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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。 图1:交并比 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: 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. 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 理解IOU、precision、recall、AP、mAP的含义 IOU计算 |
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