## 3 PID-Net-Based Yeast Counting Method
### The Structure of PID-Net  
Following the basic idea of SegNet and U-Net for image segmentation, the structure of the proposed PID-Net is shown in Figure 2, which is an end-to-end CNN structure based on the encoder and decoder. There are four blocks in the encoder network.  
The first parts in each block are two convolution operations with a kernel size of 3 \(\times\) 3 (each followed by a ReLU operation), and then the max-pooling with the size of 2 \(\times\) 2 and stride of 2 pixels is applied to reduce the size of feature maps by half, followed by a convolution and ReLU operation. The channel of feature maps is denoted as \(C\). Pixel interval down-sampling is applied for down-sampling, which is shown in Figure 2. Each pixel is sampled with the pixels apart, and the size of each feature map is replaced by half.  
The classical down-sampling methods, such as max-pooling (with the kernel size of 2) will drop \(\frac{3}{4}\) data of the original image. It can retain the main information but not be fit to the task of tiny object counting (the edge lines may be lost while max-pooling). Though there are several learnable pooling layers that have proposed, such as Fractional pooling [70], Stochastic pooling [71] and learned-norm pooling approaches [72]; however, they still cannot meet the requirement of accurate segmentation for dense tiny yeast cells.  
Thus, a new down-sampling method, pixel interval down-sampling, is proposed here, which can reduce the size of feature maps without dropping data. Afterward, four pixel interval down-sampling feature maps and the features after max-pooling are concatenated to 5C-dimensional features. Finally, a convolutional filter with \(C\) channels is applied to reduce 5C-dimensional features to \(C\)-dimensional features. Hereto, the initial feature maps with size \(H\times W\) and channel \(C\) are changed to feature maps with size \(\frac{H}{2}\times\frac{W}{2}\) and channel \(C\). The procedure is repeated four times with output resolutions of \(\frac{H}{16}\times\frac{W}{16}\) and channel of 8C.  
In the decoder network, four blocks are applied for up-sampling. Two convolution operations with a kernel size of 3 \(\times\) 3 (each followed by a ReLU operation) are applied first. Then, the transposed convolution operation with a kernel size of 3, a stride of 2 and padding of 1 is applied for up-sampling. The transposed convolution operation is widely applied in GANs to expand the size of images [73]. The count of channels after up-sampling of the bottleneck is 512, which is calculated by using the transposed convolutional filter with 512 channels [74].  
The params in the transposed convolution filter can be learned while training. After that, the high resolution feature maps of encoder network are transformed to low-resolution  
Figure 2: The structure of the proposed PID-Net.  
feature maps using 2\(\times\), 4\(\times\) and 8\(\times\) max-pooling, which is shown in Figure 2. Then, the feature maps after up-sampling and max-pooling are concatenated with the feature maps generated by the corresponding layer from the encoder.  
For instance, the 8\(\times\) max-pooling features of the first block, 4\(\times\) max-pooling of the second block and 2\(\times\) max-pooling of the third block in the encoder are concatenated with the copied features of the fourth encoder block and the features after up-sampling (five parts of feature maps are concatenated in the first decoder block). In the same way, there are 4, 3 and 2 parts of features are concatenated in the second, third and fourth level of the decoder, respectively. After the concatenated operation, two convolutions and ReLU operations are applied to change the number of channels. The up-sampling operation is repeated four times with output resolutions of \(H\times W\) and channel of \(C\), which has the same size as the encoder's input features. Finally, a Softmax layer with two output channels is applied for feature map classification.