This is the Fast R-CNN paper. It improves over R-CNN. It is a single-stage training model (i.e., only one model to train) on the Pascal VOC 2012 dataset. It is faster than R-CNN by using the RoI pooling layer.

Fast R-CNN architecture is as follows:

1. The input is the entire image and a set of object proposals. The entire image is first processed by some convolutional layer and max pooling layers to produce a conv feature map
2. RoI pooling: Each proposal is a region of interest (RoI). The RoI is matched to the conv feature map to extract a fixed-length feature vector
3. Feature vector is feed into a sequence of fully connected layers
4. The layers then branch into two sibling output layers, one for softmax probability estimate over $$K$$ object classes + 1 background class, and another for regression of $$4K$$ refined bounding box positions

The RoI pooling layer is the key to Fast R-CNN. It uses max pooling to convert each channel of a feature map into size $$H\times W$$ (e.g., 7×7). The RoI is defined by the pixel coordinate of the top-left corner $$(r,c)$$ and its size $$(h,w)$$. It first divided the $$h\times w$$ pixel grid of the RoI into windows of size $$h/H\times w/W$$, then max-pooling the values in each sub-window. The output would be a tensor of fixed size $$(H, W, C)$$. This is inspired by spatial pyramid pooling, but with only one pyramid level.

Fast R-CNN network uses a pre-trained network such as VGG-16, but

1. the last max pooling layer of the network is replaced by a RoI pooling layer with parameter $$H$$ and $$W$$ (e.g., 7×7).
2. the network’s last fully connected layer and softmax layer are replaced with two sibling layers for $$K+1$$ category classification and category-specific bounding box regression
3. the network takes two inputs: a batch is a list of images and a list of RoIs

R-CNN is slow because the ConvNet forward pass is run once per proposal, even on the same image. Fast R-CNN mitigated this by allowing the output of ConvNet reused for multiple RoI.

## Training

Each sample of input to the model is one RoI but we can design to train the model faster by taking the advantage of shared feature map.

The model is trained with SGD with mini-batches are sampled hierarchically. A mini-batch of size $$R=128$$ is from $$N=2$$ images, each image contributes $$R/N=64$$ RoIs. Therefore RoIs from the same image can share the feature map computation in both forward and backward passes, and the training can be speedup by 64x. The RoIs for training are selected IoU $$ge 0.5$$ as positive samples and IoU $$\in[0.1, 0.5)$$ as negative samples. The threshold 0.1 is selected as heuristic for hard example mining. As augmentation, the input image may be flipped horizontally with probability of 0.5.

The output of the network, for each RoI, is a propability distribution per RoI $$p=(p_0, p_1, \dots, p_K)$$ (which class 0 is conventionally for background class) and $$K$$ bounding box regression offsets $$t^k=(t_x^k, t_y^k, t_w^k, t_h^k)$$ for $$k=1,\dots,K$$. The regression offset uses the same encoding as in R-CNN.

For a ground-truth class $$u$$ and ground-truth bounding box regression target $$v$$, there is a single loss function for training

$L(p,u,t^u, v) = L_{\text{cls}}(p,u) + \lambda[u\ge 1]L_{\text{loc}}(t^u, v)$

where $$\lambda=1$$, $$L_{\text{cls}}(p,u)=-\log p_u$$ (simple log-loss for class $$u$$), The Iverson bracket indicator $$[u\ge 1]$$ is 1 if $$u\ge 1$$ and 0 otherwise (i.e., no $$L_{\text{loc}}$$ for background class). The localization loss is defined as

$L_{\text{loc}}(t^u, v) = \sum_{\ast\in\{x,y,w,h\}} L_1(t_\ast^u - v_\ast)$

where $$L_1(x)$$ is the smooth L1 function, i.e.

$L_1(x) = \begin{cases} 0.5x^2 & \text{if }\vert x\vert < 1 \\ \vert x\vert-0.5 & \text{otherwise} \end{cases}$

L1 is preferred over L2 to prevent exploding gradients.

The backpropagation at RoI pooling layer is done as follows. Assume the input to the RoI pooling is $$x_i$$ (the activation output $$i$$ from the preceding conv layer) and its output is $$y_{rj}$$ (the output $$j$$ of channel $$r$$ from RoI pooling), then we can model RoI pooling as

$y_{rj} = \max_{i\in R(r,j)} x_i$

where $$R(r,j)$$ is the set of indices of the corresponding sub-window. Let $$i^\ast = \argmax_{i\in R(r,j)} x_i$$ (which $$i^\ast$$ is a function of $$r,j$$), we can have the partial derivative

$\frac{\partial L}{\partial x_i} = \sum_r\sum_j [i=i^\ast] \frac{\partial L}{\partial y_{rj}}$

i.e., partial derivative $$\partial L/\partial y_{rj}$$ is accumulated into $$\partial L/\partial x_i$$ in the backward process if $$x_i$$ is selected for $$y_{rj}$$ by max pooling.

The SGD is run with global learning rate of 1e-3 for 30K mini-batches and then learning rate of 1e-4 for another 10K mini-batches. Per-layer learning rate of 1 for weights and 2 for biases is applied. Momentum of 0.9 and decay of 5e-4 (on weights and biases) are used.

## Inference

At inference, the network takes one image and a list of proposals (around 2000) to score. The output will be a class posterior probability distribution $$p$$ and a bounding box offset for each of the $$K$$ classes. RoI is assigned to a class with confidence based on the probability and the localization is selected using non-maximum suppression for each class, following the same algorithm as in R-CNN.

Most of the computation is at the final fully-connected layers. They are essentially matrix multiplications. To make the inference faster, the technique of truncated SVD is applied: Consider the layer’s weight as $$W$$, a $$u\times v$$ matrix, we can factorize using SVD as

$W \approx U\Sigma_t V^\top$

where $$\Sigma_t$$ is a $$t\times t$$ diagonal matrix of the top $$t$$ singular values of $$W$$. Matrix $$U$$ is $$u\times t$$ matrix of the first $$t$$ left-singular vectors of $$W$$ and matrix $$V$$ is $$v\times t$$ matrix of the first $$t$$ right-singular vectors of $$W$$. This reduces the total parameters from $$uv$$ in matrix $$W$$ into $$t(u+v)$$. We can replace one fully-connected layer (of weight $$W$$) into two (of weight $$\Sigma_tV^\top$$ without bias then of $$U$$ with the original bias), without non-linear activations in between.

## What is spatial pyramid pooling?

A fully convolutional network can work on the input image of any size but the output width and height depends on the input size. If the output of a ConvNet is the input to a fully connected network, a fixed size is required. Spatial pyramid pooling is to convert an arbitrary-sized output tensor of shape $$(H,W,C)$$ into fixed-size tensor. It is a pooling layer (e.g., average pooling or max pooling) but the receptive field is adaptive to the input size. A tensor of size $$(H, W)$$ can be pooled into output of size $$(N,N)$$ with the filter size $$(H/N, W/N)$$, where in the pyramid, $$N=1,2,3,\dots$$. The outputs of the pooling layers can be concatenated as a feature vector for the next step.

## Bibliographic data

@inproceedings{
title = "Fast R-CNN",
author = "Ross Girshick",
booktitle = "Proc ICCV",
month = "Sep",
year = "2015",
note = "arXiv:1504.08083",
}