图像分割过分割和欠分割_使用图割的图像分割
圖像分割過分割和欠分割
Image segmentation technology is an important research direction in the field of computer vision and an important part of image semantic understanding. Image segmentation refers to the process of dividing an image into several areas with similar properties. From a mathematical point of view, image segmentation is the process of dividing an image into disjoint areas.The area can be the foreground and background of the image or a single object. These areas can be constructed using features such as color, edges, or similarity of neighbors.
圖像分割技術是計算機視覺領域的重要研究方向，也是圖像語義理解的重要組成部分。 圖像分割是指將圖像劃分為多個具有相似屬性的區域的過程。 從數學的角度來看，圖像分割是將圖像劃分為不相交的區域的過程，該區域可以是圖像的前景和背景，也可以是單個對象。 可以使用諸如鄰居的顏色，邊緣或相似性之類的特征來構造這些區域。
Graph cutting algorithm is one of the classic algorithms of combinatorial graph theory. In recent years, many scholars have applied it to image and video segmentation and achieved good results. This article briefly introduces the graph cut algorithm and interactive image segmentation technology, as well as the application of graph cut algorithm in interactive image segmentation.
圖切割算法是組合圖理論的經典算法之一。 近年來，許多學者將其應用于圖像和視頻分割，并取得了良好的效果。 本文簡要介紹了圖割算法和交互式圖像分割技術，以及圖割算法在交互式圖像分割中的應用。
基本概念 (Basic concept)
We use the theories and methods in the field of graph theory to map the image into a weighted undirected graph, treat the pixels as nodes, and regard the image segmentation problem as the vertex division problem of the graph, using the smallest cutting criterion to obtains the best segmentation of the image.
我們使用圖論領域中的理論和方法，將圖像映射成加權無向圖，將像素作為節點，并將圖像分割問題視為圖的頂點劃分問題，并使用最小的切割準則獲得圖像的最佳分割。
This type of method associates the problem of image segmentation with the problem of MIN-CUT . The usual approach is to map the image to be segmented into a weighted undirected graph G=(V, E), where , V is the set of vertices, and E is the set of edges.
這種類型的方法將圖像分割問題與MIN-CUT問題聯系起來。 通常的方法是將要分割的圖像映射到加權無向圖G =(V，E)，其中，V是頂點集，E是邊緣集。
The edge formed by the connection of every two neighboring vertices are called n-links and the connection between each ordinary vertex and the two terminal vertices are called t-links.
每兩個相鄰頂點之間的連接所形成的邊稱為n-link ，每個普通頂點與兩個終端頂點之間的連接稱為t-link。
Each node corresponds to each pixel in the image, and each edge ∈ E connects a pair of adjacent pixels, and the weight of the edge is w(i,j) represents the non-negative similarity in gray, color or texture between adjacent pixels.
每個節點對應于圖像中的每個像素，每個邊緣∈E連接一對相鄰的像素，并且邊緣的權重為w(i，j)表示相鄰像素之間在灰度，顏色或紋理上的非負相似度。
Boykov and Jolly originally proposed to compute the histograms of the labeled pixels to approximate probability density functions , and to let
Boykov和Jolly最初提出計算標記像素的直方圖，以近似概率密度函數，并讓
For example, if fB is very low, then wi,F will be very high, making it much more likely that the edge between i and B is cut. The inter-node weights are computed using a simple similarity measure
例如，如果fB非常低，則wi，F將非常高，從而更有可能切割i和B之間的邊緣。 使用簡單的相似性度量來計算節點間權重
Blake et al. showed how the parameter σ could be estimated based on the local contrast of an image sample.
布萊克等。 展示了如何根據圖像樣本的局部對比度估計參數σ。
We take a two-category division as an example, divide G = (V,E) into two subsets A, B .These two subsets correspond to the foreground pixel set and the background pixel set of the image, which is equivalent to completing the image segmentation, where:
我們以兩類劃分為例，將G =(V，E)劃分為兩個子集A，B，這兩個子集分別對應于圖像的前景像素集和背景像素集，相當于完成了圖像分割，其中：
A segmentation S of an image is a cut of the image, and each region C ∈ S that is segmented corresponds to a sub-image in the image. It is normal in combinatorial optimization to define the cost of a cut as the sum of the costs of the edges that it severs.
圖像的分割S是圖像的切割，并且被分割的每個區域C∈S對應于圖像中的子圖像。 在組合優化中，將切割的成本定義為切割的邊的成本之和是正常的。
The cost of the cut is the sum of the weights of all edges in the edge set C.If a cut has the smallest sum of weights of all edges, then this cut is called a minimum cut.
切割的成本是邊集C中所有邊的權重的總和。如果切割的所有邊的權重總和最小，則此切割稱為最小切割。
Maxflow–Mincut定理 (Maxflow–Mincut Theorem)
圖流 (Flow in a graph)
We consider a directed graph (S, A), with afinite set of vertices S and a set of arcs which connect some of these vertices .
我們考慮一個有向圖(S，A)，它具有一組有限的頂點S和一組連接這些頂點中的一些的弧。
Among the vertices are distinguished the source S, and the well P.With each arc is associated a strictly positive real number, called capacitance .
在頂點之間可以區分出源S和阱P。每個圓弧都有一個嚴格的正實數，稱為電容。
We seek to pass a maximum flow of a liquid , from the source to the well — the flow in each arc not exceeding its capacity. In other words, we are looking for a function f of the set of arcs in R such that:
我們力求使液體的最大流量從源頭流向井-每個電弧中的流量均不超過其容量。 換句話說，我們正在尋找R中的一組弧的函數f，使得：
- for any arc a, 0≤f (a) ≤c (a), where c (a) is the capacity of the arc. 對于任何弧a，0≤f(a)≤c(a)，其中c(a)是弧的容量。
- for any vertex other than the source or the well, the sum of the flow rates of the incoming arcs is equal to the sum of the flows of the outgoing arcs. 對于除源或井以外的任何頂點，輸入弧的流率之和等于輸出弧的流之和。
We speak of a flow for such an application. We seek to determine a maximum flow , in the sense that
我們談到了此類應用程序的流程 。 我們試圖確定最大流量 ，
- The sum of the flow rates of the arcs leaving the source is maximum. 離開源頭的電弧的流量之和最大。
Here is an example of a flow .
這是流程的示例。
However, it is not maximum,it can for example be improved by adding a bit rate of 1 on the S-a-b-d-e-P path .
但是，它不是最大的，例如可以通過在SabdeP路徑上添加1的比特率來改善 。
There are several algorithms to achieve maximum flow, such as Dinic or ISAP algorithm.
有幾種算法可以實現最大流量，例如Dinic或ISAP算法。
最小割 (Minimum cut)
The value of a maximum flow is equal to the value of a minimum cut.
最大流量的值等于最小切割的值。
Moreover, if (A, B) is a minimal cut, and that a is an arc having its start in A and its end in B, is saturated by any maximum flow.
此外，如果(A，B)是最小切口，并且a是在A處開始而B處結束的弧，則任何最大流量都將其飽和。
結論 (Conclusion)
This lesson cover the basic, low-level operations and tools of image processing, which are necessary for understanding most of the commonly used methods and tools of computer vision.
本課程涵蓋圖像處理的基本，低級操作和工具，這對于理解計算機視覺的大多數常用方法和工具都是必需的。
提價 (Refrences)
Adelson, Edward H., and James R. Bergen (1991), “The plenoptic function and the elements of early vision”, Computational models of visual processing 1.2 (1991).
Adelson，Edward H.和James R. Bergen(1991年)，“ 全光功能和早期視覺的要素 ”，視覺處理1.2的計算模型(1991年)。
Boykov, Y., Veksler, O., and Zabih, R. (2001), “approximate energy minimization via graph cuts,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(11): 1222–1239.
Boykov，Y.，Veksler，O.和Zabih，R.(2001)，“ 通過圖割實現近似能量最小化 ”，《 IEEE模式分析和機器智能交易 》 ， 23(11)：1222-1239。
D.M. Greig, B.T. Porteous and A.H. Seheult (1989), Exact maximum a posteriori estimation for binary images, Journal of the Royal Statistical Society, Series B, 51, 271–279.
DM基利，BT波蒂厄斯和AH Seheult(1989)， 精確最大為二進制圖像后驗估計 ，雜志皇家統計學會，B系列，51，271-279的。
D. Geman and S. Geman (1984), Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6, 721–741.
D. Geman和S. Geman(1984)， 隨機松弛，Gibbs分布和圖像的貝葉斯復原 ，IEEE Trans。 模式肛門。 馬赫 INTELL，6，721-741。
J.E. Besag (1986), On the statistical analysis of dirty pictures (with discussion), Journal of the Royal Statistical Society Series B, 48, 259–302.
JE Besag(1986年)， 在骯臟的照片(與討論)的統計分析 ， 皇家統計學會的 B系列，48，259-302。
翻譯自: https://medium.com/swlh/image-segmantation-using-graph-cut-540ada07c327
圖像分割過分割和欠分割
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