Python function Explanation & Verification

由於我們需要知道 python 以及 Unet 背後真正的運算方式是什麼，因此要對 Unet 中有用到一些比較特別的 API 去做驗證以及理解

torchvision.transforms.functional.center_crop

會用在下圖中 灰色箭頭(copy and crop) 的運算，此 function 即是從圖片(from Contracting path of Unet)正中心向外擷取所需的圖片範圍，因此我們必須算出該從圖片的哪個 pixel 開始擷取並擷取至哪裡

• verification

# -*- coding: utf-8 -*-
"""
Created on Thu Sep 14 20:48:00 2023

@author: Jay
"""



import torchvision
import torch
import numpy as np
from PIL import Image

channel = 9
input_width = 64 # assume height = width
output_width = 56

torchInputArray = torch.zeros(channel, input_width, input_width, dtype=torch.int32)
sizeList = list( torchInputArray.size() )

# initial the input array
num = 1
for i in range(sizeList[0]):
    for j in range(sizeList[1]):
        for k in range(sizeList[2]):
            torchInputArray[i, j, k] = num
            num += 1

torchResult = torchvision.transforms.functional.center_crop( torchInputArray, (output_width, output_width) )

# into numpy
torchInputArray_np = torchInputArray.numpy()
torchResult_np = torchResult.numpy()

# ========== handcraft func ==========
myResult_np = np.zeros([channel, output_width, output_width], dtype=np.int32)

start_addr = ((input_width-1)//2) - (output_width - 2)//2

for i in range(channel):
    for j in range(start_addr, start_addr+output_width):
        for k in range(start_addr, start_addr+output_width): 
            myResult_np[i, (j-start_addr), (k-start_addr)] = torchInputArray_np[i, j, k]



if((myResult_np == torchResult_np).all()):
    print("function Correct!")
else:
    print("Failed...........")

left_up = (start_addr, start_addr)
right_down = (start_addr+output_width-1, start_addr+output_width-1)

print(left_up, right_down)

# ========== coordinate generator ==========
def coordinate_Gen(sizeIn):
    (input_PicWidth, out_PicWidth) = sizeIn
    #print('input_PicWidth, out_PicWidth = ', input_PicWidth, out_PicWidth)
    start_addr = ((input_PicWidth-1)//2) - (out_PicWidth - 2)//2
    left_up = (start_addr, start_addr)
    right_down = (start_addr+out_PicWidth-1, start_addr+out_PicWidth-1)
    return [left_up, right_down]


Unet_Layer_copy_and_crop = { 
    'Layer1':(568, 392), 
    'Layer2':(280, 200), 
    'Layer3':(136, 104), 
    'Layer4':(64, 56),   
    }

for Layer in range(len(Unet_Layer_copy_and_crop)):
    print("Layer" + str(Layer) + ", [pixel_addr(left_up), pixel_addr(right_down)] = ", end='')
    print( coordinate_Gen(Unet_Layer_copy_and_crop['Layer' + str(Layer+1)]) )
    

-> 實際去 call torchvision.transforms.functional.center_crop 函式並驗證擷取範圍是否一致 -> Pass!

• Result explain :
  以Layer0 為例子，輸出的
  $392\ast 392$ 應該為原圖
  $568\ast 568$ 的 (y,x) = (8, 8) 位址往右擷取到 (8, 479)，再跳行，以此類推依序到 (y,x) = (479, 479)

-> 此 code 只有保證在圖片寬、高為 even 時能 work !

---

torch.nn.ConvTranspose2d

會用在上圖中 綠色箭頭(up-conv 2x2) 的運算，此 function 即是對輸入圖像做 de-convolution

• 一般 convolution v.s. de-convolution(full convolution)
  

# -*- coding: utf-8 -*-
"""
Created on Fri Sep 15 20:28:21 2023

@author: Jay
"""

import torch
import torchvision.transforms.functional
from torch import nn
from scipy.signal import convolve2d
import numpy as np

# initial array
inputArray_np = np.array([i+1 for i in range(5*5)])
inputArray_np = np.reshape(inputArray_np, (5, 5))

kernel = np.array([i+1 for i in range(3*3)])
kernel = np.reshape(kernel, (3, 3))

# normal conv
normalConv_result = convolve2d(inputArray_np, kernel[::-1, ::-1], mode='valid')

# de-conv method1
dConv_result1 = convolve2d(normalConv_result, kernel[::-1, ::-1], mode='full')

# de-conv method2
normalConv_result_pad = np.pad(normalConv_result, 2, 'constant')
dConv_result2 = convolve2d(normalConv_result_pad, kernel[::-1, ::-1], mode='valid')

if((dConv_result1 == dConv_result2).all()):
    print("same!!")
else:
    print("different....")

-> 看不懂我上面在畫啥的也可以參考這邊

• verification
  ConvTranspose2d visualization
  ConvTranspose2d formula:
  
  $Hout=(Hin-1)\times stride[0]-2\times padding[0]+dilation[0]\times (kernelsize[0]-1)+outpu{t}_{p}adding[0]+1$
  assume: dilation = 1, padding = 0, stride = 1, the formula becomes:
  
  $Hout=Hin+kernelsize[0]-1$, width vise versa.

# -*- coding: utf-8 -*-
"""
Created on Fri Sep 15 20:28:21 2023

@author: Jay
"""

import torch
import torchvision.transforms.functional
from torch import nn
from scipy.signal import convolve2d
import numpy as np


input_channel = 3
output_channel = 10
kernel_size = 3
input_picture_size = 32

# Calculate by nn.ConvTranspose2d
input_picture = torch.ones(input_channel, input_picture_size, input_picture_size)
Layer = nn.ConvTranspose2d(input_channel, output_channel, kernel_size=kernel_size, stride=1, padding=0, bias=False)

print("input shape  = ", input_picture.shape)
output = Layer(input_picture)
print("output shape = ", output.shape)
weight = Layer.weight
print("weight shape = ", weight.size())

# get the nparray of input/output and weight
input_picture_np = input_picture.numpy()
output_np = output.detach().numpy()
weight_np = weight.detach().numpy()


# # Calculate by myself to check nn.ConvTranspose2d
myOutput = np.zeros([ output_channel, (input_picture_size+kernel_size-1), (input_picture_size+kernel_size-1) ])

for cout in range(output_channel):
    outTmp = np.zeros([ (input_picture_size+kernel_size-1), (input_picture_size+kernel_size-1) ])
    for cin in range(input_channel):
        outTmp += convolve2d(input_picture_np[cin, ], weight_np[cin, cout, :, :], mode='full')
    myOutput[cout, ] = outTmp.copy()

error = myOutput - output_np

if(np.max(error) < 1e-7):# exist mini difference
    print("same!!")
else:
    print("different....")

Note : nn 提取出來的 weight 貌似有倒過來，例如下圖

---

up-conv 2x2

上面稍微解釋了 de-convolution 的運算方式，但上面的例子 stride 為 1，而 Unet 中的 de-conv

$2\ast 2$ 是用

$2\ast 2$ 的 kernel 搭配 stride=2，所以這邊要再解釋一下 de-conv 的 stride 參數

de-conv 的 stride 參數有別於一般 conv 的 stride ，若一般 conv 的 stride 設為 1 ，則表示 kernel 在 input 上 sliding window 的移動步伐間距為 1，而 de-conv 的 stride 參數則是表示要將輸入圖片的每兩個 pixel 之間填入 0，而填入數量就是 stride - 1，而 sliding window 的移動步伐間距依舊保持為 1，如此一來達成輸出圖片長寬為輸入圖片2倍的目的。

Example:
input size:

$4\ast 4$
kernel size:

$2\ast 2$
stride =

$2$

output =

$8\ast 8$

• verification

# -*- coding: utf-8 -*-
"""
Created on Fri Sep 15 20:28:21 2023

@author: Jay
"""

import torch
import torchvision.transforms.functional
from torch import nn
from scipy.signal import convolve2d
import numpy as np

#=================================================
# stride = 2
#=================================================

input_channel = 3
output_channel = 10
kernel_size = 2
input_picture_size = 32

# Calculate by nn.ConvTranspose2d
input_picture = torch.ones(input_channel, input_picture_size, input_picture_size)
Layer = nn.ConvTranspose2d(input_channel, output_channel, kernel_size=kernel_size, stride=2, padding=0, bias=False)

print("input shape  = ", input_picture.shape)
output = Layer(input_picture)
print("output shape = ", output.shape)
weight = Layer.weight
print("weight shape = ", weight.size())

# get the nparray of input/output and weight
input_picture_np = input_picture.numpy()
output_np = output.detach().numpy()
weight_np = weight.detach().numpy()

# input expansion
input_picture_tmp_np = np.zeros([input_channel, (input_picture_size*2-1), (input_picture_size*2-1)])
for ic in range(input_channel):
    for y in range(input_picture_np.shape[1]):
        for x in range(input_picture_np.shape[2]):
            input_picture_tmp_np[ic, y*2, x*2] = input_picture_np[ic, y, x]

# Calculate by myself to check nn.ConvTranspose2d
myOutput = np.zeros([ output_channel, (input_picture_size*2), (input_picture_size*2) ])

for cout in range(output_channel):
    outTmp = np.zeros([ (input_picture_size*2), (input_picture_size*2) ])
    for cin in range(input_channel):
        outTmp += convolve2d(input_picture_tmp_np[cin, ], weight_np[cin, cout, :, :], mode='full')
    myOutput[cout, ] = outTmp.copy()

error = myOutput - output_np

if(np.max(error) < 1e-7):# exist mini difference
    print("same!!")
else:
    print("different....")