WO2022222367A1 - 基于压入与重标记可提前终止的最大流最小割求解算法 - Google Patents

基于压入与重标记可提前终止的最大流最小割求解算法 Download PDF

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WO2022222367A1
WO2022222367A1 PCT/CN2021/119508 CN2021119508W WO2022222367A1 WO 2022222367 A1 WO2022222367 A1 WO 2022222367A1 CN 2021119508 W CN2021119508 W CN 2021119508W WO 2022222367 A1 WO2022222367 A1 WO 2022222367A1
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push
algorithm
condition
relabel
early termination
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刘心哲
闫光耀
哈亚军
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上海科技大学
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    • G06V10/20—Image preprocessing
    • G06V10/26—Segmentation of patterns in the image field; Cutting or merging of image elements to establish the pattern region, e.g. clustering-based techniques; Detection of occlusion
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06V10/26—Segmentation of patterns in the image field; Cutting or merging of image elements to establish the pattern region, e.g. clustering-based techniques; Detection of occlusion
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  • the invention relates to a max-flow min-cut solution algorithm which can be terminated in advance based on a push-in and re-marking algorithm.
  • the Graph cut algorithm has been widely used to solve the minimum cut problem, which is common in computer vision tasks. These tasks map image pixels onto the nodes of the graph, and then assign the most probable label to each node of the graph. Based on this, all nodes in the graph should be cut into two disjoint sets, namely set S and set T. It should minimize the application-specific cost function.
  • the minimum cut problem can often be transformed into a maximum flow problem because it is more intuitive and easier to solve.
  • These tasks widely appear in image/video segmentation, image/video stitching, binocular vision matching, classification, image fusion, image dehazing, skeletonization, style transfer, etc.
  • the technical problem to be solved by the present invention is: the existing Push-relabel usually cannot make full use of the characteristics of application scenarios to reduce redundant computation, so it is not suitable for application scenarios with high resolution and real-time requirements.
  • a technical solution of the present invention is to provide a maximum flow minimum cut solution algorithm based on push-in and re-marking that can be terminated early, for applications that do not require an exact maximum flow, characterized in that the separation The condition and the stable condition constitute the early termination condition of the Push-relabel algorithm.
  • the separation condition is defined as: there is no augmentation path from the source point to the sink point in the residual graph;
  • the stability condition is defined as: there is no augmenting path from any active node to the sink in the residual graph.
  • Another technical solution of the present invention is to provide an application of the above-mentioned maximum flow minimum cut solution algorithm that can be terminated in advance based on push-in and re-marking. Applications with maximum flow.
  • the separation condition set by the present invention ensures that the set V formed by all nodes v has been divided into two disjoint sets S and T, that is, a (s, t)-cut has been generated.
  • the stability condition set by the present invention ensures that the generated (s, t)-cut will not change until the algorithm satisfies the original termination condition.
  • the present invention proposes a novel early termination technique that greatly eliminates redundant computations and ensures that the algorithm terminates correctly in all cases.
  • Experimental results show that using the new termination conditions can reduce the computational effort to an average of 2% in the test data.
  • Figure 1 shows the results of multiple test cases
  • Figure 2 shows the statistical results obtained with more test data.
  • the original termination condition of the push-relabel algorithm is to check that there are no active nodes in the residual graph.
  • the original termination condition of the Push-relabel algorithm creates redundant computations in applications that require only a maximum flow value or a minimum cut but not a maximum flow.
  • the set S and the set T have been separated for a long time in the later stage of the algorithm, and will not change until the end, the present invention refers to this period as the "finishing stage".
  • the Push-relabel algorithm is busy pushing excess flow in set T to sink t, and relabeling the height of active nodes in set S to allow excess flow to flow back to source s.
  • the "finishing phase" of the Push-relabel algorithm is redundant.
  • the present invention proposes that the early termination condition of the Push-relabel algorithm consists of a separation condition and a stable condition.
  • the separation condition is defined as: there is no augmentation path from the source point to the sink point in the residual graph
  • the stable condition is defined as : There is no augmenting path from any active node to the sink in the residual graph.
  • the Push-relabel algorithm includes the following:
  • V represents the set of vertices
  • E represents the set of edges
  • a pre-flow is a pseudo-flow, and the residual flow is non-negative for all non-source nodes:
  • a path is an ordered sequence of points ⁇ u, w 0 , w 1 , ..., w k , v ⁇ , for any adjacent points in the arrangement such as wi and w i+1 , there exists ( wi , w i+1 ) ⁇ E.
  • u is said to reach v if there is a path P(u, v).
  • the distance the number of points in P(u, v) with the least number of points included is reduced by one, which is called the distance from u to v, marked as d(u, v), and d(u, v) is initialized to 0.
  • T the set of all points that can reach the sink t.
  • the maximum flow minimum cut solution algorithm that can be terminated early based on the push-in and re-marking algorithm includes the following steps:
  • the height of the source point s can be initialized to
  • Any number of streaming operations, re-marking operations, and T-tree updating operations can be performed in random order.
  • the termination judgment is made at any time of the above calculation process, and it is judged whether the separation condition and the stability condition are satisfied at the same time. Among them, if there is no source point s in the set T, the separation condition is satisfied; if there is no active node in the set T, the stability condition is satisfied.
  • Each bar in Figure 1 represents a test case, and the abscissa represents the number of iteration cycles.
  • the iteration cycle represented by the leftmost part there is no (s, t)-cut.
  • the iteration cycle indicated by the second part from the left there is already a (s, t)-cut, but it is not yet a minimum cut.
  • the cycle represented by the third part from the left there is already a minimum cut, but the algorithm will still run because there are still active nodes in T, i.e. the stability condition is not satisfied.
  • the iteration period represented by the first part of the right number there are no more active nodes in T, but the algorithm with the original termination condition will continue to run because there are still active nodes in S.
  • the algorithm of the present invention will stop at the right end of the third part from the left, while the original algorithm will stop at the right end of the first part from the right. It can be seen that the algorithm of the present invention can save a lot of computation.
  • Figure 2 is the statistical result obtained with more test data.
  • the second bar from the bottom indicates that the minimum cut has been determined. This is not entirely consistent with the third item above, which states that the stability condition is satisfied.
  • the iterative gap between the averages of these two bars is called the termination gap.
  • the left and right edges of the bar graph are the minimum and maximum values, respectively, of the number of iterations that satisfy the corresponding conditions.
  • the left slash in the middle is the average.
  • the time to determine the minimum cut is close to the time to satisfy the stable condition, and is far from the original termination of the Push-relabel algorithm (first from the top).
  • the present invention is applicable to all max-flow/min-cut problems where a push-to-relabel algorithm can be used. Including running on CPU, GPU or FPGA.
  • This embodiment provides an FPGA implementation example:
  • the present invention uses the widely used Middlebury benchmark push-relabeling algorithm.
  • the baseline image is segmented into 25 ⁇ 25 tiles.
  • the tile size is also the size of the processor array.
  • the processor array can process 625 nodes in a 2D grid graph in parallel.
  • the present invention uses SystemVerilog to describe the hardware implementation of the entire system.
  • the present invention uses Xilinx's Virtex Ultrascale VU190 as an evaluation board, and uses Xilinx Vivado 2020.1 as an EDA tool.
  • the clock frequency of the VU190 evaluation board is set to 66MHz. Compared with the latest technology, the present invention can achieve at least 12 times acceleration by using the early termination function.

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Abstract

一种基于压入与重标记可提前终止的最大流最小割求解算法，用于不需要确切最大流量的应用，由分离条件和稳定条件构成Push-relabel算法的提前终止条件；在Push-relabel算法进行过程中的任意时刻，若集合T中不存在源点s，s∈S，则满足分离条件；若集合T中不存在任何活跃节点则满足稳定条件；若分离条件及稳定条件都满足，则Push-relabel算法终止。该提前终止的求解算法可以大大消除冗余计算，并确保算法在所有情况下都能正确终止。实验结果表明，使用新的终止条件可以在测试数据中将计算量平均减少到原来的2％。

Description

基于压入与重标记可提前终止的最大流最小割求解算法 技术领域
本发明涉及一种基于压入与重标记算法能够提前终止的最大流最小割求解算法。
背景技术
Graph cut算法已广泛用于解决最小切割问题，该问题在计算机视觉任务中很普遍。这些任务将图像像素映射到图的节点上，然后为图的每个节点分配最可能的标签。基于此，应将图中的所有节点切成两个不相交的集合，即集合S及集合T。切应使特定于应用场景的成本函数最小化。最小切割问题通常可以转换为最大流量问题，因为它更直观且更易于解决。这些任务广泛出现在图像/视频分割、图像/视频拼接、双目视觉匹配、分类、图像融合、图像去雾、骨架化、风格迁移等。
先前的工作表明，图割可以产生令人印象深刻的结果质量，但是，在实际应用中相对较慢。现有的实现通常不能充分利用应用场景特点来减少冗余计算。例如，在发生最大流量问题的许多应用中，仅需要最大流量的值或最小切，而不需要完整的最大流量。
先前的工作还尝试优化推入重贴标算法的计算。全局重新标记通过使用全局信息更新高度标签，减少了集合T中的冗余重新标记操作。相反，间隙重新标记通过更新高度标签的分布来减少集合S中的多余的推入和重新标记操作。此外，JF-cut提出了一种提前终止技术来去除冗余计算。只要没有找到增加路径，它就会终止推入重贴标签算法。但是，它们的提前终止条件过于激进，因为它不能确保算法在所有情况下都能正确终止。
发明内容
本发明要解决的技术问题是：现有的Push-relabel通常无法充分利用应用场景特点来减少冗余计算，因此不适合具有高分辨率和实时要求的应用程序场景。
为了解决上述技术问题，本发明的一个技术方案是提供了一种基于压入与重 标记可提前终止的最大流最小割求解算法，用于不需要确切最大流量的应用，其特征在于，由分离条件和稳定条件构成Push-relabel算法的提前终止条件，在残差图中，设可以达到汇点t的所有节点v构成集合T，其余节点构成集合S；在Push-relabel算法进行过程中的任意时刻，若集合T中不存在源点s，s∈S，则满足分离条件；若集合T中不存在任何活跃节点则满足稳定条件；若分离条件及稳定条件都满足，则Push-relabel算法终止，其中：
将分离条件定义为：残差图中不存在从源点到汇点的增广路径；
将稳定条件定义为：残差图中不存在从任一活跃节点到汇点的增广路径。
本发明的另一个技术方案是提供了一种上述的基于压入与重标记可提前终止的最大流最小割求解算法的应用，其特征在于，用于仅需要最大流量值或最小割断但不要求最大流量的应用。
本发明设定的分离条件保证了所有节点v构成的集合V已经被分割为两个不相交的集合S和集合T，也即已经产生了一个(s，t)-cut。本发明设定的稳定条件保证了产生的(s，t)-cut直到算法满足原始终止条件都不再会发生变化。
本发明提出了一种新颖的提前终止技术，可以大大消除冗余计算，并确保算法在所有情况下都能正确终止。实验结果表明，使用新的终止条件可以在测试数据中将计算量平均减少到原来的2％。
附图说明
图1为多个测试用例的结果；
图2为通过更多测试数据获得的统计结果。
具体实施方式
下面结合具体实施例，进一步阐述本发明。应理解，这些实施例仅用于说明本发明而不用于限制本发明的范围。此外应理解，在阅读了本发明讲授的内容之后，本领域技术人员可以对本发明作各种改动或修改，这些等价形式同样落于本申请所附权利要求书所限定的范围。
Push-relabel算法的原始终止条件是检查残差图中是否没有活动节点。但是，Push-relabel算法的原始终止条件在仅需要最大流量值或最小割断但不要求最大 流量的应用中创造了多余的计算。在这些应用中，集合S和集合T在算法的后期已经分开了很长时间，并且直到最后都不会改变，本发明将这段时期为“收尾阶段”。在“收尾阶段”，Push-relabel算法忙于将集合T中的多余流量推到汇点t处，并重新标记集合S中的活动节点的高度以使excess flow流回源点s。对于不需要确切最大流量的应用，Push-relabel算法的“收尾阶段”是多余的。
为此，本发明提出由分离条件和稳定条件构成Push-relabel算法的提前终止条件，将分离条件定义为：残差图中不存在从源点到汇点的增广路径，将稳定条件定义为：残差图中不存在从任一活跃节点到汇点的增广路径。
基于上述提前终止条件，Push-relabel算法包括以下内容：
对于一张图G(V，E)，V表示顶点的集合，E表示边的集合，定义源点s∈V，定义汇点t∈V，定义边的容量函数c：E→R，R表示实数集，定义流网络F＝(G，c，s，t)。
定义伪流(pseudo-flow)f p：E→R。f p满足以下两个约束：
(1)对称约束(Skew symmetry)：f(u，v)＝-f(v，u)，(u，v)表示一条从u到v的有向边，(v，u)表示从v到u的有向边；
(2)容量约束(Capacity constraint)：f(u，v)≤c(u，v)。
定义节点的剩余流ef：V→R。对节点u，剩余流ef(u)＝∑ v∈Vf(v，u)。
定义活跃节点：如果对节点u，ef(u)＞0，则称该节点u为活跃节点。
定义预流(pre-flow)f：E→R。预流是一种伪流，且对所有非源节点，剩余流都不为负：
Figure PCTCN2021119508-appb-000001
定义图G关于预流f的残差图G f(V，E f)，E f表示残差图的边集。
定义剩余容量c f：E→R。对于边e，剩余容量c f(e)＝c(e)-f(e)。
定义节点的高度标签函数h：V→N，N表示整数集。如果
Figure PCTCN2021119508-appb-000002
h(u)≤h(v)+1，则称该高度标签为“合法的”。
定义可行边：如果对于边(u，v)，h(u)≤h(v)+1，则称该边为可行边。
定义路径P(u，v)，路径是一串有序的点的排列{u，w 0，w 1，...，w k，v}，对于排列中任意相邻的点如w i和w i+1，都存在(w i，w i+1)∈E。
定义可到达：如果存在路径P(u，v)，则称u可到达v。
定义距离：所包含点的数目最少的P(u，v)中点的数目减一，称为u到v的距离，标示为d(u，v)，d(u，v)初始化为0。
定义集合T：所有可到达汇点t的点组成的集合。
则基于压入与重标记算法能够提前终止的最大流最小割求解算法包括以下步骤：
执行初始化操作：
初始化预流f(e)←0，
Figure PCTCN2021119508-appb-000003
初始化高度标签h(v)←d(v，t)；
或者还可以将源点s的高度初始化为|V|，其他点的初始化标签设定为0。此时应立即对由源点出发的所有边执行一次推流操作，而不检查它是否为可行边。
执行推流操作：
在集合T中找一活跃节点u，对该活跃节点找一可行边(u，v)，按如下顺序进行推流操作：
δ←min(ef(u)，c f(u，v))；
f(u，v)←f(u，v)+δ；
f(v，u)←f(v，u)-δ；
ef(u)←ef(u)-δ；
ef(v)←ef(v)+δ；
执行重标记操作
在集合中找一活跃节点u，如果该活跃节点不存在一条可行边，进行重标记操作：
Figure PCTCN2021119508-appb-000004
执行T树更新操作：
从汇点t出发搜索所有能到达t的节点以更新集合T
任意次的推流操作、重标记操作和T树更新的操作可以随机排序后执行。
在上述计算过程的任意时刻进行终止判断，判定是否同时满足分离条件以及稳定条件，若是，则算法终止，否则，返回推流操作。其中，若集合T中不存在源点s则满足分离条件；若集合T中不存在任何活跃节点则满足稳定条件。
图1中的每个条形表示一个测试用例，横坐标表示迭代周期数。在最左边部分所表示的迭代周期里，不存在一个(s，t)-cut。在左数第二部分所表示的迭 代周期里，已经存在了一个(s，t)-cut，但是还不是最小切。在左数第三部分所表示的周期里，已经有了一个最小切，但是算法仍将运行，因为T中仍然有活跃节点，也即不满足稳定条件。在右数第一部分所表示的迭代周期里，T中已经没有活跃节点了，但采用原始终止条件的算法仍将继续运行，因为仍然有活跃节点存在于S。本发明的算法将停止在左数第三部分的右端，而原始算法将停止在右数第一部分的右端。可以看出，本发明的算法可以节省很多计算量。
图2是通过更多测试数据获得的统计结果。下数第二条表示已确定最小切割。这与表示满足稳定条件的上数第三条并不完全一致。这两个条的平均值之间的迭代间隙称为终止间隙。条形图的左边缘和右边缘分别是满足相应条件的迭代次数的最小值和最大值。中间的左斜杠是平均值。由图可知，确定最小切割的时间与满足稳定条件的时间接近，而与Push-relabel算法的原始终止(上数第一)相去甚远。
本发明适用于所有可以使用推入重贴标签算法的最大流量/最小切割问题。包括在CPU、GPU或FPGA上运行。本实施例提供了一个FPGA实现示例：
为了评估本发明提出的方法，本发明使用了广泛使用的Middlebury基准测试推入重贴标签算法。基准图像被分割为25×25的图块。图块大小也是处理器阵列的大小。处理器阵列可以并行处理2D网格图中的625个节点。本发明使用SystemVerilog来描述整个系统的硬件实现。本发明使用Xilinx的Virtex Ultrascale VU190作为评估板，并使用Xilinx Vivado 2020.1作为EDA工具。VU190评估板的时钟频率设置为66MHz。与最新技术相比，采用了提前终止功能，本发明至少可以实现12倍的加速。

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  1. 一种基于压入与重标记可提前终止的最大流最小割求解算法，用于不需要确切最大流量的应用，其特征在于，由分离条件和稳定条件构成Push-relabel算法的提前终止条件，在残差图中，设可以达到汇点t的所有节点v构成集合T，其余节点构成集合S；在Push-relabel算法进行过程中的任意时刻，若集合T中不存在源点s，s∈S，则满足分离条件；若集合T中不存在任何活跃节点则满足稳定条件；若分离条件及稳定条件都满足，则Push-relabel算法终止，其中：
    将分离条件定义为：残差图中不存在从源点到汇点的增广路径；
    将稳定条件定义为：残差图中不存在从任一活跃节点到汇点的增广路径。
  2. 一种如权利要求1所述的基于压入与重标记可提前终止的最大流最小割求解算法的应用，其特征在于，用于仅需要最大流量值或最小割断但不要求最大流量的应用。
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