456 KiB
%matplotlib inline
import torch
from d2l import torch as d2l
torch.set_printoptions(2) # Simplify printing accuracy#@save
def multibox_prior(data, sizes, ratios):
"""Generate anchor boxes with different shapes centered on each pixel."""
in_height, in_width = data.shape[-2:]
device, num_sizes, num_ratios = data.device, len(sizes), len(ratios)
boxes_per_pixel = (num_sizes + num_ratios - 1)
size_tensor = torch.tensor(sizes, device=device)
ratio_tensor = torch.tensor(ratios, device=device)
# Offsets are required to move the anchor to the center of a pixel. Since
# a pixel has height=1 and width=1, we choose to offset our centers by 0.5
offset_h, offset_w = 0.5, 0.5
steps_h = 1.0 / in_height # Scaled steps in y axis
steps_w = 1.0 / in_width # Scaled steps in x axis
# Generate all center points for the anchor boxes
center_h = (torch.arange(in_height, device=device) + offset_h) * steps_h
center_w = (torch.arange(in_width, device=device) + offset_w) * steps_w
shift_y, shift_x = torch.meshgrid(center_h, center_w, indexing='ij')
shift_y, shift_x = shift_y.reshape(-1), shift_x.reshape(-1)
# Generate `boxes_per_pixel` number of heights and widths that are later
# used to create anchor box corner coordinates (xmin, xmax, ymin, ymax)
w = torch.cat((size_tensor * torch.sqrt(ratio_tensor[0]),
sizes[0] * torch.sqrt(ratio_tensor[1:])))\
* in_height / in_width # Handle rectangular inputs
h = torch.cat((size_tensor / torch.sqrt(ratio_tensor[0]),
sizes[0] / torch.sqrt(ratio_tensor[1:])))
# Divide by 2 to get half height and half width
anchor_manipulations = torch.stack((-w, -h, w, h)).T.repeat(
in_height * in_width, 1) / 2
# Each center point will have `boxes_per_pixel` number of anchor boxes, so
# generate a grid of all anchor box centers with `boxes_per_pixel` repeats
out_grid = torch.stack([shift_x, shift_y, shift_x, shift_y],
dim=1).repeat_interleave(boxes_per_pixel, dim=0)
output = out_grid + anchor_manipulations
return output.unsqueeze(0)img = d2l.plt.imread('../img/catdog.jpg')
h, w = img.shape[:2]
print(h, w)
X = torch.rand(size=(1, 3, h, w)) # Construct input data
Y = multibox_prior(X, sizes=[0.75, 0.5, 0.25], ratios=[1, 2, 0.5])
Y.shape561 728
torch.Size([1, 2042040, 4])
boxes = Y.reshape(h, w, 5, 4)
boxes[250, 250, 0, :]tensor([0.06, 0.07, 0.63, 0.82])
#@save
def show_bboxes(axes, bboxes, labels=None, colors=None):
"""Show bounding boxes."""
def make_list(obj, default_values=None):
if obj is None:
obj = default_values
elif not isinstance(obj, (list, tuple)):
obj = [obj]
return obj
labels = make_list(labels)
colors = make_list(colors, ['b', 'g', 'r', 'm', 'c'])
for i, bbox in enumerate(bboxes):
color = colors[i % len(colors)]
rect = d2l.bbox_to_rect(bbox.detach().numpy(), color)
axes.add_patch(rect)
if labels and len(labels) > i:
text_color = 'k' if color == 'w' else 'w'
axes.text(rect.xy[0], rect.xy[1], labels[i],
va='center', ha='center', fontsize=9, color=text_color,
bbox=dict(facecolor=color, lw=0))d2l.set_figsize()
bbox_scale = torch.tensor((w, h, w, h))
fig = d2l.plt.imshow(img)
show_bboxes(fig.axes, boxes[250, 250, :, :] * bbox_scale,
['s=0.75, r=1', 's=0.5, r=1', 's=0.25, r=1', 's=0.75, r=2',
's=0.75, r=0.5'])
[Intersection over Union (IoU)]
We just mentioned that an anchor box "well" surrounds the dog in the image.
If the ground-truth bounding box of the object is known, how can "well" here be quantified?
Intuitively, we can measure the similarity between
the anchor box and the ground-truth bounding box.
We know that the Jaccard index can measure the similarity between two sets. Given sets \mathcal{A} and \mathcal{B}, their Jaccard index is the size of their intersection divided by the size of their union:
J(\mathcal{A},\mathcal{B}) = \frac{\left|\mathcal{A} \cap \mathcal{B}\right|}{\left| \mathcal{A} \cup \mathcal{B}\right|}.
In fact, we can consider the pixel area of any bounding box as a set of pixels.
In this way, we can measure the similarity of the two bounding boxes by the Jaccard index of their pixel sets. For two bounding boxes, we usually refer their Jaccard index as intersection over union (IoU), which is the ratio of their intersection area to their union area, as shown in :numref:fig_iou.
The range of an IoU is between 0 and 1:
0 means that two bounding boxes do not overlap at all,
while 1 indicates that the two bounding boxes are equal.
🏷️fig_iou
For the remainder of this section, we will use IoU to measure the similarity between anchor boxes and ground-truth bounding boxes, and between different anchor boxes.
Given two lists of anchor or bounding boxes,
the following box_iou computes their pairwise IoU
across these two lists.
#@save
def box_iou(boxes1, boxes2):
"""Compute pairwise IoU across two lists of anchor or bounding boxes."""
box_area = lambda boxes: ((boxes[:, 2] - boxes[:, 0]) *
(boxes[:, 3] - boxes[:, 1]))
# Shape of `boxes1`, `boxes2`, `areas1`, `areas2`: (no. of boxes1, 4),
# (no. of boxes2, 4), (no. of boxes1,), (no. of boxes2,)
areas1 = box_area(boxes1)
areas2 = box_area(boxes2)
# Shape of `inter_upperlefts`, `inter_lowerrights`, `inters`: (no. of
# boxes1, no. of boxes2, 2)
inter_upperlefts = torch.max(boxes1[:, None, :2], boxes2[:, :2])
inter_lowerrights = torch.min(boxes1[:, None, 2:], boxes2[:, 2:])
inters = (inter_lowerrights - inter_upperlefts).clamp(min=0)
# Shape of `inter_areas` and `union_areas`: (no. of boxes1, no. of boxes2)
inter_areas = inters[:, :, 0] * inters[:, :, 1]
union_areas = areas1[:, None] + areas2 - inter_areas
return inter_areas / union_areasLabeling Anchor Boxes in Training Data
🏷️subsec_labeling-anchor-boxes
In a training dataset, we consider each anchor box as a training example. In order to train an object detection model, we need class and offset labels for each anchor box, where the former is the class of the object relevant to the anchor box and the latter is the offset of the ground-truth bounding box relative to the anchor box. During the prediction, for each image we generate multiple anchor boxes, predict classes and offsets for all the anchor boxes, adjust their positions according to the predicted offsets to obtain the predicted bounding boxes, and finally only output those predicted bounding boxes that satisfy certain criteria.
As we know, an object detection training set comes with labels for locations of ground-truth bounding boxes and classes of their surrounded objects. To label any generated anchor box, we refer to the labeled location and class of its assigned ground-truth bounding box that is closest to the anchor box. In the following, we describe an algorithm for assigning closest ground-truth bounding boxes to anchor boxes.
[Assigning Ground-Truth Bounding Boxes to Anchor Boxes]
Given an image,
suppose that the anchor boxes are A_1, A_2, \ldots, A_{n_a} and the ground-truth bounding boxes are B_1, B_2, \ldots, B_{n_b}, where n_a \geq n_b.
Let's define a matrix \mathbf{X} \in \mathbb{R}^{n_a \times n_b}, whose element x_{ij} in the i^\textrm{th} row and j^\textrm{th} column is the IoU of the anchor box A_i and the ground-truth bounding box B_j. The algorithm consists of the following steps:
- Find the largest element in matrix
\mathbf{X}and denote its row and column indices asi_1andj_1, respectively. Then the ground-truth bounding boxB_{j_1}is assigned to the anchor boxA_{i_1}. This is quite intuitive becauseA_{i_1}andB_{j_1}are the closest among all the pairs of anchor boxes and ground-truth bounding boxes. After the first assignment, discard all the elements in the{i_1}^\textrm{th}row and the{j_1}^\textrm{th}column in matrix\mathbf{X}. - Find the largest of the remaining elements in matrix
\mathbf{X}and denote its row and column indices asi_2andj_2, respectively. We assign ground-truth bounding boxB_{j_2}to anchor boxA_{i_2}and discard all the elements in the{i_2}^\textrm{th}row and the{j_2}^\textrm{th}column in matrix\mathbf{X}. - At this point, elements in two rows and two columns in matrix
\mathbf{X}have been discarded. We proceed until all elements inn_bcolumns in matrix\mathbf{X}are discarded. At this time, we have assigned a ground-truth bounding box to each ofn_banchor boxes. - Only traverse through the remaining
n_a - n_banchor boxes. For example, given any anchor boxA_i, find the ground-truth bounding boxB_jwith the largest IoU withA_ithroughout thei^\textrm{th}row of matrix\mathbf{X}, and assignB_jtoA_ionly if this IoU is greater than a predefined threshold.
Let's illustrate the above algorithm using a concrete
example.
As shown in :numref:fig_anchor_label (left), assuming that the maximum value in matrix \mathbf{X} is x_{23}, we assign the ground-truth bounding box B_3 to the anchor box A_2.
Then, we discard all the elements in row 2 and column 3 of the matrix, find the largest x_{71} in the remaining elements (shaded area), and assign the ground-truth bounding box B_1 to the anchor box A_7.
Next, as shown in :numref:fig_anchor_label (middle), discard all the elements in row 7 and column 1 of the matrix, find the largest x_{54} in the remaining elements (shaded area), and assign the ground-truth bounding box B_4 to the anchor box A_5.
Finally, as shown in :numref:fig_anchor_label (right), discard all the elements in row 5 and column 4 of the matrix, find the largest x_{92} in the remaining elements (shaded area), and assign the ground-truth bounding box B_2 to the anchor box A_9.
After that, we only need to traverse through
the remaining anchor boxes A_1, A_3, A_4, A_6, A_8 and determine whether to assign them ground-truth bounding boxes according to the threshold.
🏷️fig_anchor_label
This algorithm is implemented in the following assign_anchor_to_bbox function.
#@save
def assign_anchor_to_bbox(ground_truth, anchors, device, iou_threshold=0.5):
"""Assign closest ground-truth bounding boxes to anchor boxes."""
num_anchors, num_gt_boxes = anchors.shape[0], ground_truth.shape[0]
# Element x_ij in the i-th row and j-th column is the IoU of the anchor
# box i and the ground-truth bounding box j
jaccard = box_iou(anchors, ground_truth)
# Initialize the tensor to hold the assigned ground-truth bounding box for
# each anchor
anchors_bbox_map = torch.full((num_anchors,), -1, dtype=torch.long,
device=device)
# Assign ground-truth bounding boxes according to the threshold
max_ious, indices = torch.max(jaccard, dim=1)
anc_i = torch.nonzero(max_ious >= iou_threshold).reshape(-1)
box_j = indices[max_ious >= iou_threshold]
anchors_bbox_map[anc_i] = box_j
col_discard = torch.full((num_anchors,), -1)
row_discard = torch.full((num_gt_boxes,), -1)
for _ in range(num_gt_boxes):
max_idx = torch.argmax(jaccard) # Find the largest IoU
box_idx = (max_idx % num_gt_boxes).long()
anc_idx = (max_idx / num_gt_boxes).long()
anchors_bbox_map[anc_idx] = box_idx
jaccard[:, box_idx] = col_discard
jaccard[anc_idx, :] = row_discard
return anchors_bbox_map#@save
def offset_boxes(anchors, assigned_bb, eps=1e-6):
"""Transform for anchor box offsets."""
c_anc = d2l.box_corner_to_center(anchors)
c_assigned_bb = d2l.box_corner_to_center(assigned_bb)
offset_xy = 10 * (c_assigned_bb[:, :2] - c_anc[:, :2]) / c_anc[:, 2:]
offset_wh = 5 * torch.log(eps + c_assigned_bb[:, 2:] / c_anc[:, 2:])
offset = torch.cat([offset_xy, offset_wh], axis=1)
return offset#@save
def multibox_target(anchors, labels):
"""Label anchor boxes using ground-truth bounding boxes."""
batch_size, anchors = labels.shape[0], anchors.squeeze(0)
batch_offset, batch_mask, batch_class_labels = [], [], []
device, num_anchors = anchors.device, anchors.shape[0]
for i in range(batch_size):
label = labels[i, :, :]
anchors_bbox_map = assign_anchor_to_bbox(
label[:, 1:], anchors, device)
bbox_mask = ((anchors_bbox_map >= 0).float().unsqueeze(-1)).repeat(
1, 4)
# Initialize class labels and assigned bounding box coordinates with
# zeros
class_labels = torch.zeros(num_anchors, dtype=torch.long,
device=device)
assigned_bb = torch.zeros((num_anchors, 4), dtype=torch.float32,
device=device)
# Label classes of anchor boxes using their assigned ground-truth
# bounding boxes. If an anchor box is not assigned any, we label its
# class as background (the value remains zero)
indices_true = torch.nonzero(anchors_bbox_map >= 0)
bb_idx = anchors_bbox_map[indices_true]
class_labels[indices_true] = label[bb_idx, 0].long() + 1
assigned_bb[indices_true] = label[bb_idx, 1:]
# Offset transformation
offset = offset_boxes(anchors, assigned_bb) * bbox_mask
batch_offset.append(offset.reshape(-1))
batch_mask.append(bbox_mask.reshape(-1))
batch_class_labels.append(class_labels)
bbox_offset = torch.stack(batch_offset)
bbox_mask = torch.stack(batch_mask)
class_labels = torch.stack(batch_class_labels)
return (bbox_offset, bbox_mask, class_labels)ground_truth = torch.tensor([[0, 0.1, 0.08, 0.52, 0.92],
[1, 0.55, 0.2, 0.9, 0.88]])
anchors = torch.tensor([[0, 0.1, 0.2, 0.3], [0.15, 0.2, 0.4, 0.4],
[0.63, 0.05, 0.88, 0.98], [0.66, 0.45, 0.8, 0.8],
[0.57, 0.3, 0.92, 0.9]])
fig = d2l.plt.imshow(img)
show_bboxes(fig.axes, ground_truth[:, 1:] * bbox_scale, ['dog', 'cat'], 'k')
show_bboxes(fig.axes, anchors * bbox_scale, ['0', '1', '2', '3', '4']);
labels = multibox_target(anchors.unsqueeze(dim=0),
ground_truth.unsqueeze(dim=0))labels[2]tensor([[0, 1, 2, 0, 2]])
labels[1]tensor([[0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 1., 1., 0., 0., 0., 0., 1., 1.,
1., 1.]])labels[0]tensor([[-0.00e+00, -0.00e+00, -0.00e+00, -0.00e+00, 1.40e+00, 1.00e+01,
2.59e+00, 7.18e+00, -1.20e+00, 2.69e-01, 1.68e+00, -1.57e+00,
-0.00e+00, -0.00e+00, -0.00e+00, -0.00e+00, -5.71e-01, -1.00e+00,
4.17e-06, 6.26e-01]])#@save
def offset_inverse(anchors, offset_preds):
"""Predict bounding boxes based on anchor boxes with predicted offsets."""
anc = d2l.box_corner_to_center(anchors)
pred_bbox_xy = (offset_preds[:, :2] * anc[:, 2:] / 10) + anc[:, :2]
pred_bbox_wh = torch.exp(offset_preds[:, 2:] / 5) * anc[:, 2:]
pred_bbox = torch.cat((pred_bbox_xy, pred_bbox_wh), axis=1)
predicted_bbox = d2l.box_center_to_corner(pred_bbox)
return predicted_bbox#@save
def nms(boxes, scores, iou_threshold):
"""Sort confidence scores of predicted bounding boxes."""
B = torch.argsort(scores, dim=-1, descending=True)
keep = [] # Indices of predicted bounding boxes that will be kept
while B.numel() > 0:
i = B[0]
keep.append(i)
if B.numel() == 1: break
iou = box_iou(boxes[i, :].reshape(-1, 4),
boxes[B[1:], :].reshape(-1, 4)).reshape(-1)
inds = torch.nonzero(iou <= iou_threshold).reshape(-1)
B = B[inds + 1]
return torch.tensor(keep, device=boxes.device)#@save
def multibox_detection(cls_probs, offset_preds, anchors, nms_threshold=0.5,
pos_threshold=0.009999999):
"""Predict bounding boxes using non-maximum suppression."""
device, batch_size = cls_probs.device, cls_probs.shape[0]
anchors = anchors.squeeze(0)
num_classes, num_anchors = cls_probs.shape[1], cls_probs.shape[2]
out = []
for i in range(batch_size):
cls_prob, offset_pred = cls_probs[i], offset_preds[i].reshape(-1, 4)
conf, class_id = torch.max(cls_prob[1:], 0)
predicted_bb = offset_inverse(anchors, offset_pred)
keep = nms(predicted_bb, conf, nms_threshold)
# Find all non-`keep` indices and set the class to background
all_idx = torch.arange(num_anchors, dtype=torch.long, device=device)
combined = torch.cat((keep, all_idx))
uniques, counts = combined.unique(return_counts=True)
non_keep = uniques[counts == 1]
all_id_sorted = torch.cat((keep, non_keep))
class_id[non_keep] = -1
class_id = class_id[all_id_sorted]
conf, predicted_bb = conf[all_id_sorted], predicted_bb[all_id_sorted]
# Here `pos_threshold` is a threshold for positive (non-background)
# predictions
below_min_idx = (conf < pos_threshold)
class_id[below_min_idx] = -1
conf[below_min_idx] = 1 - conf[below_min_idx]
pred_info = torch.cat((class_id.unsqueeze(1),
conf.unsqueeze(1),
predicted_bb), dim=1)
out.append(pred_info)
return torch.stack(out)anchors = torch.tensor([[0.1, 0.08, 0.52, 0.92], [0.08, 0.2, 0.56, 0.95],
[0.15, 0.3, 0.62, 0.91], [0.55, 0.2, 0.9, 0.88]])
offset_preds = torch.tensor([0] * anchors.numel())
cls_probs = torch.tensor([[0] * 4, # Predicted background likelihood
[0.9, 0.8, 0.7, 0.1], # Predicted dog likelihood
[0.1, 0.2, 0.3, 0.9]]) # Predicted cat likelihoodfig = d2l.plt.imshow(img)
show_bboxes(fig.axes, anchors * bbox_scale,
['dog=0.9', 'dog=0.8', 'dog=0.7', 'cat=0.9'])
output = multibox_detection(cls_probs.unsqueeze(dim=0),
offset_preds.unsqueeze(dim=0),
anchors.unsqueeze(dim=0),
nms_threshold=0.5)
outputtensor([[[ 0.00, 0.90, 0.10, 0.08, 0.52, 0.92],
[ 1.00, 0.90, 0.55, 0.20, 0.90, 0.88],
[-1.00, 0.80, 0.08, 0.20, 0.56, 0.95],
[-1.00, 0.70, 0.15, 0.30, 0.62, 0.91]]])fig = d2l.plt.imshow(img)
for i in output[0].detach().numpy():
if i[0] == -1:
continue
label = ('dog=', 'cat=')[int(i[0])] + str(i[1])
show_bboxes(fig.axes, [torch.tensor(i[2:]) * bbox_scale], label)