第2章 二维运动估计
第三章_二维运动估计之二
动矢量是从每个块中独立预测得到的。
–解决方法：基于网格的运动估计
当块内有多物体运动或者因照明产生变化时，
DBMA效果不好
–三模型法：
• 先对每个块实行EBMA • EBMA误差小的块具有平移运动 • EBMA误差大的块具有非平移运动 –对这些块运用 DBMA
目录
EBMA的问题 可变形块匹配算法 (DBMA) – 基于节点的运动模型 基于网格的运动估计 基于区域的运动估计 全局运动估计 – 直接估计法 – 间接估计法 多分辨率运动估计 – 分层块匹配算法(HBMA) 小结
过程： 1. 将锚定帧分割成多个规整的块 2. 对每个块采用较复杂的运动模型
xm,1
（如仿射，双线性或投影映射） 3. 对每个块进行独立的运动参数估计
6.5 可变形块匹配算法
仿射 (6 个参数):
– 三角形映射
d x ( x, y ) a0 a1 x a2 y d ( x, y ) b b x b y 2 y 0 1
基于网格的运动估计
当前帧被分割为不重 叠的多边形单元（网 格），运动估计为网 格的每个节点在参考 帧找到对应点（即得 到其运动矢量），其 余点的位移矢量由节 点的运动矢量插值得 到。
(a) 使用三角形网格
(b) 使用四边形网格
基于网格的运动估计
(a) 基于块匹配的运动估计
(b) 基于网格的运动估计
全搜索块匹配 (半像素精度)
预测图像
基于网格
基于网格的运动估计与块匹配法的比较
网格
– 当前帧可以分成规则的或不规则的网格 – 参考帧中的不规则网格是不重叠的 – 每个节点只有一个运动矢量
可变形块
第三章二维运动估计之一PPT课件
运动参数，但是投影会造成信息丢失（不可逆），导致估计误差。
.
3
特征对应：运动物体上的特征与其在二维平面上的投影坐标的对应关系。 见下图示：
设t1→t2时，物体由P运动至P’，即： 空 间：P(X,Y,Z) → P’(X’,Y’,Z’) 像平面： p(x,y) → p’(x’,y’) 二维位移（△x, △y ）称为二维运动矢量，
❖ 映射函数： w(x, a) = x + d(x, a) , x
❖ 运动参数矢量: a 存在问. 题：遮挡
14
.
15
2.2 运动估计的一般方法
❖ 两种主要的方法:
– 基于特征（常用在物体跟踪上，从2D构建出3D） – 基于亮度（基于恒定亮度假设或光流方程，常用在
运动补偿预测，视频编码和插值方面） -> 重点
v vnen vtet
vn ||
||
t
0
.
13
2.2 运动估计的一般方法
后向运动估计
Time t
Time t +t
Time t - t x
d(x, t - t) 当前帧
x 参考帧
d(x, t + t) x
当前帧 前向运动估计
❖ 参考帧： 1(x, t1) ❖ 当前帧: 2(x, t2)
❖ 前向运动估计 /后向运动估计： t2 > t1 / t2 < t1 ❖ 运动场： d(x, a), x
一致。如计算机视觉、目标跟踪、工业监视。
非真实运动估计：在不被察觉的情况下允许有估计误差，
从而最大限度降低信息量和传输带宽。如广播电视中的视
二维位姿估计的定义
二维位姿估计的定义二维位姿估计是指通过使用传感器数据和算法来推断或估计目标物体在二维平面上的位置和姿态。
在机器人技术、计算机视觉和自动驾驶等领域，二维位姿估计起着至关重要的作用。
它为机器人和计算机系统提供了实时感知和定位的能力，从而使它们能够快速准确地与周围环境进行交互和导航。
位姿估计主要分为两个方面：位置估计和姿态估计。
位置估计是指确定目标物体在平面上的位置坐标，通常使用二维坐标系表示。
姿态估计是指确定目标物体的旋转方向或朝向，常用欧拉角或四元数表示。
通过综合位置和姿态信息，能够得到目标物体的完整位姿。
在二维位姿估计中，常用的传感器包括摄像头、激光雷达、惯性测量单元（IMU）等。
摄像头可以通过图像处理技术来提取目标物体在图像中的特征点或者轮廓线，进而计算出其位置和姿态。
激光雷达可以通过发射激光束并测量其反射时间来获取目标物体的距离信息，从而实现三角定位或者点云匹配来估计位姿。
IMU则可以通过测量物体的加速度和角速度来推断其位置和姿态变化。
在位姿估计的算法中，常用的方法有：1. 特征点匹配方法：通过提取图像中的特征点，并通过计算特征点间的相对位置关系来估计位姿。
常用的特征点描述算法包括SIFT、ORB、SURF等。
2. 预测-校正方法：通过先验模型对目标物体的位置和姿态进行预测，然后根据实际观测数据进行校正。
预测模型可以基于上一时刻的位姿信息和运动模型，校正可以使用卡尔曼滤波、扩展卡尔曼滤波等方法。
3. 点云匹配方法：通过将激光雷达获取的点云数据与地图或者模型进行匹配，来估计位姿。
常用的点云匹配算法有ICP（Iterative Closest Point）算法和NDT（Normal Distributions Transform）算法等。
4. 深度学习方法：近年来，随着深度学习的发展，一些基于神经网络的方法也被应用到位姿估计中。
这些方法通过训练网络来学习从传感器数据到位姿的映射关系。
二维位姿估计在各个领域都有广泛的应用。
三维和二维平移运动估计算法分析研究
第 ６卷 第 ６期
２０ ０ ７年 １ １月
杭州 师 范学 院学报 （ 然科 学版 ） 自
Ｊ ｕｎ ｌ ｆ ａ ｇ ｈ ｕＴ ａ ｈ ｒ ｏｌ ｅ Ｎ ｔｒ ｌ ｃ ｎ ｅ Ｅ ｉｏ ） ｏ ｒ ａ ｏ ｎ ｚ ｏ ｅ ｃ ｅ ｓＣ ｌ ｇ （ ａｕ ａ Ｓ ｉ ｃ ｄ ｉｎ Ｈ ｅ ｅ ｔ
采 用 二 维运 动 估 计 或 三维 平 移 运 动 估计 ， 高 压 缩 编 码 性 能 ． 提
关键 词 ： 移 运 动 估 计 ； 维 运 动 ； 维 运 动 ； 算 复 杂性 ； ２ ４ 平 三 二 计 Ｈ． ６
中 图分 类 号 ：ＴＰ３ １ ９ 文献 标 志 码 ：Ａ
０ 引 言
ＶＯ ． Ｉ ６ ＮＯ ６ ． ＮＯ ． ２ ０７ Ｖ ０
文 章 编 号 ：０ ８ ９ ０ （ ０ ７ ０ —０ ３ —０ １０— ４３２０）６ ４５ ５
三 维 和 二维 平移 运 动 估 计 算 法 分 析 研 究
陈 丽 娅
（ 江 行 政 学 院 信 息 管 理 部 ， 江 杭 州 ３０ １ ） 浙 浙 １０ ２
法研究 ． １ １ 传 统 平移 运 动估计 算法 ．
基 于块 匹配 的搜索算 法是 常用 的运动 估计 算法 ， 把每 帧分 成若 干个块 ， 在参 考帧 中寻 找这些 块 的最 佳 匹配 ， 穷举 搜索 算法是 比较 搜索 窗 口内所有 点找 到最 小参 差 点 ， 算 法虽 然 能够 找 到 窗 口内的最 佳 匹 配 ， 该 便 于硬件 的并行 实现 ， 但是计 算 量非 常大 ， 在硬件 编码 时需要 相 当大 的功耗 ． 针对 这个 问题 ， 多快 速运 动 很
摘
要 ：运 动 估 计 和 补 偿 预 测 编 码 是 视 频 编 码 标 准 中 的一 项 关 键 技 术 ， 核 心 问 题 在 于 如 何 利 用 当前 帧 和 其
二维动量守恒公式
二维动量守恒公式好嘞，以下是为您生成的关于二维动量守恒公式的文章：在咱们学习物理的这个奇妙旅程中，二维动量守恒公式就像是一把神奇的钥匙，能帮咱们打开好多知识的大门。
先来说说啥是二维动量守恒公式。
其实啊，它就是描述在一个平面内，物体之间相互作用时动量总和保持不变的规律。
比如说，咱们想象一下打台球的场景。
台球桌上，球A 以一定的速度和方向撞击球B，在这个碰撞过程中，如果没有外力的干扰，那球 A 和球 B 的动量之和在碰撞前后是不会改变的。
我记得有一次在物理实验室里，老师让我们做一个有关二维动量守恒的实验。
那是一组简单但又很能说明问题的装置。
一个小滑块在光滑的平面上滑动，另一个滑块从侧面撞过来。
我们得通过测量它们碰撞前后的速度和质量，来验证二维动量守恒公式是不是真的那么灵。
当时我特别紧张，手都有点抖，就怕自己测的数据不准。
我小心翼翼地调整着测量工具，眼睛紧紧盯着滑块的运动，心里默默念叨着：“可千万别出错啊！”当我把最后一组数据记录下来，开始计算的时候，心都提到嗓子眼了。
结果发现，计算出来的数据和二维动量守恒公式的预测几乎一模一样，那种兴奋和成就感，真的没法形容。
那二维动量守恒公式到底怎么用呢？假设一个物体在 x 方向上的速度是 vx，质量是 m，在 y 方向上的速度是 vy，那它在 x 方向上的动量就是 m*vx，y 方向上的动量就是 m*vy。
如果有两个物体相互作用，那在这两个方向上，它们的动量之和在作用前后都是相等的。
比如说，一个小球以 3m/s 的速度水平向右运动，质量是 2kg，同时以 2m/s 的速度竖直向上运动。
那它在水平方向上的动量就是 2×3 = 6 kg·m/s，竖直方向上的动量就是 2×2 = 4 kg·m/s。
在实际生活中，二维动量守恒公式也有好多用处呢。
像火箭发射，燃料燃烧产生的气体向后喷射，给火箭一个向前的反作用力，这就可以用二维动量守恒来解释。
第二章+二维运动估计
2 2 误差函数： E ( v ) = ∫ x∈Α (eof (v) + ws eds (v))dx
2 1 1 2 2 当F=0， δ = 1 时， e ( v) = ( ∇vx + ∇v y ) = es ( v) 2 2 2 ds
∂ψ ∂t ∂ψ ∂t
对 E (v )求关于 v x 和 v y 的偏微 分，并令其为0，可得：
v 表示对像素邻域（不包括当
前像素）求平均。 表示迭代次数，初始光流矢 量 v ( 0 )可设为零。
( v xl +1)
l
v (yl +1)
∂ψ (l ) ∂ψ (l ) ∂ψ vx + vy + ∂ψ ∂x ∂y ∂t = vx(l ) − 2 2 ∂x ∂ψ ∂ψ ws + + ∂y ∂x ∂ψ (l ) ∂ψ (l ) ∂ψ v + vy + ∂ψ ∂x x ∂y ∂t = v y(l ) − 2 2 ∂y ∂ψ ∂ψ ws + + ∂x ∂y
∂ψ ∂ψ ∂ψ dx + dy + dt = 0 ∂x ∂y ∂t ∂ψ ∂ψ ∂ψ ∂ψ vx + vy + = 0 或 ∇ψ T v + =0 ∂x ∂y ∂t ∂t
T
∂x
∂y
∂t
∂ψ ∂ψ , ∇ψ = 其中 为空间梯度向量，v = (v x , v y ) 为光流场。 ∂x ∂y
提纲
概述 基于光流的运动估计 基于像素的运动估计 基于块的运动估计 基于网格的运动估计 基于区域的运动估计 全局运动估计 多分辨率运动估计
二维的Mahler猜想
J.Korean Math.Soc.48(2011),No.5,pp.985–999/10.4134/JKMS.2011.48.5.985A FILLED FUNCTION METHOD FOR BOX CONSTRAINEDNONLINEAR INTEGER PROGRAMMINGYoujiang Lin and Yongjian YangAbstract.A newfilled function method is presented in this paper tosolve box-constrained nonlinear integer programming problems.It isshown that for a given non-global local minimizer,a better local min-imizer can be obtained by local search starting from an improved initialpoint which is obtained by locally solving a box-constrained integer pro-gramming problem.Several illustrative numerical examples are reportedto show the efficiency of the present method.1.IntroductionIn[13],Zhu showed that over unbounded domain,the integer programming is undecidable;i.e.,there cannot be any algorithm for the problem.So we consider the following box-constrained discrete global optimization problem: (1.1)min{f(x):x∈X⊂Z n},where f:Z n→R,and Z n is the set of integer points in R n,and X is box;i.e., X={x∈Z n:a≤x≤b,a,b∈Z n}.Like the continuous global optimization problems,the existence of multiple local minima of a general nonconvex objective function makes discrete global optimization a great challenge.For continuous global optimization problems, many deterministic methods(see[1],[7],[8],[10])have been proposed to search for a globally optimal solution of a function of several variables.Thefilled function algorithm(see[2],[4])is an effective and practical method among determinate algorithms.The primaryfilled function was proposed by Ge in paper[2].The definition of thefilled function is as follows:Definition1.1.Assume that x∗1is a current minimizer of f(x).A continuous function P(x)is said to be afilled function of f(x)at x∗1if it satisfies the following properties:Received March8,2010;Revised July26,2010.2010Mathematics Subject Classification.52A10,52A40.Key words and phrases.global optimization,filled function method,nonlinear integer programming,local minimizer,global minimizer.The authors would like to acknowledge the support from the National Natural Science Foundation of China(10971128),Shanghai Leading Academic Discipline Project(S30104).c⃝2011The Korean Mathematical Society985986YOUJIANG LIN AND YONGJIAN YANG(1)x∗1is a maximizer of P(x)and the whole basin B∗1of f(x)at x∗1becomesa part of a hill of P(x);(2)P(x)has no minimizers or saddle points in any higher basin of f(x)than B∗1;(3)If f(x)has a lower basin than B∗1,then there is a point x′in such abasin that minimizes P(x)on the line through x and x∗1.For the definitions of basin and hill,refer to Ge(1990),paper[2].Thefilled function given at x∗1in the paper[2]has the following form:(1.2)P(x,x∗1,r,ρ)=1r+f(x)exp(−∥x−x∗1∥2ρ2),where the parameters r andρneed to be chosen appropriately.The main idea of thefilled function method is to construct an auxiliary func-tion calledfilled function via the current local minimizer of the original opti-mization problem,with the property that the current local minimizer is a local maximizer of the constructedfilled function and a better initial point of the primal optimization problem can be obtained by minimizing the constructed filled function locally.By the same idea as that of solving continuous global optimization problems,We try to solve discrete global optimization problem (1.1).In[3],Ge adapted thefilled function method by Ge[2]for continuous global optimization to the discrete case.However,thefilled function algorithm de-scribed in the paper[2]still has some unexpected features:(i)The efficiency of thefilled function algorithm strongly depend on two parameters r and q.They are not so easy to be adjusted to make them satisfy the needed conditions;(ii)Thefilled function includes exponential terms.If the value of1/ρbe-comes large as iterations proceed,as required to preserve thefilling property, numerical illness may result in failure of computation;(iii)The termination criteria is not good because it requires a large amount of computation before a global minimizer has been found.In this paper,we provide a new definition of the discretefilled function,and a discretefilled function satisfying the definition is presented.An algorithm is developed from the newfilled function.The newfilled function algorithm overcomes the disadvantages mentioned above in a certain extent.Specifically, it has the following several advantages:(i)The newfilled function includes neither exponential terms nor logarithmic terms.Elementary functionϕ(t)=arctan t is used in thefilled function,which possesses many good properties and is efficient in numerical implementations.(ii)The parameters q and r in the newfilled function are easier to be ap-propriately chosen than those of the originalfilled function(1.2).(iii)The new algorithm has a simple termination criteria and the compu-tational results show that this algorithm is more efficient than the original algorithm.A FILLED FUNCTION METHOD987The rest of this paper is organized as follows.In Section2,wefirst re-call some definitions in discrete analysis and discrete optimization.In Section 3,a definition of discretefilled function is given,a discretefilled function is presented,and we investigate its properties.An algorithm is developed and numerical experiments are presented in Section4.Finally,some conclusions are drawn in Section5.2.PreliminaryFirst,we recall some definitions in discrete analysis and discrete optimization (see[11],[12]).Definition2.1.A sequence{x i}u i=−1is called a discrete path in X between two distinct point x∗and x∗∗in X if x−1=x∗,x u=x∗∗,x i∈X for all i; x i=x j for i=j;and∥x0−x∗∥=∥x i+1−x i∥=∥x∗∗−x u−1∥=1for all i.If such a discrete path exists,then x∗and x∗∗are said to be pathwise connected in X.Furthermore,if every two distinct points in X are pathwise connected in X,then X is called a pathwise connected set.Definition2.2.The set of all axial directions in Z n is defined by D={±e i: i=1,2,...,n},where e i is the i th unit vector;The set of all feasible directions at x∈X is defined by D x={d∈D:x+d∈X},where D is the set of axial directions.Definition2.3.For any x∈Z n,the discrete neighborhood of x is defined by N(x)={x,x±e i:i=1,2,...,n};The discrete interior of X is defined by int X={x∈X:N(x)⊆X}.While,the discrete boundary of X is denoted by∂X=X/int X.Definition2.4.A point x∗∈X is called a discrete local minimizer of f over X if f(x∗)≤f(x)for all x∈X∩N(x∗).If,in addition,f(x∗)<f(x)for all x∈X∩N(x∗)/{x∗},then x∗is called a strict discrete local minimizer of over X;A point x∗∈X is called a discrete global minimizer of f over X if f(x∗)≤f(x)for all x∈X.If,in addition,f(x∗)<f(x)for all x∈X,then x∗is called a strict discrete global minimizer of f over X.Definition2.5.For any x∈X,d∈D is said to be a discrete descent direction of f at x over X if x+d∈X and f(x+d)<f(x);beside,d∗∈D is called a discrete steepest descent direction of f at x over X if f(x+d∗)≤f(x+d)for all d∈D∗,where D∗is the set of all descent direction of f at x over X. Algorithm2.6.(Discrete steepest descent method).1.Start from the initial point x∈X.2.If x is a local minimizer of f over X,then stop.Otherwise,a discrete steepest descent direction d∗of f at x over X can be found.3.Let x:=x+λd∗,whereλ∈Z+is the step length such that f has maximum reduction in direction d∗,and go to Step2.988YOUJIANG LIN AND YONGJIAN YANGAlgorithm 2.7.(Modified discrete descent method).1.Start from the initial point x ∈X .2.If x is a local minimizer of f over X ,then stop.Otherwise,letd ∗=argmin {f (x +d i ):d i ∈D x ,f (x +d i )<f (x )},where D x denotes the set of feasible directions at x .3.Let x :=x +d ∗,and go to Step 2.Obviously,by Algorithms 2.6and 2.7,we can only find a discrete local minimizer.Finally,for the discrete global optimization problem (1.1),we make the following assumptions in this paper:Assumption 2.8.X ⊆Z n is a bounded set which contains more than one point.This implies that there exists a constant K >0such that1≤K =max x,y ∈X∥x −y ∥≤∞,where ∥·∥is the usual Euclidean norm.Assumption 2.9.f :∪x ∈X N (x )→R satisfies the following Lipschiz condi-tion for every x,y ∈∪x ∈X N (x ):|f (x )−f (y )|≤L ∥x −y ∥,where 0<L <∞is a constant,N (x )is the discrete neighborhood of x .3.A filled function and its propertiesIn this section,we propose a filled function of f (x )at a current local mini-mizer and discuss its properties.LetS 1={x :|f (x )≥f (x ∗1),x ∈X \{x ∗1}},S 2={x |f (x )<f (x ∗1),x ∈X }.Definition 3.1.P (x,x ∗1)is called a discrete filled function of f (x )at a discrete local minimizer x ∗1if P (x,x ∗1)has the following properties:(1)x ∗1is a strict discrete local maximizer of P (x,x ∗1)on X ;(2)P (x,x ∗1)has no discrete local minimizers in the region S 1;(3)If x ∗1is not a discrete global minimizer of f (x ),then P (x,x ∗1)does havea discrete minimizer in the region S 2.Now,we give a discrete filled function for problem (1.1)at a local minimizer x ∗1as follows:F (x,x ∗1,q,r )=1q +∥x −x ∗1∥ϕq (max {f (x )−f (x ∗1)+r,0}),(3.1)whereϕq (t )={arctan (−qt )+π2,if t =0,0,if t =0.(3.2)r=0.2q>0and r satisfies0<r<maxx∗,x∗1∈L(P)f(x∗)<f(x∗1)(f(x∗1)−f(x∗))where L(P)stand for the set of discrete local minimizers of f(x).A simple example is given in Figure1.Next we will show that the function F(x,x∗1,q,r)is a discretefilled function satisfying Definition3.1under certain conditions on the parameters q and r. Theorem3.2.Suppose that X holds Assumption2.8.Further suppose that x∗1 is a discrete local minimizer of f(x).For any r>0,when q>0is satisfactorily small,x∗1is a strict discrete local maximizer of F(x,x∗1,q,r).Proof.Since x∗1is a discrete local minimizer of f(x)for any x∈N(x∗1)∩X, f(x)≥f(x∗1)and∥x−x∗1∥=1.Hence,we haveF(x,x∗1,q,r)=1q+1(arctan(−qf(x)−f(x∗1)+r)+π2)andF(x∗1,x∗1,q,r)=1q(arctan(−qr)+π2),990YOUJIANG LIN AND YONGJIAN YANGwhen x ≥y ,arctan(x )−arctan(y )≤x −y ;let q <r ,we haveF (x,x ∗1,q,r )−F (x ∗1,x ∗1,q,r )=1q +1(arctan(−q f (x )−f (x ∗1)+r )+π2)−1q (arctan(−q r )+π2)=1q +1arctan(−q f (x )−f (x ∗1)+r )−1q arctan(−q r )+−1q (q +1)π2=1q +1×(arctan(q r )−arctan(q f (x )−f (x ∗1)+r ))+1q ×(q +1)(arctan(q r )−π2)<1q ×(q +1)×(q ×(q r −q f (x )−f (x ∗1)+r )+(arctan 1−π2))=1q ×(q +1)×(q 2×f (x )−f (x ∗1)r ×(f (x )−f (x ∗1)+r )−π4)≤1q ×(q +1)×(q 2×L ∥x −x ∗1∥r 2−π4)=1q ×(q +1)×(q 2×L r 2−π4)≤1q ×(q +1)×(q ×L r −π4).Therefore,when 0<q <min {r,πr 4L },we have F (x,x ∗1,q,r )−F (x ∗1,x ∗1,q,r )<0for any x ∈N (x ∗1)∩X ,x ∗1is a strict discrete local maximizer of F (x,x ∗1,q,r ).□Lemma 3.3.For every x ,x ∗∈X ,if there exits i ∈{1,2,...,n }such that x ±e i ∈X ,then there exists d ∈D such that∥x +d −x ∗∥>∥x −x ∗∥.Proof.If there is an i ∈{1,2,...,n }such that x ±e i ∈X ,then either ∥x +e i −x ∗∥>∥x −x ∗∥or ∥x −e i −x ∗∥>∥x −x ∗∥,therefore we let d =e i or d =−e i ,this completes the proof.□Theorem 3.4.Suppose that Assumptions 2.8-2.9are satisfied.If x ∗1is a discrete local minimizer of f (x ),then the function F (x,x ∗1,q,r )has no discrete local minimizers in the region S 1={x |f (x )≥f (x ∗1),x ∈X/{x ∗1}}when r >0and q >0are satisfactorily small.Proof.Let ˜X =∪x ∈X N (x ),obviously,˜X holds Assumptions 2.8-2.9,and wehave S 1⊆X ⊆int ˜X.For every x ∈S 1,by Lemma 3.3,there must exists a direction d ∈D such that x +d ∈X and∥x +d −x ∗1∥>∥x −x ∗1∥.A FILLED FUNCTION METHOD991 Consider the following two cases:(1)f(x+d)≥f(x∗1):Since f(x+d)≥f(x∗1),we haveF(x+d,x∗1,q,r)−F(x,x∗1,q,r)=1q+∥x+d−x∗1∥(arctan(−qf(x+d)−f(x∗1)+r)+π2)−1q+∥x−x∗1∥(arctan(−qf(x)−f(x∗1)+r)+π2)=1q+∥x+d−x∗1∥arctan(−qf(x+d)−f(x∗1)+r)−1q+∥x−x∗1∥arctan(−qf(x)−f(x∗1)+r)+∥x−x∗1∥−∥x+d−x∗1∥(q+∥x+d−x∗1∥)(q+∥x−x∗1∥)π2=1q+∥x+d−x∗1∥×(arctan(q∗1)−arctan(q∗1))+∥x+d−x∗1∥−∥x−x∗1∥(q+∥x+d−x∗1∥)(q+∥x−x∗1∥)(arctan(qf(x)−f(x∗1)+r)−π2).If f(x)>f(x+d),thenarctan(qf(x)−f(x∗1)+r)−arctan(qf(x+d)−f(x∗1)+r)<0therefore when q<r,we have arctan(qf(x)−f(x∗1)+r)−π2<0,henceF(x+d,x∗1,q,r)−F(x,x∗1,q,r)<0.If f(x)≤f(x+d)for any x,y∈R,when x≤y,we have arctan y−arctan x≤y−x,let q<r and q<1,we haveF(x+d,x∗1,q,r)−F(x,x∗1,q,r)≤1q+∥x+d−x∗1∥×(qf(x)−f(x∗1)+r−qf(x+d)−f(x∗1)+r)+∥x+d−x∗1∥−∥x−x∗1∥(q+∥x+d−x∗1∥)(q+∥x−x∗1∥)(arctan1−π2)≤qq+∥x+d−x∗1∥×(f(x+d)−f(x)(f(x)−f(x∗1)+r)(f(x+d)−f(x∗1)+r))+∥x+d−x∗1∥−∥x−x∗1∥(q+∥x+d−x∗1∥)(2∥x−x∗1∥)(arctan1−π2)≤1(q+∥x+d−x∗1∥)(2∥x−x∗1∥)992YOUJIANG LIN AND YONGJIAN YANG×(2qL∥d∥∥x−x∗1∥r2−π4(∥x+d−x∗1∥−∥x−x∗1∥)).Hence,for any given x∈S1and d,when q is satisfactorily small thatq<πr2(∥x+d−x∗1∥−∥x−x∗1∥)8L∥x−x∗1∥,we haveF(x+d,x∗1,q,r)<F(x,x∗1,q,r).(2)f(x+d)<f(x∗1):In this case,it is clear that f(x+d)<f(x),sincefunction f(t)=arctan(−qt )+π2is increasing about t,hence,we haveF(x+d,x∗1,q,r)<F(x,x∗1,q,r).The above two cases imply that any x∈S1is not the discrete local minimizer of F(x,x∗1,q,r)when q is satisfactorily small.□Theorem3.5.If x∗1is not a discrete global minimizer of f(x)in X,then there exists a discrete minimizer¯x1∗of F(x,x∗1,q,r)in the region S2={x|f(x)< f(x∗1),x∈X}.Proof.Since x∗1is not a discrete global minimizer and F(x,x∗1,q,r)≥0,there exist a point¯x1∗∈S2and r such that f(¯x1∗)<f(x∗1)−r.Hence,F(¯x1∗,x∗1,q,r) =0,it implies that¯x∗1∈S2is a discrete minimizer of F(x,x∗1,q,r).□Theorem3.2,Theorem3.4,and Theorem3.5show that the function F(x, x∗1,q,r)at point x∗1is a discretefilled function satisfying Definition3.1with satisfactorily small q and r.The following theorems further show that the proposedfilled function has some good properties which classical functions have.Theorem3.6.Suppose that Assumption2.9is satisfied.If x1,x2∈X and satisfy the following conditions:(1)f(x1)≥f(x∗1)and f(x2)≥f(x∗1),(2)∥x2−x∗1∥>∥x1−x∗1∥.Then,when r>0and q>0are satisfactorily small,F(x2,x∗1,q,r)<F(x1,x∗1,q,r).Proof.Consider the following two cases:(1)If f(x∗1)≤f(x2)≤f(x1),then it is obvious that the result follows.(2)If f(x∗1)≤f(x1)<f(x2),the result also holds(see the proof process in Theorem3.4).□Theorem3.7.If x1,x2∈X and satisfy the following conditions:(1)∥x2−x∗∥>∥x1−x∗∥,(2)f(x1)≥f(x∗1)>f(x2),and f(x2)−f(x∗1)+r>0.Then,we have F(x2,x∗1,r,q)<F(x1,x∗1,r,q).A FILLED FUNCTION METHOD993 Proof.By Conditions1and2,we have1q+∥x2−x∗1∥<1q+∥x1−x∗1∥and0<f(x2)−f(x∗1)+r<f(x1)−f(x∗1)+r.Hence F(x2,x∗1,r,q)<F(x1,x∗1,r,q).□Now we make some remarks.Firstly,in the phase of minimizing the new discretefilled function,Theorems3.6and3.7guarantee that the current dis-crete local minimizer x∗1of the objective function is escaped and the minimum of the new discretefilled function will be always achieved at a point where the objective function value is less than the current discrete minimum.Secondly, the parameters q and r are easier to be appropriately chosen.In the next section,a new discretefilled function algorithm is given.4.Algorithm and numerical resultsBased on the theoretical results in the previous section,a global optimization algorithm over X is proposed as follows:AlgorithmInitialization:(1)Choose any x0∈X as an initial point.(2)Letε=10−5and q0=0.01.(3)Let D0={±e i:i=1,2,...,n}.Main Program:(1)Starting from initial point x0,minimize f(x)(x∈X)by the discretesteepest descent method(see Algorithm2.1),we can obtain the discretelocal minimizer x∗1.Let r=1,q=q0and D=D0.(2)Construct the discretefilled function:F(x,x∗1,q,r)=1q+∥x−x∗∥ϕq(max{f(x)−f(x∗1)+r,0}),whereϕq(t)={arctan(−qt)+π2,if t=0, 0,if t=0.(3)If r≤ε,then terminate the iteration,the x∗k is the global minimizer off(x),otherwise,the next step.(4)If D=∅,then goto(6),otherwise the next step.(5)If q<ε×10−2,then let r=r/10,q=q0/10and D=D0,goto(2);otherwise let q=q/10,goto(2).(6)Take a direction d∈D,and D←D/{d},turn to Inner Loop.994YOUJIANG LIN AND YONGJIAN YANGInner Loop:(1)k=0.(2)Let y k=x∗1+d.(3)Minimize F(x,x∗1,q,r),starting from the point y k,by implementing themodified discrete descent method(see Algorithm2.2).y k+1denotes thenext iterative point.(4)If y k+1/∈X,then return Main Program(4),otherwise next step.(5)If f(y k+1)≤f(x∗1),then let x0=y m+1and return Main program(1),otherwise let k=k+1and goto Inner Loop(3).In the following part,several test problems are given and results of the algorithm in solving these problems are reported.Through out the tests,we use the modified discrete descent method as shown in Algorithm2.7to perform local searches,in the initialization of the algorithm we let q=0.01and r=1. The algorithm in Fortran95is successfully used tofind the global minimizers of these test problems.The main iterative results are summarized in tables for each function.The symbols used are shown as follows:k:The iteration number infinding the k th local minimizer.x0 k or y0k:The k th initial point.f(x0k )or f(y0k):The function value of the k th initial point.x∗k or y∗k:The k th local minimizer.f(x∗k )or f(y∗k):The function value of the k th local minimizer.Problem4.1.min f(x)=100(x2−x21)2+(1−x1)2+90(x4−x23)2+(1−x3)2,+10.1[(x2−1)2+(x4−1)2]+19.8(x2−1)(x4−1), s.t.10≤x i≤10,i=1,2,3,4.This problem is a discrete counterpart of the problem38in[6].It is a box constrained nonlinear integer programming problem.It has214≈1.94×105 feasible points where41of them are discrete local minimizers but only one of those discrete local minimizers is the discrete global minimum solution:x∗global =(1,1,1,1)with f(x∗global)=0.Let x01=(9,6,5,6)and x01=(9,−9,−9,9),a summary of the computational results are displayed in Tables 1and2.Problem4.2.min f(x)=g(x)h(x),s.t.x i=0.001y i,−2000<y i<2000,i=1,2,where y i(i=1,2)is integer,andg(x)=1+(x1+x2+1)2(19−14x1+3x21−14x2+6x1x2+3x2),h(x)=30+(2x1−3x2)2(18−32x1+12x21+48x2−36x1x2+27x22).A FILLED FUNCTION METHOD995 Table1.Problem4.1initial point is(9,6,5,6)k x0k f(x0k)x∗kf(x∗k)1(9,6,5,6)596070.0(3,8,3,8)2158.0000 2(3,10,2,5)1887.5000(3,8,2,3)1007.5000 3(3,10,1,1)922.1000(3,8,0,−1)453.1000 4(2,5,0,−1)235.6000(2,4,0,0)43.6000 5(1,1,0,0)11.1000(1,1,1,1)0.0000 Table2.Problem4.1initial point is(9,−9,−9,9)k x0k f(x0k)x∗kf(x∗k)1(9,−9,−9,9)1276796.40(3,8,−3,8)2170.0000 2(3,10,−2,5)1895.5000(3,8,−2,3)1015.5000 3(3,10,−1,1)926.1000(3,8,0,−1)453.1000 4(2,5,0,−1)235.6000(2,4,0,0)43.6000 5(1,1,0,0)11.1000(1,1,1,1)0.0000Table3.Problem4.2k y0k f(y0k)y∗kf(y∗k)1(1000,1000)1876.000(1280,890)954.13822(2000,632)951.2482(1609,71)92.34253(1379,2000)19.4895(85,2000)−210795.62531(−1000,−1000)1890.000(0,−1000) 3.0000002(0,1740)−256.5294(85,2000)−210795.62531(2000,2000)35028.000(1280,890)954.13822(2000,632)951.2482(1609,71)92.34263(1379,2000)19.4895(85,2000)−210795.62531(−2000,−2000)20811.9999(0,−1000) 3.0000002(0,1740)−256.5294(85,2000)−210795.6253 This problem is a discrete counterpart of the Goldstein and Price’s function in[5].It is a box constrained nonlinear integer programming problem.It has 40012≈1.60×107feasible points.More precisely,it has207and2discrete local minimizers in the interior and the boundary of box−2.00≤x i≤2.00,i=1,2, respectively.Nevertheless,it has only one discrete global minimum solution:y∗global =(85,2000)with f(y∗global)=−210795.6253.We used four initial pointsin our experiment:(1000,1000),(−1000,−1000),(2000,2000),(−2000,−2000), a summary of the computational results are displayed in Table3.Problem4.3(Beale’s function).min f(x)=[1.5−x1(1−x2)]2+[2.25−x1(1−x22)]2+[2.625−x1(1−x32)]2,996YOUJIANG LIN AND YONGJIAN YANGTable4.Problem4.3k y0k f(y0k)y∗kf(y∗k)1(3000,5000)146039.2031(−3,9735) 6.06732(10000,963) 6.0183(3015,504)3.7589×10−5 1(−3000,−5000)150120.7031(11,−5964)9.02302(10000,788)8.9179(3015,504)3.7589×10−5 1(8000,8000)16992808.2031(−3,9735) 6.06732(10000,963) 6.0183(3015,504)3.7589×10−5 1(−8000,−8000)17121524.2031(5,−7842)8.67812(10000,790)8.6132(3015,504)3.7589×10−5Table5.Problem4.4k y0k f(y0k)y∗kf(y∗k)1(−5000,−5000,−5000,−5000)3650.0000(−116,12,55,−56)3.7213×10−4 2(−78,8,38,−38)7.9387×10−5(−79,8,38,−38)7.9045×10−5 3(11,−1,11,11)1.2798×10−6(10,−1,11,11)2.7985×10−7 4(0,0,3,3)2.1060×10−9(0,0,1,1)2.6000×10−11 1(5000,5000,5000,5000)3650.0000(116,−12,56,57)3.7859×10−4 2(78,−8,38,38)7.9387×10−5(79,−8,38,38)7.9045×10−5 3(−11,1,−11,−11)1.2798×10−6(−10,1,11,−11)2.7985×10−7 4(0,0,3,−3)2.1060×10−9(0,0,1,−1)2.6000×10−11Table6P N DN IN T I F N1450.236049532230.070732317673220.05216224364444278.245675801758525247.49104950368555021147.47298451600510024406.341276667658625242.6725336724326502249.264535436658610023087.764275437056s.t.x i=0.001y i,−104≤y i≤104,i=1,2,where y i(i=1,2)is integer.This problem is discrete counterpart if the problem203in[9].It is a box constrained nonlinear integer programming problem.It has200012≈4.00×108 feasible points and many discrete local minimizers,but it has only one discreteA FILLED FUNCTION METHOD 997Table parison of the resultsGe’s algorithm New algorithm PN DN IN TI F N RA IN TI F N RA 1.41073.667707461.22−4638.564466530.47−5321837.5366175644.73−311328.102295761.78−512657.0751126852.03−38375.807522360.38−435866.2673855076.12−418406.542453513.80−618659.5161008410.24−415315.345500551.67−62.21157.36723520.10−4648.0512303.15−613168.74265900.67−56123.0985675.03−71389.52763501.25−4631.9165850.72−63.263454.3553674598.07−412982.440134223.90−567098.6216532109.12−4101752.812179839.92−64.42012873.31053243183.19−3162577.1205453901.10−5205952.54630876415.00−416214.3423166343.00−65.253228750.21783421091.05−6181524.7083452972.13−7326532.0972*******.09−618352.761965312.20−7316098.73230678124.56−619213.450882064.12−76.25181105.5503680560.06−525428.650309874.31−818963.2712987675.01−525320.710756183.00−8182544.3268021142.10−523484.56289457.00−7183245.71123547700.38−525145.3542345528.29−8global minimum solution y ∗global =(3000,500)with f (y ∗global )=0.We used four initial points in our experiment:(3000,5000),(−3000,−5000),(8000,8000),(−8000,−8000),a summary of the computational results are displayed in Table 4.Problem 4.4(Powell’s singular function).min f (x )=(x 1+10x 2)2+5(x 3−x 4)2+(x 2−2x 3)4+10(x 1−x 4)4,s .t .x i =0.001y i ,−104≤y i ≤104,i =1,2,3,4,where y i (i =1,2,3,4)is integer.It is a box constrained nonlinear integer programming problem.It has 200014≈1.60×1017feasible points and many local minimizers,but it hasonly one global minimum solution:y ∗global =(0,0,0,0)with f (y ∗global )=0.We used two initial points in our experiment:(−5000,−5000,−5000,−5000),(5000,5000,5000,5000),a summary of the computational results are displayed in Table 5.Problem 4.5.min f (x )=(x 1−1)2+5(x n −1)2+n −1∑i =1(n −i )(x 2i −x i +1)2,s .t .−5≤y i ≤5,i =1,2,...,n.998YOUJIANG LIN AND YONGJIAN YANGThis problem is a generalization of the problem282in[9].It is a box con-strained nonlinear integer programming problem.It has11n feasible points andmany local minimizers,but it has only one global minimum solution:x∗global =(1,...,1)with f(x∗global )=0.For all problem with different size,we used fourinitial points in our experiment:(5,...,5),(−5,...,−5),(−5,...,−5,5,...,5), (5,...,5,−5,...,−5).For every experiment,the proposed algorithm succeeded in identifying the discrete global minimum.Let x01=(5,...,5),for n= 25,50,100,respectively,the summary of the computational results are dis-played in Table6.Problem4.6(Rosenbrock’s function).min f(x)=n−1∑i=1[100(x i+1−x2i)2+(1−x i)2],s.t.−5≤y i≤5,i=1,2,...,n.It is a box constrained/unconstrained nonlinear integer programming prob-lem.It has11n feasible points and many local minimizers,but it has onlyone global minimum solution:x∗global =(1,1,...,1)with f(x∗global)=0.Forall problems with different sizes,we used four initial points in our experiment: (5,...,5),(−5,...,−5),(−5,...,−5,5,...,5),(5,...,5,−5,...,−5).For ev-ery experiment,the proposed algorithm succeeded in identifying the discrete global minimum.Let x01=(5,...,5),for n=25,50,100,respectively,the summary of the computational results are displayed in Table6.In Table6,we give the experiment results for Problems4.1-4.6.In Table7, Problems4.1-4.6.with2to25variables are tested and the table shows that in most cases the newfilled function algorithm works better than Ge’sfilled function algorithm.The symbols used in Tables6-7are shown as follows: P N:The N th problem.DN:The dimension of objective function of a problem.IN:The number of iteration cycles.T I:The CPU time in seconds for the algorithm to stop.F N:The number of objective function evaluations for algorithm to stop.RA:The ratio of the number of function evaluations to the number of feasible points.5.ConclusionsThis paper gives a definition of thefilled function for the nonlinear integer programming problem,and presents a newfilled function.Afilled function algorithm based on this givenfilled function is designed.The implementation of the algorithm on several test problems is reported with satisfactory numerical results.A FILLED FUNCTION METHOD999 Acknowledgment.The authors are most grateful to the referee for his many excellent suggestions for improving the original manuscript.References[1] B.C.Cetin,J.Barhen,and J.W.Burdick,Terminal repeller unconstrained subenergytunneling(TRUST)for fast global optimization,J.Optim.Theory Appl.77(1993),no.1,97–126.[2]R.P.Ge,Afilled function method forfinding a global minimizer of a function of severalvariables,Math.Programming46(1990),no.2,(Ser.A),191–204.[3]R.P.Ge and C.B.Huang,A continuous approach to nonlinear integer programming,put.34(1989),no.1,part I,39–60.[4]R.P.Ge and Y.F.Qin,The globally convexizedfilled functions for global optimization,put.35(1990),no.2,part II,131–158.[5] A.A.Goldstein and J.F.Price,On descent from local minima,p.25(1971),569–574.[6]W.Hock and K.Schittkowski,Test Examples for Nonlinear Programming Codes,Springer,NewYork,1981.[7]R.Horst,P.M.Pardalos,and N.V.Thoai,Introduction to Global Optimization,seconded.,Kluwer Academic Publishers,Dordrecht,2000.[8] A.V.Levy and A.Montalvo,The tunneling algorithm for the global minimization offunctions,SIAM put.6(1985),no.1,15–29.[9]K.Schittkowski,More Test Examples for Nonlinear Programming Codes,Springer,NewYork,1987.[10]Y.Yao,Dynamic tunneling algorithm for global optimization,IEEE Trans.SystemsMan Cybernet.19(1989),no.5,1222–1230.[11]L.S.Zhang,F.Gao,and W.X.Zhu,Nonlinear integer programming and global opti-mization,put.Math.17(1999),no.2,179–190.[12]W.X.Zhu,An approximate algorithm for nonlinear integer programming,Appl.Math.Comput.93(1998),no.2-3,183–193.[13],Unsolvability of some optimization problems,put.174(2006),no.2,921–926.Youjiang LinDepartment of MathematicsShanghai UniversityShanghai200444,P.R.ChinaE-mail address:linyoujiang@Yongjian YangDepartment of MathematicsShanghai UniversityShanghai200444,P.R.ChinaE-mail address:yjyang@。
运动估计综述
运动估计综述1.定义这里指基于块的运动估计，基本思想是将图像序列的每一帧分成许多互不重叠的块，并认为块内所有像素的位移量都相同，然后对每个宏块到参考帧某一给定特定搜索范围内根据一定的块匹配准则找出与当前块最相似的块，即匹配块，匹配块与当前块的相对位移即为运动矢量。
2.运动估计算法2.1全搜索每一点都要比较，需计算（2*d+1）*(2*d+1)次（d是搜索范围）。
对分辨率360x288，帧率30fps的视频，设d=21，每秒要计算1.09E10次，计算量太大，需要研究相应的快速算法。
2.2早期的快速算法（固定模式法）这些算法假设匹配误差随着离全局误差最小点的距离增加而单调增加。
一般从原点开始，采用固定的搜索模板和搜索策略得到最佳匹配块。
常见的有：三步法（TSS）、四步法（FSS）、菱形法（DS）、六边形法(HEXBS)等。
三步法（TSS）四步法（FSS ）菱形法（DS ）：六边形法（HEXBS ）：早期算法的不足：∙ 没有利用图像本身的相关信息，不能根据物体运动的剧烈程度自适应的改变搜索起点和搜索半径；∙ 以菱形法为例，对背景图像，也要经历从大模板到小模板的转换过程，至少需要13个搜索点，搜索速度还有待改进；∙ 对于运动剧烈的图像，从原点开始搜索时，要经过多次搜索才能找到匹配点，搜索点过多，且容易陷入局部最优点。
2.3近年来提出的新算法针对以上不足，近几年来，针对序列图像的时空相关性和人眼视觉特性，提出了许多改进算法，主要从以下几个方面着手：∙预测搜索起点利用相邻块之间的运动相关性选择一个反映当前块运动趋势的预测点作为初始搜索点，这个预测点一般比原点更靠近全局最小点。
从预测点开始搜索可以在一定程度上提高搜索速度和搜索精度。
∙中止判别条件利用相邻块的相关性自适应的调整终止阀值，当搜索值小于该值时，则认为满足条件，跳出后面的搜索过程。
∙搜索模板的选择在序列图像中，大多数的运动矢量都位于水平或垂直方向，因此可以设计相应的搜索模板（非对称搜索模板）来加快搜索速度。
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W
F I tr （ F I ）
2．3 基于像素的运动估计
• 2.3.1 位移帧差
• 定义像素x从t1时刻到t1+dt时刻的位移帧差为：
e ( x , d ) ( x d , t1 d t ) ( x , t1 )
2
(x d ) 1 (x)
) (
2
v y x
) (
2
v y y
)
2
迭代更新公式
vx
( l 1)
vx
(l )
x
x ws
vx
(l )
y
2
vy
(l )
t
2
x
(l )
y
(l )
vy
( l 1)
数字视频处理
黎洪松 主编 北京邮电大学出版社
第2章 二维运动估计
• 2．1概述
• 根据摄象机和目标物体的运动状态,运动 可分为：摄像机不动物体不动，摄像机 不动物体动，摄像机动物体不动和摄像 机动物体动4种。
2.1.1几个基本概念
• 1．时间序列图像
• 运动估计主要是研究空间中的物体运动
情况。为了观察物体的运动，人们将摄
vy
(l )
y
x ws
vx
y
2
vy
t
2
x
y
2.2.4 有向平滑约束
• 有向平滑约束可表示为
e ds ( v ) ( v x ) W ( v x ) ( v y ) W ( v y )
• 投影运动：
a 0 a1 x a 2 y x ( x, y) 1 c1 x c 2 y x b b x b y ( x, y) 1 2 0 y y 1 c x c y 1 2
d d
• 仿射运动：
2.4.2 相位相关法
• 相位相关法广泛用于图像配准，即两幅图像 只发生全局平移的情况。假设两帧图像 和
2 (x)
1
(x)
之间只是纯平移关系，有
1 (x) 2 (x d )
• 互相关函数为
C 1, 2 ( f x , f y ) 1 ( f x , f y ) 2 ( f x , f y )
差函数求解运动矢量di。
• 邻域B(xi)上的误差函数可表示为
E (d i ) 1 2
w ( x )(
2
(x d i ) 1 (x ))
2
xB ( x i )
• 误差函数关于运动矢量的梯度为
E (d i ) d E d
i
w ( x ) e ( x , d i ) x
3．运动估计的基本问题
• 由摄像机模型知道，摄像机的投影过程
是不可逆的，投影过程中将不可避免地
丢失一些有用信息，例如物体的深度信
息等，因此估计物体的真实运动和结构 是比较困难的。
O
X
x Y
y
p ( x, y )
(x, y )
p' ( x' , y ' )
P ' ( X ' ,Y ' , Z ' )
xB
xB
w (x) x y w (x) y y
1
xB
xB
w (x ) x t w (x ) y t
2.2.3 运动平滑约束
• 目标误差函数为
t dt
x
t
t dt
x
d(x, t; t dt )
x
d(x, t; t dt )
前向运动估计
图2-3 前向运动估计和后向运动估计
二维运动估计存在几个问题：
1. 遮挡问题，即图像中的某些像素可能不存在
对应点。
2. 孔径问题，也称为多义性问题。
3. 噪声问题，在生成视频图像的过程中，噪声
中一个同样大小的块
1 ( x ) | xB 2 ( x d )
1 ( x)
(a)非重叠块
2 ( x)
1 ( x)
(b)重叠块
2 ( x)
图2-10块平移模型
2．可变形块运动模型
• 可变形块运动模型可以对物体的旋转、
缩放、变形等建模。块的运动参数不再 是简单的一个平移参数，而是一些空间 变换参数，常用的可变形块运动模型有 投影运动、仿射运动、双线性运动等。
x 1 4
( m 1, n , k ) ( m , n , k ) ( m 1, n 1, k )
( m , n 1, k ) ( m 1, n , k 1 ) ( m , n , k 1 ) ( m 1, n 1, k 1 ) ( m , n 1, k 1 )
是不可避免的。
2.1.3 二维运动模型及估计方法
• 为了有效地进行运动估计，需要对真实
运动施加一些约束条件，即对运动建模。 我们可以将二维运动模型粗略地分为参
数模型和非参数模型。
(a) (b) (c) (d)
(a)基于像素的运动; (b)基于块的运动; (c)全局运动; (d)基于区域的运动
图2-6
t
1 4
( m , n , k 1 ) ( m , n , k ) ( m 1, n , k 1 )
( m 1, n , k ) ( m , n 1, k 1 ) ( m , n 1, k ) ( m 1, n 1, k 1 ) ( m 1, n 1, k )
(X , Y , Z )
P( X , Y , Z )
Z
图2-1特征点对应运动模型的几何图形
4．运动分析方法
1. 根据时间相邻的两幅或多幅图像求解
物体的运动参数和三维结构信息； 2. 图像序列的光流分析方法。
(a)
(b)
图2-2 真实运动与光流不一致的情况
2.1.2二维运动估计
后向运动估计
数最小化的方法来更新运动矢量。可表示为
d (x, t; d t ) d p (x, t; d t ) u (x, t; d t )
2．基于梯度下降的算法
• 像素递归方程为
d
( l 1)
d
(l )
e(x, d
(l )
) x 2 ( x d
(l )
)
• Caffario 和 Rocca 提出另一个类似的步 长表达式
d x ( x, y) a 0 a1 x a 2 y d ( x, y) b 0 b1 x b 2 y y
• 双线性运动：
d x ( x, y) a 0 a 1 x a 2 y a 3 xy d y ( x, y) b b x b y b xy 1 2 3 0
(b) 8像素邻域及其团
图2-9 吉布斯（Gibbs）随机场的团
• 运动场的先验概率密度函数可定义为
c p (d | 1 ) exp V d ( d | 1 ) Q c C 1
• 对于一个包含两个像素的团，势函数可 写成如下形式
V d ( d ( x i ), d ( x j )) d ( x i ) d ( x j )
E (v)
( e of ( v ) w s e s ( v ) )d x
2
2
x A
e of
2
(v) v v x y x y t
2
es ( v ) v x
2
2
vy
2
(
vx x
) (
2
vx y
c 2
2．4 基于块的运动估计
• 2.4.1 概述
• 在基于块的运动估计中，先将视频图像 分成一个个规则的图像块，然后对每个
图像块估计运动参数。
1．块平移模型
• 块平移模型假定每个块只做二维平移运动。 给定两帧视频图像
1
(x)
和
2
(x)
ቤተ መጻሕፍቲ ባይዱ
，对于
1
(x)
中
的一个块 B ，可由
重建，即
2
(x)
1 x 2 ( x d
(l ) 2
)
2
3．多点邻域约束
1 5 2 6 3 x 4
1～6 为先前估计过的像素， x 为当前像素
图2-8 因果邻域
E (x, d )
xB
[ e ( x , d )]
2
xB
[
( x d i ) 1 ( x )] 2
2
2.3.4基于贝叶斯准则的方法
• 贝叶斯准则是基于运动估计问题的概率
公式，视频图像函数和运动场均视为随 机场，由概率密度函数建模，分别称为 观察模型和运动场模型。
• 给定两帧视频图像 1 ( x ) 和
2
(x )
，运动场估
计可以看成一个最大后验概率（MAP）问题， 即
d MAP arg max p ( d | 1 , 2 )
d
p ( d | 1 , 2 )
p (
2
| d , 1 ) p ( d | 1 ) p (