手动绘制方向梯度直方图（HOG）

HOG（Histogram of Oriented Gradients）——方向梯度直方图，是一种表示图像特征量的方法，特征量是表示图像的状态等的向量集合。

在图像识别（图像是什么）和检测（物体在图像的哪个位置）中，我们需要：

1. 从图像中获取特征量（特征提取）；
2. 基于特征量识别和检测（识别和检测）；
   由于深度学习通过卷积网络自动执行特征提取和识别，所以看不到HOG，但在深度学习变得流行之前，HOG经常被用作特征量表达。

获得HOG算法如下：

• 图像灰度化，然后在x方向和y方向上求出亮度的梯度：
  • x方向：
    
  • y方向：
    
• 从g_x和g_y确定梯度幅值和梯度方向：
  • 梯度幅值：
    
  • 梯度方向：
    
• 将梯度方向[0,180]进行9等分量化。也就是说，对于[0,20]量化为index 0，对于[20,40]量化为index 1......
• 将图像划分为N x N个区域（该区域称为cell），并作出cell内上一步得到的index的直方图。
• C x C个cell被称为一个block，对每个block内的cell的直方图通过下面的式子进行归一化。由于归一化过程中窗口一次移动一个cell来完成，因此一个cell会被归一化多次，通常ε=1。
  
• 以上，求出HOG特征值。
• 将得到的特征量可视化。为 cell 内的每个 index 的方向画一条线段，并且值越大，线段越白，
  值越小，线段越黑。

import cv2
import numpy as np
import matplotlib.pyplot as plt

# get HOG
def HOG(img):
    # Grayscale
    def BGR2GRAY(img):
        gray = 0.2126 * img[..., 2] + 0.7152 * img[..., 1] + 0.0722 * img[..., 0]
        return gray

    # Magnitude and gradient
    def get_gradXY(gray):
        H, W = gray.shape

        # padding before grad
        gray = np.pad(gray, (1, 1), 'edge')

        # get grad x
        gx = gray[1:H+1, 2:] - gray[1:H+1, :W]
        # get grad y
        gy = gray[2:, 1:W+1] - gray[:H, 1:W+1]
        # replace 0 with 
        gx[gx == 0] = 1e-6

        return gx, gy

    # get magnitude and gradient
    def get_MagGrad(gx, gy):
        # get gradient maginitude
        magnitude = np.sqrt(gx ** 2 + gy ** 2)

        # get gradient angle
        gradient = np.arctan(gy / gx)

        gradient[gradient < 0] = np.pi / 2 + gradient[gradient < 0] + np.pi / 2

        return magnitude, gradient

    # Gradient quantization
    def quantization(gradient):
        # prepare quantization table
        gradient_quantized = np.zeros_like(gradient, dtype=np.int)

        # quantization base
        d = np.pi / 9

        # quantization
        for i in range(9):
            gradient_quantized[np.where((gradient >= d * i) & (gradient <= d * (i + 1)))] = i

        return gradient_quantized


    # get gradient histogram
    def gradient_histogram(gradient_quantized, magnitude, N=8):
        # get shape
        H, W = magnitude.shape

        # get cell num
        cell_N_H = H // N
        cell_N_W = W // N
        histogram = np.zeros((cell_N_H, cell_N_W, 9), dtype=np.float32)

        # each pixel
        for y in range(cell_N_H):
            for x in range(cell_N_W):
                for j in range(N):
                    for i in range(N):
                        histogram[y, x, gradient_quantized[y * 4 + j, x * 4 + i]] += magnitude[y * 4 + j, x * 4 + i]

        return histogram

		# histogram normalization
    def normalization(histogram, C=3, epsilon=1):
        cell_N_H, cell_N_W, _ = histogram.shape
        ## each histogram
        for y in range(cell_N_H):
    	    for x in range(cell_N_W):
       	    #for i in range(9):
                histogram[y, x] /= np.sqrt(np.sum(histogram[max(y - 1, 0) : min(y + 2, cell_N_H),
                                                            max(x - 1, 0) : min(x + 2, cell_N_W)] ** 2) + epsilon)

        return histogram

    # 1. BGR -> Gray
    gray = BGR2GRAY(img)

    # 1. Gray -> Gradient x and y
    gx, gy = get_gradXY(gray)

    # 2. get gradient magnitude and angle
    magnitude, gradient = get_MagGrad(gx, gy)

    # 3. Quantization
    gradient_quantized = quantization(gradient)

    # 4. Gradient histogram
    histogram = gradient_histogram(gradient_quantized, magnitude)
    
    # 5. Histogram normalization
    histogram = normalization(histogram)

    return histogram


# draw HOG
def draw_HOG(img, histogram):
    # Grayscale
    def BGR2GRAY(img):
        gray = 0.2126 * img[..., 2] + 0.7152 * img[..., 1] + 0.0722 * img[..., 0]
        return gray

    def draw(gray, histogram, N=8):
        # get shape
        H, W = gray.shape
        cell_N_H, cell_N_W, _ = histogram.shape
        
        ## Draw
        out = gray[1 : H + 1, 1 : W + 1].copy().astype(np.uint8)

        for y in range(cell_N_H):
            for x in range(cell_N_W):
                cx = x * N + N // 2
                cy = y * N + N // 2
                x1 = cx + N // 2 - 1
                y1 = cy
                x2 = cx - N // 2 + 1
                y2 = cy
                
                h = histogram[y, x] / np.sum(histogram[y, x])
                h /= h.max()
        
                for c in range(9):
                    #angle = (20 * c + 10 - 90) / 180. * np.pi
                    # get angle
                    angle = (20 * c + 10) / 180. * np.pi
                    rx = int(np.sin(angle) * (x1 - cx) + np.cos(angle) * (y1 - cy) + cx)
                    ry = int(np.cos(angle) * (x1 - cx) - np.cos(angle) * (y1 - cy) + cy)
                    lx = int(np.sin(angle) * (x2 - cx) + np.cos(angle) * (y2 - cy) + cx)
                    ly = int(np.cos(angle) * (x2 - cx) - np.cos(angle) * (y2 - cy) + cy)

                    # color is HOG value
                    c = int(255. * h[c])

                    # draw line
                    cv2.line(out, (lx, ly), (rx, ry), (c, c, c), thickness=1)

        return out
    

    # get gray
    gray = BGR2GRAY(img)

    # draw HOG
    out = draw(gray, histogram)

    return out

# Read image
img = cv2.imread("../baby.png").astype(np.float32)

# get HOG
histogram = HOG(img)

# draw HOG
out = draw_HOG(img, histogram)


# Save result
cv2.imwrite("out.jpg", out)
cv2.imshow("result", out)
cv2.waitKey(0)
cv2.destroyAllWindows()