目标检测——交并比（Intersection of Union，IoU）计算

超、凢脫俗 2022-10-29 04:28 280阅读 0赞

在检测任务中，使用交并比（Intersection of Union，IoU）作为衡量指标,来描述两个框之间的重合度。这一概念来源于数学中的集合，用来描述两个集合 A A A和 B B B之间的关系，它等于两个集合的交集里面所包含的元素个数，除以它们的并集里面所包含的元素个数，具体计算公式如下：

I o U = A ∩ B A ∪ B IoU = \\frac\{A\\cap B\}\{A \\cup B\} IoU=A∪BA∩B

两个框可以看成是两个像素的集合，它们的交并比等于两个框重合部分的面积除以它们合并起来的面积。  
图“交集”中青色区域是两个框的重合面积，  
图“并集”中蓝色区域是两个框的相并面积。  
用这两个面积相除即可得到它们之间的交并比，如 **图1** 所示。

![c2095c01997044f8a054d676ab585f3beed4400961ea40379771a1fd6d8bf2ea][]  
图1：交并比  
  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzI3Mjc4MQ_size_16_color_FFFFFF_t_70][]  
图2：交并比公式

假设两个矩形框A和B的位置分别为：  
 A : \[ x a 1 , y a 1 , x a 2 , y a 2 \] A: \[x\_\{a1\}, y\_\{a1\}, x\_\{a2\}, y\_\{a2\}\] A:\[xa1,ya1,xa2,ya2\]

B : \[ x b 1 , y b 1 , x b 2 , y b 2 \] B: \[x\_\{b1\}, y\_\{b1\}, x\_\{b2\}, y\_\{b2\}\] B:\[xb1,yb1,xb2,yb2\]

假如位置关系如 **图3** 所示：

![8abb3525989044fa8f512f82dfcd0f2f244594ff79cc4f14a900a09d464b22b8][]  
图3：计算交并比

如果二者有相交部分，则相交部分左上角坐标为：  
 x 1 = m a x ( x a 1 , x b 1 ) ,       y 1 = m a x ( y a 1 , y b 1 ) x\_1 = max(x\_\{a1\}, x\_\{b1\}), \\ \\ \\ \\ \\ y\_1 = max(y\_\{a1\}, y\_\{b1\}) x1=max(xa1,xb1),     y1=max(ya1,yb1)

相交部分右下角坐标为：  
 x 2 = m i n ( x a 2 , x b 2 ) ,       y 2 = m i n ( y a 2 , y b 2 ) x\_2 = min(x\_\{a2\}, x\_\{b2\}), \\ \\ \\ \\ \\ y\_2 = min(y\_\{a2\}, y\_\{b2\}) x2=min(xa2,xb2),     y2=min(ya2,yb2)

计算先交部分面积：  
 i n t e r s e c t i o n = m a x ( x 2 − x 1 + 1.0 , 0 ) ⋅ m a x ( y 2 − y 1 + 1.0 , 0 ) intersection = max(x\_2 - x\_1 + 1.0, 0) \\cdot max(y\_2 - y\_1 + 1.0, 0) intersection=max(x2−x1\+1.0,0)⋅max(y2−y1\+1.0,0)

矩形框A和B的面积分别是：  
 S A = ( x a 2 − x a 1 + 1.0 ) ⋅ ( y a 2 − y a 1 + 1.0 ) S\_A = (x\_\{a2\} - x\_\{a1\} + 1.0) \\cdot (y\_\{a2\} - y\_\{a1\} + 1.0) SA=(xa2−xa1\+1.0)⋅(ya2−ya1\+1.0)

S B = ( x b 2 − x b 1 + 1.0 ) ⋅ ( y b 2 − y b 1 + 1.0 ) S\_B = (x\_\{b2\} - x\_\{b1\} + 1.0) \\cdot (y\_\{b2\} - y\_\{b1\} + 1.0) SB=(xb2−xb1\+1.0)⋅(yb2−yb1\+1.0)

计算相并部分面积：  
 u n i o n = S A + S B − i n t e r s e c t i o n union = S\_A + S\_B - intersection union=SA\+SB−intersection

计算交并比：

I o U = i n t e r s e c t i o n u n i o n IoU = \\frac\{intersection\}\{union\} IoU=unionintersection

--------------------

交并比计算程序如下：

# 计算IoU，矩形框的坐标形式为xyxy，这个函数会被保存在box_utils.py文件中
    def box_iou_xyxy(box1, box2):
        # 获取box1左上角和右下角的坐标
        x1min, y1min, x1max, y1max = box1[0], box1[1], box1[2], box1[3]
        # 计算box1的面积
        s1 = (y1max - y1min + 1.) * (x1max - x1min + 1.)
        # 获取box2左上角和右下角的坐标
        x2min, y2min, x2max, y2max = box2[0], box2[1], box2[2], box2[3]
        # 计算box2的面积
        s2 = (y2max - y2min + 1.) * (x2max - x2min + 1.)
        
        # 计算相交矩形框的坐标
        xmin = np.maximum(x1min, x2min)
        ymin = np.maximum(y1min, y2min)
        xmax = np.minimum(x1max, x2max)
        ymax = np.minimum(y1max, y2max)
        # 计算相交矩形行的高度、宽度、面积
        inter_h = np.maximum(ymax - ymin + 1., 0.)
        inter_w = np.maximum(xmax - xmin + 1., 0.)
        intersection = inter_h * inter_w
        # 计算相并面积
        union = s1 + s2 - intersection
        # 计算交并比
        iou = intersection / union
        return iou
    
    
    bbox1 = [100., 100., 200., 200.]
    bbox2 = [120., 120., 220., 220.]
    iou = box_iou_xyxy(bbox1, bbox2)
    print('IoU is {}'.format(iou))

# 计算IoU，矩形框的坐标形式为xywh
    def box_iou_xywh(box1, box2):
        x1min, y1min = box1[0] - box1[2]/2.0, box1[1] - box1[3]/2.0
        x1max, y1max = box1[0] + box1[2]/2.0, box1[1] + box1[3]/2.0
        s1 = box1[2] * box1[3]
    
        x2min, y2min = box2[0] - box2[2]/2.0, box2[1] - box2[3]/2.0
        x2max, y2max = box2[0] + box2[2]/2.0, box2[1] + box2[3]/2.0
        s2 = box2[2] * box2[3]
    
        xmin = np.maximum(x1min, x2min)
        ymin = np.maximum(y1min, y2min)
        xmax = np.minimum(x1max, x2max)
        ymax = np.minimum(y1max, y2max)
        inter_h = np.maximum(ymax - ymin, 0.)
        inter_w = np.maximum(xmax - xmin, 0.)
        intersection = inter_h * inter_w
    
        union = s1 + s2 - intersection
        iou = intersection / union
        return iou
    
    bbox1 = [100., 100., 200., 200.]
    bbox2 = [120., 120., 220., 220.]
    iou = box_iou_xywh(bbox1, bbox2)
    print('IoU is {}'.format(iou))

为了直观的展示交并比的大小跟重合程度之间的关系，**图4** 示意了不同交并比下两个框之间的相对位置关系，从 IoU = 0.95 到 IoU = 0.

![52faaa317c6c4de59088de70e7ba0d478969aacffa6e40b4998a3ca9203a9ee2][]  
图4：不同交并比下两个框之间相对位置示意图

--------------------

## 参考文章 ##

[目标检测基础介绍][Link 1]  
[理解IOU、precision、recall、AP、mAP的含义][IOU_precision_recall_AP_mAP]  
[IOU计算][IOU]

[c2095c01997044f8a054d676ab585f3beed4400961ea40379771a1fd6d8bf2ea]: /images/20221024/8fb8bcd324c34144991bbe5df3033916.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzI3Mjc4MQ_size_16_color_FFFFFF_t_70]: /images/20221024/144755c5f5da43aba84f431c7dec63ef.png
[8abb3525989044fa8f512f82dfcd0f2f244594ff79cc4f14a900a09d464b22b8]: /images/20221024/f0d4a8e93c924105854e86e84e9f0705.png
[52faaa317c6c4de59088de70e7ba0d478969aacffa6e40b4998a3ca9203a9ee2]: /images/20221024/08dcd24af0b941dd8820ca32b646e6a6.png
[Link 1]: https://aistudio.baidu.com/aistudio/projectdetail/1524437
[IOU_precision_recall_AP_mAP]: https://blog.csdn.net/qq_16540387/article/details/81912289
[IOU]: https://zhuanlan.zhihu.com/p/83591574