一、概念

在目标检测算法中，通常会在输入图像中采样大量的区域（生成多个边缘框），锚框就是从多个边缘框中判断最接近真实预测目标的区域。

整个过程分为两次预测：

1. 类别：预测锚框中所含物体的类别
2. 位置：预测锚框到边缘框的位置偏移

IoU-交叉比

IoU用来计算两个框之间的相似度：0 表示无重叠，1 表示重合。

赋予锚框标号

在训练集中，将每个锚框视为一个训练样本，为了训练目标检测模型，需要每个锚框的类别和偏移量(真实边缘框相对于锚框的偏移量)标签。

在预测的时候，首先为每个图像生成多个锚框，预测所有锚框的类别和偏移量，根据预测的偏移量调整它们的位置以获得预测的边缘框，最后只输出符合特定条件的预测边缘框

过程

如下图所示，我们有4个边缘框，对应4个类别物体，每一列对应7个锚框。

首先我们要求每个锚框与边缘框的IoU值，蓝色的代表已经算出IoU。假设M22是最大的IoU值，就说明锚框2的任务是去预测边缘框2，接下来把M22的行列删除，如右图。

现在M51是最大的IoU，同理说明锚框5的任务是去预测边缘框1。以此类推，直到所有的锚框都有了标号。

非极大值抑制(NMS)输出

每个锚框都会预测一个边缘框，这些预测值之间可能存在比较相似的预测，NMS就是去合并相似的预测，让输出干净一点。

过程：

1. 选中非背景的最大预测值
2. 去掉所有其它和它IoU值大于θ的预测（也就是去掉和最大预测值相似度比较高的其它锚框）
3. 重复上面过程直到所有预测要么被选中，要么被去掉

二、代码

1 生成锚框

以每个像素点为中心，生成高、宽不同的锚框。锚框的宽度和高度分别是$$ws\sqrt{(}r)$$和$$hs/\sqrt{(}r)$$，w和h是高、宽，s表示锚框占图片的大小，r是锚框的高宽比。一般的组合是$$(s1,r1),(s1,r2),...,(s1,rm),(s2,r1),(s3,r1),...,(sn,r1)$$

def multibox_prior(data, sizes, ratios):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)offset_h, offset_w = 0.5, 0.5steps_h = 1.0 / in_heightsteps_w = 1.0 / in_widthcenter_h = (torch.arange(in_height, device=device) + offset_h) * steps_hcenter_w = (torch.arange(in_width, device=device) + offset_w) * steps_wshift_y, shift_x = torch.meshgrid(center_h, center_w)shift_y, shift_x = shift_y.reshape(-1), shift_x.reshape(-1)w = torch.cat((size_tensor * torch.sqrt(ratio_tensor[0]),sizes[0] * torch.sqrt(ratio_tensor[1:])))\* in_height / in_widthh = torch.cat((size_tensor / torch.sqrt(ratio_tensor[0]),sizes[0] / torch.sqrt(ratio_tensor[1:])))anchor_manipulations = torch.stack((-w, -h, w, h)).T.repeat(in_height * in_width, 1) / 2out_grid = torch.stack([shift_x, shift_y, shift_x, shift_y],dim=1).repeat_interleave(boxes_per_pixel, dim=0)output = out_grid + anchor_manipulationsreturn output.unsqueeze(0)

展示一个锚框

img = d2l.plt.imread('catdog.jpg')
h, w = img.shape[:2]  # 561 728
X = torch.rand(size=(1, 3, h, w))
Y = multibox_prior(X, sizes=[0.75, 0.5, 0.25], ratios=[1, 2, 0.5]) # torch.Size([1, 2042040, 4])
boxes = Y.reshape(h, w, 5, 4) # tensor([0.06, 0.07, 0.63, 0.82])def show_bboxes(axes, bboxes, labels=None, colors=None):"""显示所有边界框。"""def _make_list(obj, default_values=None):if obj is None:obj = default_valueselif not isinstance(obj, (list, tuple)):obj = [obj]return objlabels = _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'
])
d2l.plt.show()

2 IoU(交互比)

def box_iou(boxes1, boxes2):"""计算两个锚框或边界框列表中成对的交并比。"""box_area = lambda boxes: ((boxes[:, 2] - boxes[:, 0]) *(boxes[:, 3] - boxes[:, 1]))areas1 = box_area(boxes1) # 算出锚框1的面积areas2 = box_area(boxes2) # 算出锚框2的面积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)inter_areas = inters[:, :, 0] * inters[:, :, 1] # 交集框union_areas = areas1[:, None] + areas2 - inter_areas # 各自的面积减去交集return inter_areas / union_areas

3 将真实边界框分配给锚框

def assign_anchor_to_bbox(ground_truth, anchors, device, iou_threshold=0.5):num_anchors, num_gt_boxes = anchors.shape[0], ground_truth.shape[0]jaccard = box_iou(anchors, ground_truth)anchors_bbox_map = torch.full((num_anchors,), -1, dtype=torch.long,device=device)max_ious, indices = torch.max(jaccard, dim=1)anc_i = torch.nonzero(max_ious >= 0.5).reshape(-1)box_j = indices[max_ious >= 0.5]anchors_bbox_map[anc_i] = box_jcol_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)box_idx = (max_idx % num_gt_boxes).long()anc_idx = (max_idx / num_gt_boxes).long()anchors_bbox_map[anc_idx] = box_idxjaccard[:, box_idx] = col_discardjaccard[anc_idx, :] = row_discardreturn anchors_bbox_map

4 标记类和偏移

def offset_boxes(anchors, assigned_bb, eps=1e-6):"""对锚框偏移量的转换。"""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 offsetdef multibox_target(anchors, labels):"""使用真实边界框标记锚框。"""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)class_labels = torch.zeros(num_anchors, dtype=torch.long,device=device)assigned_bb = torch.zeros((num_anchors, 4), dtype=torch.float32,device=device)indices_true = torch.nonzero(anchors_bbox_map >= 0)bb_idx = anchors_bbox_map[indices_true]class_labels[indices_true] = label[bb_idx, 0].long() + 1assigned_bb[indices_true] = label[bb_idx, 1:]offset = offset_boxes(anchors, assigned_bb) * bbox_maskbatch_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)

在图像中绘制这些地面真相边界框和锚框

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'])
d2l.plt.show()

5 应用逆偏移变换来返回预测的边界框坐标

def offset_inverse(anchors, offset_preds):"""根据带有预测偏移量的锚框来预测边界框。"""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

6 nms

def nms(boxes, scores, iou_threshold):"""对预测边界框的置信度进行排序。"""B = torch.argsort(scores, dim=-1, descending=True)keep = []while B.numel() > 0:i = B[0]keep.append(i)if B.numel() == 1: breakiou = 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)

7 将非极大值抑制应用于预测边界框

def multibox_detection(cls_probs, offset_preds, anchors, nms_threshold=0.5,pos_threshold=0.009999999):"""使用非极大值抑制来预测边界框。"""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)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] = -1class_id = class_id[all_id_sorted]conf, predicted_bb = conf[all_id_sorted], predicted_bb[all_id_sorted]below_min_idx = (conf < pos_threshold)class_id[below_min_idx] = -1conf[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)

最后的输出只保留了最大置信度的锚框