# 13.3. Object Detection and Bounding Boxes¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in SageMaker Studio Lab

In earlier sections (e.g., Section 7.1Section 7.4), we introduced various models for image classification. In image classification tasks, we assume that there is only one major object in the image and we only focus on how to recognize its category. However, there are often multiple objects in the image of interest. We not only want to know their categories, but also their specific positions in the image. In computer vision, we refer to such tasks as object detection (or object recognition).

Object detection has been widely applied in many fields. For example, self-driving needs to plan traveling routes by detecting the positions of vehicles, pedestrians, roads, and obstacles in the captured video images. Besides, robots may use this technique to detect and localize objects of interest throughout its navigation of an environment. Moreover, security systems may need to detect abnormal objects, such as intruders or bombs.

In the next few sections, we will introduce several deep learning methods for object detection. We will begin with an introduction to positions (or locations) of objects.

%matplotlib inline
from mxnet import image, np, npx
from d2l import mxnet as d2l

npx.set_np()

%matplotlib inline
import torch
from d2l import torch as d2l

%matplotlib inline
import tensorflow as tf
from d2l import tensorflow as d2l


We will load the sample image to be used in this section. We can see that there is a dog on the left side of the image and a cat on the right. They are the two major objects in this image.

d2l.set_figsize()
d2l.plt.imshow(img);

d2l.set_figsize()
d2l.plt.imshow(img);

d2l.set_figsize()
d2l.plt.imshow(img);


## 13.3.1. Bounding Boxes¶

In object detection, we usually use a bounding box to describe the spatial location of an object. The bounding box is rectangular, which is determined by the $$x$$ and $$y$$ coordinates of the upper-left corner of the rectangle and the such coordinates of the lower-right corner. Another commonly used bounding box representation is the $$(x, y)$$-axis coordinates of the bounding box center, and the width and height of the box.

Here we define functions to convert between these two representations: box_corner_to_center converts from the two-corner representation to the center-width-height presentation, and box_center_to_corner vice versa. The input argument boxes should be a two-dimensional tensor of shape ($$n$$, 4), where $$n$$ is the number of bounding boxes.

#@save
def box_corner_to_center(boxes):
"""Convert from (upper-left, lower-right) to (center, width, height)."""
x1, y1, x2, y2 = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
cx = (x1 + x2) / 2
cy = (y1 + y2) / 2
w = x2 - x1
h = y2 - y1
boxes = np.stack((cx, cy, w, h), axis=-1)
return boxes

#@save
def box_center_to_corner(boxes):
"""Convert from (center, width, height) to (upper-left, lower-right)."""
cx, cy, w, h = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
x1 = cx - 0.5 * w
y1 = cy - 0.5 * h
x2 = cx + 0.5 * w
y2 = cy + 0.5 * h
boxes = np.stack((x1, y1, x2, y2), axis=-1)
return boxes

#@save
def box_corner_to_center(boxes):
"""Convert from (upper-left, lower-right) to (center, width, height)."""
x1, y1, x2, y2 = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
cx = (x1 + x2) / 2
cy = (y1 + y2) / 2
w = x2 - x1
h = y2 - y1
boxes = torch.stack((cx, cy, w, h), axis=-1)
return boxes

#@save
def box_center_to_corner(boxes):
"""Convert from (center, width, height) to (upper-left, lower-right)."""
cx, cy, w, h = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
x1 = cx - 0.5 * w
y1 = cy - 0.5 * h
x2 = cx + 0.5 * w
y2 = cy + 0.5 * h
boxes = torch.stack((x1, y1, x2, y2), axis=-1)
return boxes

#@save
def box_corner_to_center(boxes):
"""Convert from (upper-left, lower-right) to (center, width, height)."""
x1, y1, x2, y2 = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
cx = (x1 + x2) / 2
cy = (y1 + y2) / 2
w = x2 - x1
h = y2 - y1
boxes = tf.stack((cx, cy, w, h), axis=-1)
return boxes

#@save
def box_center_to_corner(boxes):
"""Convert from (center, width, height) to (upper-left, lower-right)."""
cx, cy, w, h = boxes[:, 0], boxes[:, 1], boxes[:, 2], boxes[:, 3]
x1 = cx - 0.5 * w
y1 = cy - 0.5 * h
x2 = cx + 0.5 * w
y2 = cy + 0.5 * h
boxes = tf.stack((x1, y1, x2, y2), axis=-1)
return boxes


We will define the bounding boxes of the dog and the cat in the image based on the coordinate information. The origin of the coordinates in the image is the upper-left corner of the image, and to the right and down are the positive directions of the $$x$$ and $$y$$ axes, respectively.

# Here bbox is the abbreviation for bounding box
dog_bbox, cat_bbox = [60.0, 45.0, 378.0, 516.0], [400.0, 112.0, 655.0, 493.0]


We can verify the correctness of the two bounding box conversion functions by converting twice.

boxes = np.array((dog_bbox, cat_bbox))
box_center_to_corner(box_corner_to_center(boxes)) == boxes

array([[ True,  True,  True,  True],
[ True,  True,  True,  True]])

boxes = torch.tensor((dog_bbox, cat_bbox))
box_center_to_corner(box_corner_to_center(boxes)) == boxes

tensor([[True, True, True, True],
[True, True, True, True]])

boxes = tf.constant((dog_bbox, cat_bbox))
box_center_to_corner(box_corner_to_center(boxes)) == boxes

<tf.Tensor: shape=(2, 4), dtype=bool, numpy=
array([[ True,  True,  True,  True],
[ True,  True,  True,  True]])>


Let us draw the bounding boxes in the image to check if they are accurate. Before drawing, we will define a helper function bbox_to_rect. It represents the bounding box in the bounding box format of the matplotlib package.

#@save
def bbox_to_rect(bbox, color):
"""Convert bounding box to matplotlib format."""
# Convert the bounding box (upper-left x, upper-left y, lower-right x,
# lower-right y) format to the matplotlib format: ((upper-left x,
# upper-left y), width, height)
return d2l.plt.Rectangle(
xy=(bbox[0], bbox[1]), width=bbox[2]-bbox[0], height=bbox[3]-bbox[1],
fill=False, edgecolor=color, linewidth=2)


After adding the bounding boxes on the image, we can see that the main outline of the two objects are basically inside the two boxes.

fig = d2l.plt.imshow(img)

fig = d2l.plt.imshow(img)

fig = d2l.plt.imshow(img)

2. Why is the innermost dimension of the input argument boxes of box_corner_to_center and box_center_to_corner always 4?