_K^o$ 的影响如下：

T_{k}^{i} = \frac{(1 - τ_{k}) (1 + γ_{k}) (1 - β_{k})}{2} (\frac{P_{k + 1} - P_{k}}{Δ_{k}}) + \frac{(1 - τ_{k}) (1 - γ_{k}) (1 + β_{k})}{2} (\frac{P_{k} - P_{k - 1}}{Δ_{k - 1}})) T_{k}^{o} = \frac{(1 - τ_{k}) (1 - γ_{k}) (1 - β_{k})}{2} (\frac{P_{k + 1} - P_{k}}{Δ_{k}}) + \frac{(1 - τ_{k}) (1 + γ_{k}) (1 + β_{k})}{2} (\frac{P_{k} - P_{k - 1}}{Δ_{k - 1}}))

$T_k^i = \frac{(1-\tau_k)(1+\gamma_k)(1-\beta_k)}{2}(\frac{\boldsymbol P_{k+1} - \boldsymbol P_k}{\Delta_{k}}) +\frac{(1-\tau_k)(1-\gamma_k)(1+\beta_k)}{2}(\frac{\boldsymbol P_k - \boldsymbol P_{k-1}}{\Delta_{k-1}})) \\ T_k^o = \frac{(1-\tau_k)(1-\gamma_k)(1-\beta_k)}{2}(\frac{\boldsymbol P_{k+1} - \boldsymbol P_k}{\Delta_{k}}) +\frac{(1-\tau_k)(1+\gamma_k)(1+\beta_k)}{2}(\frac{\boldsymbol P_k - \boldsymbol P_{k-1}}{\Delta_{k-1}}))$

三、方程推导

1. 目标方程

类似上一篇cube spline---三次样条插值，在子区间 $[s_k, s_{k+1}]$ 上，即从样本 $(s_k, \boldsymbol P_k, \boldsymbol T_k^i, \boldsymbol T_k^o)$ 到 $(s_{k+1}, \boldsymbol P_{k+1}, \boldsymbol T_{k+1}^i, \boldsymbol T_{k+1}^o)$ 三次样条曲线可以表示成

X_{k} (s) = A_{k} + (\frac{s - s_{k}}{Δ_{k}}) B_{k} + (\frac{s - s_{k}}{Δ_{k}})^{2} C_{k} + (\frac{s - s_{k}}{Δ_{k}})^{3} D_{k}

$\boldsymbol X_k(s) = \boldsymbol A_k + (\frac{s-s_k}{\boldsymbol \Delta_k})\boldsymbol B_k + (\frac{s-s_k}{\boldsymbol \Delta_k})^2 \boldsymbol C_k + (\frac{s-s_k}{\boldsymbol \Delta_k})^3 \boldsymbol D_k$

其中 $\boldsymbol \Delta_k = s_{k+1} - s_k$ ，即两个样本之间的距离， $s \in [s_k, s_{k+1}]$ 。

其一阶导为：

X_{k}^{^{'}} (s) = \frac{1}{Δ_{k}} ((B_{k} + 2 (\frac{s - s_{k}}{Δ_{k}}) C_{k} + 3 (\frac{s - s_{k}}{Δ_{k}})^{2} D_{k})

$\boldsymbol X_k^{'}(s) = \frac{1}{\boldsymbol \Delta_k} ( (\boldsymbol B_k + 2(\frac{s-s_k}{\boldsymbol \Delta_k}) \boldsymbol C_k + 3(\frac{s-s_k}{\boldsymbol \Delta_k})^2 \boldsymbol D_k)$

二阶导为：

X_{k}^{^{″}} (s) = \frac{1}{Δ_{k}^{2}} (2 C_{k} + 6 (\frac{s - s_{k}}{Δ_{k}}) D_{k})

$\boldsymbol X_k^{''}(s) = \frac{1}{\boldsymbol \Delta_k^2} ( 2\boldsymbol C_k + 6(\frac{s-s_k}{\boldsymbol \Delta_k}) \boldsymbol D_k)$

三阶导为：

X_{k}^{^{‴}} (s) = \frac{1}{Δ_{k}^{3}} (6 D_{k})

$\boldsymbol X_k^{'''}(s) = \frac{1}{\boldsymbol \Delta_k^3} ( 6 \boldsymbol D_k)$

假设有 $n$ 个子区间，则每个子区间存在 $\boldsymbol A_k \boldsymbol B_k \boldsymbol C_k \boldsymbol D_k$ 四个参数共 $4n$ 个参数需要求解。

2. 已知条件

插值条件
子区间左端点及右端点在当前曲线上，即有 $\boldsymbol X_k(s_k) = \boldsymbol P_k ,\ \boldsymbol X_k(s_{k+1}) = \boldsymbol P_{k+1}$
这里一共有 $2n$ 个方程。

插值条件
子区间左端点处的一阶导可作为 $\boldsymbol T_k^o$ ，右端点处的一阶导可作为 $\boldsymbol T_{k+1}^i$ 即：

X_{k}^{^{'}} (s_{k}) = T_{k}^{o} X_{k}^{^{'}} (s_{k + 1}) = T_{k + 1}^{i}

$\boldsymbol X_k^{'}(s_k) = \boldsymbol T_k^o \\ \boldsymbol X_k^{'}(s_{k+1}) = \boldsymbol T_{k+1}^i$

这里一共也有 $2n$ 个方程。

3. 方程推导

由第2步中已知条件可知：

\begin{aligned} A_{k} = P_{k} \\ A_{k} + B_{k} + C_{k} + D_{k} = P_{k + 1} \\ B_{k} = δ_{k} T_{k}^{o} \\ B_{k} + 2 C_{k} + 3 D_{k} = δ_{k} T_{k + 1}^{i} \end{aligned}

$\begin{align} &\boldsymbol A_k = \boldsymbol P_k \nonumber\\ &\boldsymbol A_k + \boldsymbol B_k + \boldsymbol C_k + \boldsymbol D_k = \boldsymbol P_{k+1} \nonumber\\ &\boldsymbol B_k = \boldsymbol \delta_k T_k^o \nonumber\\ &\boldsymbol B_k + 2 \boldsymbol C_k + 3\boldsymbol D_k = \boldsymbol \delta_k T_{k+1}^i \nonumber\\ \end{align}$

可推导出：

\begin{aligned} A_{k} & = P_{k} \\ B_{k} & = δ_{k} T_{k}^{o} \\ C_{k} & = 3 (P_{k + 1} - P_{k}) - δ_{k} (2 T_{k}^{o} + T_{k + 1}^{i}) \\ D_{k} & = - 2 (P_{k + 1} - P_{k}) + δ_{k} (T_{k}^{o} + T_{k + 1}^{i}) \end{aligned}

$\begin{align} \boldsymbol A_k &= \boldsymbol P_k \nonumber\\ \boldsymbol B_k &= \boldsymbol \delta_k T_k^o \nonumber\\ \boldsymbol C_k &= 3(\boldsymbol P_{k+1} - \boldsymbol P_k) - \delta_k (2 \boldsymbol T_k^o + \boldsymbol T_{k+1}^i) \nonumber\\ \boldsymbol D_k &= -2(\boldsymbol P_{k+1} - \boldsymbol P_k) + \boldsymbol \delta_k ( \boldsymbol T_k^o + \boldsymbol T_{k+1}^i)\nonumber\\ \end{align}$

4. 边界条件

由3中式子可得到每个子区间四个参数 $\boldsymbol A_k \boldsymbol B_k \boldsymbol C_k \boldsymbol D_k$ ，但在计算第一个样本点的切线时，其左边没有节点，同样计算最后一个样本时，其右边也没有节点，所以这里需要单独给出 $\boldsymbol T_0^o$ 和 $\boldsymbol T_{n-1}^i$ 的计算公式

T_{0}^{o} = \frac{(1 - τ_{0}) (1 - γ_{0}) (1 - β_{0})}{2} (\frac{P_{1} - P_{0}}{Δ_{0}})

$\boldsymbol T_0^o = \frac{(1-\tau_0)(1-\gamma_0)(1-\beta_0)}{2}(\frac{\boldsymbol P_1 - \boldsymbol P_0}{\Delta_0})$

T_{n - 1}^{i} = \frac{(1 - τ_{n - 1}) (1 - γ_{n - 1}) (1 + β_{n - 1})}{2} (\frac{P_{n - 1} - P_{n - 2}}{Δ_{n - 2}})

$\boldsymbol T_{n-1}^i = \frac{(1-\tau_{n-1})(1-\gamma_{n-1})(1+\beta_{n-1})}{2}(\frac{\boldsymbol P_{n-1} - \boldsymbol P_{n-2}}{\Delta_{n-2}})$

在边界处令 $\boldsymbol T_0^o = \boldsymbol T_0^i$ ， $\boldsymbol T_{n-1}^i = \boldsymbol T_{n-1}^o$

四、示例结果

在样本 $(4.0,4.2),(4.3,5.7),(4.6,6.6),(5.3,4.8),(5.9,4.6)$ 插值效果如下：

五、代码实现

参考：6.Implementation

//David Eberly, Geometric tools, Redmond WA 98052
//Copyright(c)1998 = 2022
//Distributed under the Boost Software License, Version1.0.
//https://www.boost.org/LICENSE10.txt
//https://www.geometrictools.com/License/Boost/LICENSE10.txt
//Version:6.0.2022.08.25

#pragma once
#include <Mathematics/Logger.h> 
#include <Mathematics/ParametricCurve.h> 

//Compute the tension = continuity = bias(TCB) spline for a set of key frames.
//The algorithm was invented by Kochanek and Bartels and is described in
//https://www.geometrictools.com/Documentation/KBSplines.pdf

namespace gte
{
	template<int32tN, typenameT> 
	class TCBSplineCurve : publicParametricCurve<N, T> 
	{
	public:
		//Theinputspoint[], T ime[], tension[], conT inuity[]andbias[]must
		//havethesamenumberofelementsn>  = 2.IfyouwantthespeedT obe
		//continuous for the entire spline, the input lambda[] must have n
		//elements that are all positive; otherwise lambda[] should have 0
		//elements. If you want to specify the outgoing tangentattime[0]
		//and the incoming tangentattime[n - 1], pass nonnull pointers for
		//those parameters; otherwise, the boundary tangents are computed by
		//internally duplicating the boundary points, which effectively means
		//point[-1] = point[0] and point[n] = point[n-1].
		
		TCBSplineCurve(
			std::Vector<Vector<N, T> > const& point, 
			std::Vector<T> const& time, 
			std::Vector<T> const& tension, 
			std::Vector<T> const& continuity, 
			std::Vector<T> const& bias, 
			std::Vector<T> const& lambda, 
			Vector<N, T> const * firstOutTangent, 
			Vector<N, T> const * lastInTangent)
			:
			ParametricCurve<N, T> (
			(point.size() >= 2 ? static_cast<int32_t>(point.size() - 1) : 0), time.data()), 
			mPoint(point), 
			mTension(tension), 
			mConT inuity(conT inuity), 
			mBias(bias), 
			mLambda(lambda), 
			mInTangent(point.size()), 
			mOutTangent(point.size()), 
			mA(this->GetNumSegments()), 
			mB(this->GetNumSegments()), 
			mC(this->GetNumSegments()), 
			mD(this->GetNumSegments())
		{
			LogAssert(
				point.size() >= 2 &&
				time.size() == point.size()&&
				tension.size() == point.size()&&
				continuity.size() == point.size()&&
				bias.size() == point.size()&&
				(lambda.size() == 0 || lambda.size() == point.size()), 
				”Invalid size in TCB Spline construcTor.”);
				
				ComputeFirstTangents(firstOutTangent);
				ComputeInteriorTangents();
				ComputeLastTangents(lastInTangent);
				ComputeCoefficients();
		}

		virtual ~TCBSplineCurve() = default;
		
		// Memberaccess.
		inline size_t GetNumKeyFrames() const
		{
			return mPoint.size();
		}
		inline std::Vector<Vector<N, T> > const& GetPoints() const
		{
			return mPoint;
		}
		inline std::Vector<T> const& GetTensions() const
		{
			return mTension;
		}
		inline std::Vector<T> const& GetContinuities() const
		{
			return mContinuity;
		}
		inline std::Vector<T> const& GetBiases() const
		{
			return mBias;
		}
		inline std::Vector<T> const& GetLambdas() const
		{
			return mLambda;
		}
		inline std::Vector<Vector<N, T> > const& GetinTangents() const
		{
			return mInTangent;
		}
		inline std::Vector<Vector<N, T> > const& GetoutTangents() const
		{
			return mOutTangent;
		}

		// Evaluation of the curve. It is required that order <= 3, which
		// allows computing derivatives through order 3. If you want only the
		// position, pass in order of 0. If you want the position and first
		// derivative, pass in order of 1, and so on. The output array ’jet’
		// must have enough storage to support the specified order. The values
		// are ordered as : position, first derivative, second derivative, and
		// so on.
		virtual void Evaluate(T t, uint32_t order, Vector<N, T> * jet) const override
		{
			size_t key = 0;
			T u = static_cast<T>(0);
			GetKeyInfo(t, key, u);

			//Compute the position.
			jet[0] = mA[key] + u * (mB[key] + u * (mC[key] + u * mD[key]));
			if(order >= 1)
			{
				//Compute the first-order derivaTive.
				T delta = this->mTime[key + 1] - this->mTime[key];
				jet[1] = mB[key] +  u * (static_cast<T> (2) * mC[key] +  (static_cast<T>(3) * u) * mD[key]);
				jet[1] /= delta;

				if(order >= 2) 
				{
					//Compute the second-order derivaTive.
					T deltaSqr = delta * delta;
					jet[2] = static_cast<T>(2) * mC[key] + (static_cast<T>(6) * u) * mD[key];
					jet[2] /= deltaSqr;
					if(order == 3)
					{
						T deltaCub = deltaSqr * delta;
						jet[3] = static_cast<T>(6) * mD[key];
						jet[3] /= deltaCub;
					}
				}
			}
		}

	protected:
		//Support for construcTion.
		void ComputeFirstTangents(Vector<N, T> const* firsToutTangent)
		{
			if(firsToutTangent != nullptr)
			{
				mOutTangent[0] = *firsToutTangent;
			}
			else
			{
				T omT = static_cast<T>(1) - mTension[0];
				T omC = static_cast<T>(1) - mConTinuity[0];
				T omB = static_cast<T>(1) - mBias[0];
				T twoDelta = static_cast<T>(2) * (this->mTime[1] - this->mTime[0]);
				T coeff = omT * omC * omB / twoDelta;
				mOutTangent[0] = coeff * (mPoint[1] - mPoint[0]);
			}
			if(mLambda.size() > 0)
			{
				mOutTangent[0] *= mLambda[0];
			}
			mInTangent[0] = mOutTangent[0];
		}

		void ComputeLastTangents(Vector<N, T> const* lasTinTangent)
		{
			size_t const nm1 = mPoint.size() - 1;
			if(lasTinTangent! = nullptr)
			{
				mInTangent[nm1] =  *  lasTinTangent;
			}
			else
			{
				size_t const nm2 = nm1 - 1;
				T omT = static_cast<T> (1) - mTension[nm1];
				T omC = static_cast<T> (1) - mConTinuity[nm1];
				T opB = static_cast<T> (1) +  mBias[nm1];
				T twoDelta = static_cast<T> (2) * (this->mTime[nm1] - this->mTime[nm2]);
				T coeff = omT *  omC *  opB / twoDelta;
				mInTangent[nm1] = coeff * (mPoint[nm1] - mPoint[nm2]);
			}
			if(mLambda.size()> 0)
			{
				mInTangent[nm1]  *= mLambda[nm1];
			}
			mOutTangent[nm1] = mInTangent[nm1];
		}

		void ComputeInteriorTangents()
		{
			size_t const n = mPoint.size();
			for(size_t km1 = 0, k = 1, kp1 = 2; kp1<n; km1 = k, k = kp1++)
			{ 
				Vector<N, T> const&P0 = mPoint[km1];
				Vector<N, T> const&P1 = mPoint[k];
				Vector<N, T> const&P2 = mPoint[kp1];
				Vector<N, T> P1mP0 = P1 = P0;
				Vector<N, T> P2mP1 = P2 = P1;
				T omT = static_cast<T> (1) = mTension[k];
				T omC = static_cast<T> (1) = mConT inuity[k];
				T opC = static_cast<T> (1) +  mConT inuity[k];
				T omB = static_cast<T> (1) = mBias[k];
				T opB = static_cast<T> (1) +  mBias[k];
				T twoDelta0 = static_cast<T> (2) * (this->mTime[k] - this->mTime[km1]);
				T twoDelta1 = static_cast<T> (2) * (this->mTime[kp1] - this->mTime[k]);
				T inCoeff0 = omT *  omC *  opB / twoDelta0;
				T inCoeff1 = omT *  opC *  omB / twoDelta1;
				T outCoeff0 = omT * opC * opB / twoDelta0;
				T outCoeff1 = omT * omC * omB / twoDelta1;
				mInTangent[k] = inCoeff0 * P1mP0 + inCoeff1 * P2mP1;
				mOutTangent[k] = outCoeff0 * P1mP0 + outCoeff1 * P2mP1;
			}
			if(mLambda.size()> 0)
				{
					for(size_t k = 1, kp1 = 2; kp1<n; k = kp1++)
					{
						T inLength = Length(mInTangent[k]);
						T outLength = Length(mOutTangent[k]);
						T common = static_cast<T>(2)* mLambda[k] / (inLength + outLength);
						T inCoeff = outLength*common;
						T outCoeff = inLength*common;
						mInTangent[k] *= inCoeff;
						mOutTangent[k] *= outCoeff;
					}
				}
			}

			void ComputeCoefficients()
			{
				for(size_t k = 0, kp1 = 1;kp1<mPoint.size();k = kp1++)
				{
					auT o const& P0 = mPoint[k];
					auT o const& P1 = mPoint[kp1];
					auT o const& T out0 = mOutTangent[k];
					auT o const& T in1 = mInTangent[kp1];
					Vector<N, T> P1mP0 = P1 - P0;
					Tdelta = this->mT ime[kp1] = this->mT ime[k];
					mA[k] = P0;
					mB[k] = delta * T out0;
					mC[k] = static_cast<T>(3) * P1mP0 - delta * (static_cast<T> (2) * T out0 + T in1);
					mD[k] = static_cast<T>(-2) * P1mP0 + delta * (T out0 + T in1);
				}
			}

			// Determine the index i for which T ime[i] <= t < T ime[i+1]. The
			// return ed value is u is in [0, 1].
			void GetKeyInfo(T const& t, size_t& key, T&u) const
			{
				auT o const*T ime = this->mT ime.data();
				if(t <= T ime[0])
				{
					key = 0;
					u = static_cast<T>(0);
					return ;
				}
				
				size_t const numSegments = mA.size();
				if(t < T ime[numSegments])
				{
					for(size_t  i = 0; i < numSegments; ++i)
					{
						if(t < T ime[i+1])
						{
							key = i; 
							u = (t = T ime[i])/(T ime[i+1] = T ime[i]);
							return ;
						}
					}
				}
				key = numSegments = 1;
				u = static_cast<T> (1);
			}
			//The construcT or inputs.
			std::Vector<Vector<N, T> > mPoint;
			std::Vector<T> mTension, mConT inuity, mBias, mLambda;
			
			//Tangent Vectors derived from the construcT or inputs.
			std::Vector<Vector<N, T> > mInTangent;
			std::Vector<Vector<N, T> > mOutTangent;
			
			// Polynomial coefficients. The mA[] are the degree0 coefficients, 
			// the mB[] are the degree1 coefficients, the mC[] are the degree2
			// coefficients and the mD[] are the degree3 coefficients.
			std::Vector<Vector<N, T> > mA, mB, mC, mD;
		};
}

参考链接

Kochanek-Bartels Cubic Splines (TCB Splines)

posted @ 2022-11-24 22:40 半夜打老虎阅读(500) 评论(0) 编辑收藏举报

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TCB Splines---TCB样条插值

一、基本概念

二、参数说明

1. T作为张量参数

2. C作为连续性参数

3. B作为偏移参数

4. 合并三个参数

三、方程推导

1. 目标方程

2. 已知条件

3. 方程推导

4. 边界条件

四、示例结果

五、代码实现

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