一、IOU

论文：UnitBox: An Advanced Object Detection Network

definition

The full name of IOU is Intersection over Union, which is what we often say. IOU can be used as the basis for the distribution of positive and negative samples in target detection, and can also be used to measure the difference between the prediction frame and the GT frame (AP (average precision) in target detection is obtained based on IOU). The calculation formula of IOU is also very simple, that is, the ratio of the intersection and union of two bounding boxes,

features

Non-negativity, identity, symmetry, scale invariance
Cannot accurately reflect the coincidence of two bounding boxes

If the two bounding boxes do not intersect, the gap between the two bounding boxes cannot be reflected

code example

import torch


def cal_iou(box1, box2, eps=1e-7):
    b1_x1, b1_y1, b1_x2, b1_y2 = box1[0], box1[1], box1[2], box1[3]
    b2_x1, b2_y1, b2_x2, b2_y2 = box2[0], box2[1], box2[2], box2[3]
    # Intersection area
    inter = (torch.min(b1_x2, b2_x2) - torch.max(b1_x1, b2_x1)).clamp(0) * \
            (torch.min(b1_y2, b2_y2) - torch.max(b1_y1, b2_y1)).clamp(0)
    # Union Area
    w1, h1 = b1_x2 - b1_x1, b1_y2 - b1_y1 + eps
    w2, h2 = b2_x2 - b2_x1, b2_y2 - b2_y1 + eps
    union = w1 * h1 + w2 * h2 - inter + eps

    iou = inter / union

    return iou

二、GIOU

论文：Generalized Intersection over Union: A Metric and A Loss for Bounding Box Regression

definition

In order to overcome the shortcomings of IOU, GIOU has been modified on the basis of IOU, the formula is as follows,

Among them, C: the smallest circumscribing rectangle containing the bounding boxes A and B (the green rectangle in the figure below)

features

Based on the IOU, the minimum bounding rectangle is introduced to solve the problem that the IOU is 0 when there is no intersection between the prediction frame and the GT frame
GIoU not only focuses on overlapping areas, but also on other non-overlapping areas, which can better reflect the overlap between the two
The value range of GIOU is [-1, 1]. When the two overlap, take the maximum value of 1, and take the minimum value of -1 when the two have no intersection and are infinitely far away.
When the prediction box and GT box contain phenomenon, GIOU degenerates into IOU

code example

import torch


def cal_giou(box1, box2, eps=1e-7):
    b1_x1, b1_y1, b1_x2, b1_y2 = box1[0], box1[1], box1[2], box1[3]
    b2_x1, b2_y1, b2_x2, b2_y2 = box2[0], box2[1], box2[2], box2[3]
    # Intersection area
    inter = (torch.min(b1_x2, b2_x2) - torch.max(b1_x1, b2_x1)).clamp(0) * \
            (torch.min(b1_y2, b2_y2) - torch.max(b1_y1, b2_y1)).clamp(0)
    # Union Area
    w1, h1 = b1_x2 - b1_x1, b1_y2 - b1_y1 + eps
    w2, h2 = b2_x2 - b2_x1, b2_y2 - b2_y1 + eps
    union = w1 * h1 + w2 * h2 - inter + eps

    iou = inter / union

    cw = torch.max(b1_x2, b2_x2) - torch.min(b1_x1, b2_x1)  # smallest enclosing box width
    ch = torch.max(b1_y2, b2_y2) - torch.min(b1_y1, b2_y1)  # smallest enclosing box height
    c_area = cw * ch + eps  # convex area
    
    giou = iou - (c_area - union) / c_area

    return giou

三、DIOU

论文：Distance-IoULoss:FasterandBetterLearningforBoundingBoxRegression

definition

In order to overcome the shortcomings of GIOU, DIOU has been improved on the basis of IOU, the formula is as follows,

Among them, $\rho ^{2}$ : the sum of the squares of the Euclidean distance between two points, , $center_{A}$ : $center_{B}$ the center point coordinates of the bounding boxes A and B, $c$ : the diagonal length of the smallest circumscribed rectangular box containing A and B

features

DIOU can directly optimize the distance between two bounding boxes, which converges faster than GIOU Loss
If the two bounding boxes coincide perfectly, d=0, IOU=1, DIOU=1-0=1. If the two bounding boxes are far apart, $\frac{d^{2}}{c^{2}}$ approaching 1, IOU=0, DIOU=0-1=-1. Therefore, the value range of DIOU is also [-1, 1]
Even if the c value does not change when one box contains another, the d value can be a valid measure
When the center points of the two bounding boxes coincide, there may be cases where both c and d remain unchanged

code example

import torch


def cal_diou(box1, box2, eps=1e-7):
    b1_x1, b1_y1, b1_x2, b1_y2 = box1[0], box1[1], box1[2], box1[3]
    b2_x1, b2_y1, b2_x2, b2_y2 = box2[0], box2[1], box2[2], box2[3]
    # Intersection area
    inter = (torch.min(b1_x2, b2_x2) - torch.max(b1_x1, b2_x1)).clamp(0) * \
            (torch.min(b1_y2, b2_y2) - torch.max(b1_y1, b2_y1)).clamp(0)
    # Union Area
    w1, h1 = b1_x2 - b1_x1, b1_y2 - b1_y1 + eps
    w2, h2 = b2_x2 - b2_x1, b2_y2 - b2_y1 + eps
    union = w1 * h1 + w2 * h2 - inter + eps

    iou = inter / union

    cw = torch.max(b1_x2, b2_x2) - torch.min(b1_x1, b2_x1)  # smallest enclosing box width
    ch = torch.max(b1_y2, b2_y2) - torch.min(b1_y1, b2_y1)  # smallest enclosing box height
    c2 = cw ** 2 + ch ** 2 + eps  # the diagonal of smallest enclosing box
    rho2 = ((b2_x1 + b2_x2 - b1_x1 - b1_x2) ** 2 + (b2_y1 + b2_y2 - b1_y1 - b1_y2) ** 2) / 4  # center distance squared
    
    diou = iou - rho2 / c2

    return diou

4. CIOU

Complete IOU

definition

CIOU is to increase the aspect ratio of the bounding box on the basis of DIOU, the formula is as follows,

Therefore, the three items of CIOU correspond exactly to IOU, the ratio of the center point distance to the diagonal distance of the smallest circumscribed rectangular frame, and the gap between the aspect ratios of the two bounding boxes

features

CIOU is to increase the aspect ratio of the bounding box on the basis of DIOU, so that the prediction box is closer to the GT box
The aspect ratio of the bounding box describes a relative value, and there is some ambiguity

code example

import torch
import math


def cal_ciou(box1, box2, eps=1e-7):
    b1_x1, b1_y1, b1_x2, b1_y2 = box1[0], box1[1], box1[2], box1[3]
    b2_x1, b2_y1, b2_x2, b2_y2 = box2[0], box2[1], box2[2], box2[3]
    # Intersection area
    inter = (torch.min(b1_x2, b2_x2) - torch.max(b1_x1, b2_x1)).clamp(0) * \
            (torch.min(b1_y2, b2_y2) - torch.max(b1_y1, b2_y1)).clamp(0)
    # Union Area
    w1, h1 = b1_x2 - b1_x1, b1_y2 - b1_y1 + eps
    w2, h2 = b2_x2 - b2_x1, b2_y2 - b2_y1 + eps
    union = w1 * h1 + w2 * h2 - inter + eps

    iou = inter / union

    cw = torch.max(b1_x2, b2_x2) - torch.min(b1_x1, b2_x1)  # smallest enclosing box width
    ch = torch.max(b1_y2, b2_y2) - torch.min(b1_y1, b2_y1)  # smallest enclosing box height
    c2 = cw ** 2 + ch ** 2 + eps
    rho2 = ((b2_x1 + b2_x2 - b1_x1 - b1_x2) ** 2 + (b2_y1 + b2_y2 - b1_y1 - b1_y2) ** 2) / 4  # center distance squared
    
    v = (4 / math.pi ** 2) * torch.pow(torch.atan(w2 / (h2 + eps)) - torch.atan(w1 / (h1 + eps)), 2)

    with torch.no_grad():
        alpha = v / (v - iou + (1 + eps))
    ciou = iou - (rho2 / c2 + v * alpha)

    return ciou

【Reference article】

https://www.cnblogs.com/wujianming-110117/p/13019343.html

Detailed explanation of IOU, GIOU, DIOU, CIOU loss function

Detailed explanation and code implementation of IOU, GIOU, DIOU, CIOU - Programmer Sought

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