一文读懂IoU，GIoU, DIoU, CIoU, Alpha-IoU (代码非常优雅)

IoU就是就是我们说的交并比 Intersection over Union ，具体就是两个box的交集除以并集。

当我们计算我们的anchors 或者 proposals 与 ground truth bounding boxes 的损失的时候，就需要用到IoU。不同的IoU有不同的特性。

IoU loss：

IoU计算了最简单的情况：

IoU的损失函数公式：至于求loss为啥用1减去，是因为：iou越大 代表拟合效果越好，我们应让模型的loss越小。iou最大为1，也就是重合的情况。因此用1-IoU来代表loss。
$$Los{s}_{IoU}=1-IoU$$

GIoU loss：

当两个anchor与gt box都不相交的时候，IoU的loss是一样大的，我们理论认为anchor距离gt box越近，loss应该越小，不应该一样大。这样GIoU就提出来了。GIoU通过计算两个box的最小闭包区域ac来计算loss。底色为红色的范围是Anchor2与Gt box的最小闭包区域，底色为黄色的范围是Anchor1与Gt box的最小闭包区域。明显Anchor2的最小闭包区域小，u代表并集，ac代表最小闭包区域，ac越大， $L_{GIoU}$ 值越大。Anchor1的ac大，所以Anchor1的损失更高
$$L_{GIoU}=1-IoU+\frac{ac-u}{ac}$$

DIoU loss：

黄色为proposal，蓝色为gt box。当propsals与gt box重叠时，我们认为下面的两种情况左边的效果好，因为它位于gt box中心。GIoU并不能解决这个问题，所以DIoU loss被提出来了，DIoU loss用，两个box中心点距离平方 除以 最小闭包区域对角线距离平方，来衡量预测的proposal是否位于gt box中心。
$$Los{s}_{DIoU}=1-（IoU+\frac{{\rho }^2(b,b^{gt})}{c^2}）$$

CIoU loss：

当两个proposals都位于gt box中心时，我们还是认为坐标的效果比较好，因为左边的宽高比跟我们的gt box一致，所以CIoU loss在DIoU loss的基础上改进了宽高比。
$$Los{s}_{CIoU}=1-IoU+\frac{{\rho }^2(b,b^{gt})}{c^2}+\alpha v$$
v用来度量长宽比的相似性，$\frac{4}{{\pi }^2}$是作者感觉这个值取得的效果不错，就用了这个值。
$$v=\frac{4}{{\pi }^2}(arctan\frac{w^{gt}}{h^{gt}}-arctan\frac{w}{h}{)}^2$$
alpha是权重值，衡量ciou公式中第三项和第四项的权重，当IoU越大，alpha就越大，alpha大就优先考虑v； IoU越小时，alpha越小，alpha小优先考虑第三项，距离比。
$$\alpha =\frac{v}{(1-IoU)+v}$$

Alpha-IoU loss：

Alpha-IoU loss主要是考虑IoU大于0.5的时候的梯度，因为在普通 $L_{IoU}=1-IoU$ 中，IoU的梯度一直是-1。但是在Alpha-IoU loss中，当iou大于0.5的时候，loss的梯度是大于-1的，收敛的更快，在map0.7/map0.9有提升效果。
$$Los{s}_{\alpha -IoU}=1-Io{U}^{\alpha }$$
$\alpha >0$，当 $\alpha =2，iou=0.6$ 时，$L_{\alpha -IoU}$ 的梯度已经是-1.2了。训练时经验所得alpha取3比较好。

下面是常见的loss function的代码实现。

import torch
import math
import torch.nn.functional as F
# 放到编辑器里收缩每个方法，直接看main函数，然后再对应loss去看。
#################################   分类损失  ########################################
def BCE_loss(proposals, gt_boxes):
    # 对所求概率进行 clamp ，不然当某一概率过小时，进行 log ，会让 loss 变为 nan
    proposals = proposals.clamp(min=0.0001, max=1.0)
    diff = gt_boxes * torch.log(proposals) + (1 - gt_boxes) * (torch.log(1 - proposals))
    loss = -torch.mean(diff)
    return loss

def CE_loss(proposals, gt_boxes_classid):
    loss = 0
    # 对所求概率进行 clamp ，不然当某一概率过小时，进行 log ，会让 loss 变为 nan
    proposals = proposals.clamp(min=0.0001, max=1.0)

    # method1:
    for i in range(proposals.shape[0]):
        fenzi = torch.exp(proposals[i][gt_boxes_classid[i]])  # proposals中对应真实类别的置信度
        fenmu = torch.sum(torch.exp(proposals[i]))  # 所有类别的置信度
        loss += -torch.log(fenzi / fenmu)
    # method2:
    # for i in range(proposals.shape[0]):
    #     first=-proposals[i][gt_boxes_classid[i]]
    #     second=torch.log(torch.sum(torch.exp(proposals[i])))
    #     loss+=first+second

    return loss / proposals.shape[0]

#################################   位置损失  ########################################
def L1_loss(proposals, gt_boxes):
    # 优点：鲁棒性好，因为梯度各个地方都为1，所以对异常值不是那么敏感。
    # 缺点：不稳定解，达不到最优解，也就是函数最低点。
    # 也叫MAE，平均绝对误差，预测值和真实值之间距离的平均值
    diff = torch.abs(gt_boxes - proposals)
    loss = torch.mean(diff)
    return loss

def L2_loss(proposals, gt_boxes):
    # 优点：稳定解，能够达到最优解，也就是函数最低点。
    # 缺点：鲁棒性差，对异常值敏感，容易形成梯度爆炸。
    # 也叫MSE，均方误差。预测值和真实值之差的平方的平均值。
    diff = torch.pow(gt_boxes - proposals, 2)
    loss = torch.mean(diff)
    return loss

def Smooth_l1_loss(proposals, gt_boxes):
    diff = torch.abs(gt_boxes - proposals)
    diff = torch.where(diff < 1, 0.5 * diff * diff, diff - 0.5)
    loss = torch.mean(diff)
    return loss

def IoU_loss(boxa, boxb):
    """
    boxa/boxb:Tensor [x1,y1,x2,y2],    x2,y2保证大于x1,y1
    loss = 1 - iou
    """
    inter_x1, inter_y1 = torch.maximum(boxa[:, 0], boxb[:, 0]), torch.maximum(boxa[:, 1], boxb[:, 1])
    inter_x2, inter_y2 = torch.minimum(boxa[:, 2], boxb[:, 2]), torch.minimum(boxa[:, 3], boxb[:, 3])
    inter_h = torch.maximum(torch.tensor([0]), inter_y2 - inter_y1)
    inter_w = torch.maximum(torch.tensor([0]), inter_x2 - inter_x1)
    inter_area = inter_w * inter_h
    union_area = ((boxa[:, 3] - boxa[:, 1]) * (boxa[:, 2] - boxa[:, 0])) + \
                 ((boxb[:, 3] - boxb[:, 1]) * (boxb[:, 2] - boxb[:, 0])) - inter_area + 1e-8  # + 1e-8 防止除零
    iou = inter_area / union_area
    iou_loss = 1 - iou
    return iou_loss

def GIoU_loss(boxa, boxb):
    """
    # 为了解决当两个bbox不相交时，距离远的和距离近的损失值一样大。我们认为距离近的损失应该小一点。
    # 注意：划分anchor是否是正样本的时候，anchor与label不一定相交，这样giou能够起到积极的作用
    # 当用正样本计算与label的iou损失时，这时候正样本与label都是相交的情况，这时候GIoU不一定起到积极的作用。
    giou = iou-(|ac-u|)/|ac|   ac最小闭包区域，u并集
    loss = 1 - giou
    """
    inter_x1, inter_y1 = torch.maximum(boxa[:, 0], boxb[:, 0]), torch.maximum(boxa[:, 1], boxb[:, 1])
    inter_x2, inter_y2 = torch.minimum(boxa[:, 2], boxb[:, 2]), torch.minimum(boxa[:, 3], boxb[:, 3])
    inter_h = torch.maximum(torch.tensor([0]), inter_y2 - inter_y1)
    inter_w = torch.maximum(torch.tensor([0]), inter_x2 - inter_x1)
    inter_area = inter_w * inter_h
    union_area = ((boxa[:, 3] - boxa[:, 1]) * (boxa[:, 2] - boxa[:, 0])) + \
                 ((boxb[:, 3] - boxb[:, 1]) * (boxb[:, 2] - boxb[:, 0])) - inter_area + 1e-8  # + 1e-8 防止除零

    # 求最小闭包区域的x1,y1,x2,y2,h,w,area
    ac_x1, ac_y1 = torch.minimum(boxa[:, 0], boxb[:, 0]), torch.minimum(boxa[:, 1], boxb[:, 1])
    ac_x2, ac_y2 = torch.maximum(boxa[:, 2], boxb[:, 2]), torch.maximum(boxa[:, 3], boxb[:, 3])
    ac_w = ac_x2 - ac_x1
    ac_h = ac_y2 - ac_y1
    ac_area = ac_w * ac_h

    giou = (inter_area / union_area) - (torch.abs(ac_area - union_area) / ac_area)
    giou_loss = 1 - giou
    return giou_loss

def DIoU_loss(boxa, boxb):
    """
    # 当boxes与真实box重合时，一个在中间重合，一个在边缘重合，我们认为在中间重合的是比较好的，
    # 所以提出计算两个box中心点的距离，因为预测小目标的中心点box与真实值box本来距离就很小，
    # 所以再除以一个最小闭包区域对角线长度，来平衡小目标和大目标的diou。都用平方不开根号减少计算量和精度损失。
    diou=iou-两个box中心点距离平方/最小闭包区域对角线距离平方
    loss=1-diou
    """
    # 求交集
    inter_x1, inter_y1 = torch.maximum(boxa[:, 0], boxb[:, 0]), torch.maximum(boxa[:, 1], boxb[:, 1])
    inter_x2, inter_y2 = torch.minimum(boxa[:, 2], boxb[:, 2]), torch.minimum(boxa[:, 3], boxb[:, 3])
    inter_h = torch.maximum(torch.tensor([0]), inter_y2 - inter_y1)
    inter_w = torch.maximum(torch.tensor([0]), inter_x2 - inter_x1)
    inter_area = inter_w * inter_h

    # 求并集
    union_area = ((boxa[:, 3] - boxa[:, 1]) * (boxa[:, 2] - boxa[:, 0])) + \
                 ((boxb[:, 3] - boxb[:, 1]) * (boxb[:, 2] - boxb[:, 0])) - inter_area + 1e-8  # + 1e-8 防止除零

    # 求最小闭包区域的x1,y1,x2,y2
    ac_x1, ac_y1 = torch.minimum(boxa[:, 0], boxb[:, 0]), torch.minimum(boxa[:, 1], boxb[:, 1])
    ac_x2, ac_y2 = torch.maximum(boxa[:, 2], boxb[:, 2]), torch.maximum(boxa[:, 3], boxb[:, 3])

    # 把两个bbox的x1,y1,x2,y2转换成ctr_x,ctr_y
    boxa_ctrx, boxa_ctry = boxa[:, 0] + (boxa[:, 2] - boxa[:, 0]) / 2, boxa[:, 1] + (boxa[:, 3] - boxa[:, 1]) / 2
    boxb_ctrx, boxb_ctry = boxb[:, 0] + (boxb[:, 2] - boxb[:, 0]) / 2, boxb[:, 1] + (boxb[:, 3] - boxb[:, 1]) / 2

    # 求两个box中心点距离平方length_box_ctr，最小闭包区域对角线距离平方length_ac，以及diou
    length_box_ctr = (boxb_ctrx - boxa_ctrx) * (boxb_ctrx - boxa_ctrx) + \
                     (boxb_ctry - boxa_ctry) * (boxb_ctry - boxa_ctry)
    length_ac = (ac_x2 - ac_x1) * (ac_x2 - ac_x1) + (ac_y2 - ac_y1) * (ac_y2 - ac_y1)
    # 求平方，相乘是最快的
    iou = inter_area / (union_area + 1e-8)
    diou = iou - length_box_ctr / length_ac
    diou_loss = 1 - diou
    return diou_loss

def CIoU_loss(boxa, boxb):
    """
    # 当boxes与真实box重合时，且都在在中心点重合时，一个长宽比接近真实box，一个差异很大
    # 我们认为长宽比接近的是比较好的，损失应该是比较小的。所以ciou增加了对box长宽比的考虑
    ciou=iou+两个box中心点距离平方/最小闭包区域对角线距离平方+alpha*v
    loss=1-iou+两个box中心点距离平方/最小闭包区域对角线距离平方+alpha*v
    注意loss跟上边不一样，这里不是1-ciou
    v用来度量长宽比的相似性,4/(pi *pi)*(arctan(boxa_w/boxa_h)-arctan(boxb_w/boxb_h))^2
    alpha是权重值，衡量ciou公式中第二项和第三项的权重，
    alpha大优先考虑v，alpha小优先考虑第二项距离比,alpha = v / ((1 - iou) + v)。
    """
    # 求交集
    inter_x1, inter_y1 = torch.maximum(boxa[:, 0], boxb[:, 0]), torch.maximum(boxa[:, 1], boxb[:, 1])
    inter_x2, inter_y2 = torch.minimum(boxa[:, 2], boxb[:, 2]), torch.minimum(boxa[:, 3], boxb[:, 3])
    inter_h = torch.maximum(torch.tensor([0]), inter_y2 - inter_y1)
    inter_w = torch.maximum(torch.tensor([0]), inter_x2 - inter_x1)
    inter_area = inter_w * inter_h

    # 求并集
    union_area = ((boxa[:, 3] - boxa[:, 1]) * (boxa[:, 2] - boxa[:, 0])) + \
                 ((boxb[:, 3] - boxb[:, 1]) * (boxb[:, 2] - boxb[:, 0])) - inter_area + 1e-8  # + 1e-8 防止除零

    # 求最小闭包区域的x1,y1,x2,y2
    ac_x1, ac_y1 = torch.minimum(boxa[:, 0], boxb[:, 0]), torch.minimum(boxa[:, 1], boxb[:, 1])
    ac_x2, ac_y2 = torch.maximum(boxa[:, 2], boxb[:, 2]), torch.maximum(boxa[:, 3], boxb[:, 3])

    # 把两个bbox的x1,y1,x2,y2转换成ctr_x,ctr_y
    boxa_ctrx, boxa_ctry = boxa[:, 0] + (boxa[:, 2] - boxa[:, 0]) / 2, boxa[:, 1] + (boxa[:, 3] - boxa[:, 1]) / 2
    boxb_ctrx, boxb_ctry = boxb[:, 0] + (boxb[:, 2] - boxb[:, 0]) / 2, boxb[:, 1] + (boxb[:, 3] - boxb[:, 1]) / 2
    boxa_w, boxa_h = boxa[:, 2] - boxa[:, 0], boxa[:, 3] - boxa[:, 1]
    boxb_w, boxb_h = boxb[:, 2] - boxb[:, 0], boxb[:, 3] - boxb[:, 1]

    # 求两个box中心点距离平方length_box_ctr，最小闭包区域对角线距离平方length_ac
    length_box_ctr = (boxb_ctrx - boxa_ctrx) * (boxb_ctrx - boxa_ctrx) + \
                     (boxb_ctry - boxa_ctry) * (boxb_ctry - boxa_ctry)
    length_ac = (ac_x2 - ac_x1) * (ac_x2 - ac_x1) + (ac_y2 - ac_y1) * (ac_y2 - ac_y1)

    v = (4 / (math.pi * math.pi)) * (torch.atan(boxa_w / boxa_h) - torch.atan(boxb_w / boxb_h)) \
        * (torch.atan(boxa_w / boxa_h) - torch.atan(boxb_w / boxb_h))
    iou = inter_area / (union_area + 1e-8)
    alpha = v / ((1 - iou) + v)
    # ciou = iou - length_box_ctr / length_ac - alpha * v
    ciou_loss = 1 - iou + length_box_ctr / length_ac + alpha * v
    return ciou_loss

def AlphaIoU_loss(boxa, boxb, alpha):
    """
    # 除了alpha-iou,还有alpha-giou, alpha-diou, alpha-ciou，这里就不写了。
    # alpha-iou的优点是，例如alpha取2，当iou大于0.5的时候，loss的梯度是大于1的，
    # 相比iou的loss一直等于-1，收敛的更快，map0.7/map0.9有提升效果。
    loss = 1 - iou^alpha   alpha>0，取3效果比较好
    """
    inter_x1, inter_y1 = torch.maximum(boxa[:, 0], boxb[:, 0]), torch.maximum(boxa[:, 1], boxb[:, 1])
    inter_x2, inter_y2 = torch.minimum(boxa[:, 2], boxb[:, 2]), torch.minimum(boxa[:, 3], boxb[:, 3])
    inter_h = torch.maximum(torch.tensor([0]), inter_y2 - inter_y1)
    inter_w = torch.maximum(torch.tensor([0]), inter_x2 - inter_x1)
    inter_area = inter_w * inter_h
    union_area = ((boxa[:, 3] - boxa[:, 1]) * (boxa[:, 2] - boxa[:, 0])) + \
                 ((boxb[:, 3] - boxb[:, 1]) * (boxb[:, 2] - boxb[:, 0])) - inter_area + 1e-8  # + 1e-8 防止除零
    iou = inter_area / union_area

    alpha_iou = torch.pow(iou, alpha)
    alpha_iou_loss = 1 - alpha_iou
    return alpha_iou_loss

if __name__ == '__main__':
    # 定义一些输入的tensor
    proposals = torch.tensor([0., 0., 2., 2.], dtype=torch.float32)
    gt_boxes = torch.tensor([1., 1., 5., 5.], dtype=torch.float32)

    # 专门用于bce loss的输入
    bce_prop = torch.tensor([0.2, 0.7, 0.99, 0.5], dtype=torch.float32)
    bce_gt = torch.tensor([0, 1, 0, 1], dtype=torch.float32)

    # 专门用于ce loss的输入，4个边界框，每个边界框对应2个类别的置信度
    ce_prop = torch.randn([4, 2], dtype=torch.float32)
    ce_prop = F.softmax(ce_prop, dim=1)  # 对每个bbox的置信度进行softmax
    # 4个边界框的真实类别id
    ce_gt_boxes_classid = torch.randint(0, 2, [4], dtype=torch.int64)

    # 专门用于iou loss的输入
    iou_proposals = torch.tensor([[0, 0, 2, 2], [0, 0, 2, 2]])
    iou_gt_boxes = torch.tensor([[1, 1, 3, 3], [1, 1, 2, 4]])

    ########################################### our methods  #############################################
    # 分类损失：
    bce_loss = BCE_loss(bce_prop, bce_gt)
    ce_loss = CE_loss(ce_prop, ce_gt_boxes_classid)

    # 位置损失：
    l1_loss = L1_loss(proposals, gt_boxes)  # 也叫MAE
    l2_loss = L2_loss(proposals, gt_boxes)  # 也叫MSE
    smooth_l1_loss = Smooth_l1_loss(proposals, gt_boxes)
    iou_loss = IoU_loss(iou_proposals, iou_gt_boxes)
    giou_loss = GIoU_loss(iou_proposals, iou_gt_boxes)
    diou_loss = DIoU_loss(iou_proposals, iou_gt_boxes)
    ciou_loss = CIoU_loss(iou_proposals, iou_gt_boxes)  # proposals和gt_boxes宽高比一样，所以ciou等于diou
    alphaiou1_loss = AlphaIoU_loss(iou_proposals, iou_gt_boxes, alpha=1)
    alphaiou3_loss = AlphaIoU_loss(iou_proposals, iou_gt_boxes, alpha=3)

    ########################################### official methods  #############################################
    # 分类损失：
    bce_loss_ = F.binary_cross_entropy(bce_prop, bce_gt)
    ce_loss_ = F.cross_entropy(ce_prop, ce_gt_boxes_classid)

    # 位置损失：
    l1_loss_ = F.l1_loss(proposals, gt_boxes)  # 也叫MAE
    l2_loss_ = F.mse_loss(proposals, gt_boxes)  # 也叫MSE
    smooth_l1_loss_ = F.smooth_l1_loss(proposals, gt_boxes)

    # 输出结果对比一下：
    print("bce:",bce_loss)
    print("bce_:",bce_loss_)
    print("ce:",ce_loss)
    print("ce_:",ce_loss_)
    print("l1_loss:",l1_loss)
    print("l1_loss_:",l1_loss_)
    print("l2_loss:",l2_loss)
    print("l2_loss_:",l2_loss_)
    print("smooth_l1_loss:",smooth_l1_loss)
    print("smooth_l1_loss_:",smooth_l1_loss_)


    # 自己计算一下，看写的iou loss函数对不对，下面是手动计算的结果：
    # 下面我把并集中1e-8省略了，所以会有略微差距。下面手动计算的是[0, 0, 2, 2]与[1, 1, 3, 3]的各种iou loss
    # box1 area=4, box2 area=4,inter area=1, union area=7, ac area=9, iou=1/7
    print("iou loss:",iou_loss)
    print("iou loss:", 1 - 1 / 7)
    print("giou loss:",giou_loss)
    print("giou loss:", 1 - (1 / 7 - (9 - 7) / 9))
    print("diou loss:",diou_loss)
    print("diou loss:", 1 - (1 / 7 - (1 * 1) / (3 * 3)))
    print("ciou loss:",ciou_loss)
    v = 4 / (math.pi * math.pi) * ((math.atan(2 / 2) - math.atan(2 / 2)) * (math.atan(2 / 2) - math.atan(2 / 2)))
    print("ciou loss:", 1 - 1 / 7 + (1 * 1) / (3 * 3) + v / ((1 - 1 / 7) + v) * v)
    print("alpha1 iou loss:",alphaiou1_loss)
    print("alpha1 iou loss:", 1 - 1 / 7)
    print("alpha3 iou loss:",alphaiou3_loss)
    print("alpha3 iou loss:1", 1 - math.pow((1 / 7), 3))

输出结果：

bce: tensor(1.4695)
bce_: tensor(1.4695)
ce: tensor(0.9038)
ce_: tensor(0.9038)
l1_loss: tensor(2.)
l1_loss_: tensor(2.)
l2_loss: tensor(5.)
l2_loss_: tensor(5.)
smooth_l1_loss: tensor(1.5000)
smooth_l1_loss_: tensor(1.5000)
iou loss: tensor([0.8571, 0.8333])
iou loss: 0.8571428571428572
giou loss: tensor([1.0794, 1.0833])
giou loss: 1.0793650793650793
diou loss: tensor([0.9683, 0.9583])
diou loss: 0.9682539682539683
ciou loss: tensor([0.9683, 0.9666])
ciou loss: 0.9682539682539684
alpha1 iou loss: tensor([0.8571, 0.8333])
alpha1 iou loss: 0.8571428571428572
alpha3 iou loss: tensor([0.9971, 0.9954])
alpha3 iou loss:1 0.9970845481049563