第3章 Python 数字图像处理(DIP) - 灰度变换与空间滤波8 - 直方图处理 - 直方

直方图均衡化

灰度映射函数：

s=T(r),0≤r≤L−1(3.8)s = T(r), \quad 0\leq r \leq L -1 \tag{3.8}s=T(r),0≤r≤L−1(3.8)

假设：

(1) T(r)T(r)T(r)在区间0≤r≤L−10 \leq{r} \leq{L-1}0≤r≤L−1 上是一个单调递增函数。

(2) 对于0≤r≤L−10 \leq{r} \leq{L-1}0≤r≤L−1 ，有0≤T(r)≤L−10 \leq{T(r)} \leq{L-1}0≤T(r)≤L−1 。

逆变换

r=T−1(s),0≤s≤L−1(3.9)r = T^{-1}(s), \quad 0 \leq s \leq L-1 \tag{3.9} r=T−1(s),0≤s≤L−1(3.9)

将条件(1)改为：

(1’) T(r)T(r)T(r)在区间0≤r≤L−10 \leq{r} \leq{L-1}0≤r≤L−1 上是一个严格单调递增函数。

(1)中T(r)T(r)T(r)单调递增的条件保证输出灰度值从不小于对应的输入值，从而防止灰度反转而产生伪像。

(2)保证输出灰度的范围与输入的范围相同。

(1’)保证从sss返回到rrr的映射是一对一的，从而防止出现歧义。

图像的灰度可视为区间[0,L−1][0, L-1][0,L−1]内的一个随机变量。令pr(r)p_{r}(r)pr​(r)和ps(s)p_{s}(s)ps​(s)表示两幅不同图像中灰度值rrr和sss的PDF（概率密度函数）。ppp的下标表明pr(r)p_{r}(r)pr​(r)和ps(s)p_{s}(s)ps​(s)是不同的函数。若已知pr(r)p_{r}(r)pr​(r)和T(r)T(r)T(r)，且T(r)T(r)T(r)是连续的且在感兴趣的值域上是可微的，则变换（映射）后的变量sss的PDF是

ps(s)=pr(r)∣drds∣(3.10)p_{s}(s) = p_{r}(r) \bigg\lvert \frac{\text{d}r}{\text{d}s}\bigg\rvert \tag{3.10}ps​(s)=pr​(r)∣∣∣∣​dsdr​∣∣∣∣​(3.10)

可以看到输出灰度变量sss的PDF是由输入灰度的PDF和所有的变换函数确定的

图像处理中的一个特别重要的变换函数是

s=T(r)=(L−1)∫0rpr(w)dw(3.11)s = T(r) = (L-1) \int_{0}^{r} p_r(w) \text{d} w \tag{3.11} s=T(r)=(L−1)∫0r​pr​(w)dw(3.11)

www是一个假积分变量，右侧的积分是随机变量rrr的累积分布函数（CDF）

根据莱布尼茨积分法则可知

drds=dT(r)dr=(L−1)ddr[∫0rpr(w)dw]=(L−1)pr(r)(3.12)\frac{\text{d}r}{\text{d}s} = \frac{\text{d}T(r)}{\text{d}r} = (L - 1) \frac{\text{d}}{\text{d}r} \Bigg[\int_{0}^{r} p_r(w) \text{d} w \Bigg] = (L-1)p_{r}(r)\tag{3.12}dsdr​=drdT(r)​=(L−1)drd​[∫0r​pr​(w)dw]=(L−1)pr​(r)(3.12)

用这个结果代替式（3.10）的drds\frac{\text{d}r}{\text{d}s}dsdr​，并且所有的概率值都是正的，有：

ps(s)=pr(r)∣drds∣=pr(r)∣1(L−1)pr(r)∣=1L−1,0≤s≤L−1(3.10)p_{s}(s) = p_{r}(r) \bigg\lvert \frac{\text{d}r}{\text{d}s}\bigg\rvert = p_{r}(r) \bigg\lvert \frac{1}{(L-1)p_{r}(r)}\bigg\rvert = \frac{1}{L - 1}, \;0 \leq s \leq L - 1\tag{3.10}ps​(s)=pr​(r)∣∣∣∣​dsdr​∣∣∣∣​=pr​(r)∣∣∣∣​(L−1)pr​(r)1​∣∣∣∣​=L−11​,0≤s≤L−1(3.10)

直方图均衡化或直方图线性变换

式（3.11)变换的离散形式为：

sk=T(rk)=(L−1)∑j=0kpr(rj),k=0,1,2,…,L−1(3.15)s_{k} = T(r_{k}) = (L -1) \sum_{j=0}^k p_{r}(r_{j}),\quad k = 0, 1, 2, \dots, L-1 \tag{3.15}sk​=T(rk​)=(L−1)j=0∑k​pr​(rj​),k=0,1,2,…,L−1(3.15)

def my_calhist(img):"""histogram equalizationparam: input img: uint8[0, 255] grayscale imagereturn uint8[0, 255] grayscale image after histogram equalization"""hist, bins = my_hist(img, bins=256, normalized=True)#--------------------------------Numpy-------------------hist_cumsum = np.round(np.cumsum(hist) * 255).astype(int)img_dst = img.copy()img_dst = hist_cumsum[img_dst]#--------------------------------loop----------------------#map_dict = {}#map_dict = map_dict.fromkeys(bins.astype(int), 0) #for i in bins:# s = np.round(255 * hist[:i].sum()).astype(int)# map_dict[i] = s#height, width = img.shape[:2]#img_dst = np.zeros([height, width], np.uint8)#for h in range(height):# for w in range(width):# img_dst[h, w] = map_dict[img[h, w]] # dict 用[ ]#img_dst = np.clip(img_dst, 0, 255).astype(np.uint8)#return img_dst, map_dict return img_dst, _

# 直方图均衡化import timeimg_1st = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(1)(top_left).tif', 0)img_2nd = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(2)(2nd_from_top).tif', 0)img_3rd = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(3)(third_from_top).tif', 0)img_4th = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(4)(bottom_left).tif', 0)img_list =['img_1st', 'img_2nd', 'img_3rd', 'img_4th']fig = plt.figure(figsize=(20, 20))start_time = time.time()for i in range(len(img_list)):# plot orginal imageax = fig.add_subplot(4, 4, i+1)ax.imshow(eval(img_list[i]), cmap='gray', vmin=0, vmax=255), ax.set_title('Original')# plot histogram equalization imageimg_dst, _ = my_calhist(eval(img_list[i]))ax1 = fig.add_subplot(4, 4, i+5)ax1.imshow(img_dst, cmap='gray', vmin=0, vmax=255), ax1.set_title('After Histogram Equalization')# plot orginal histax2 = fig.add_subplot(4, 4, i+9)hist, bins = my_hist(eval(img_list[i]), bins=256)ax2.bar(bins, hist), ax2.set_title('Original Histogram')# plot hist after histogram equalizationax3 = fig.add_subplot(4, 4, i+13)hist, bins = my_hist(img_dst, bins=256)ax3.bar(bins, hist), ax3.set_title('Hist After Histogram Equalization')elapse = time.time() - start_timeprint(f'Elapse -> {elapse}s') plt.tight_layout()plt.show()

Elapse -> 2.3357810974121094s

def my_calhist(img):"""histogram equalizationparam: input img: uint8[0, 255] grayscale imagereturn uint8[0, 255] grayscale image after histogram equalization"""hist, bins = my_hist(img, bins=256, normalized=True)hist_cumsum = np.round(np.cumsum(hist) * 255).astype(int)height, width = img.shape[:2]img_dst = np.zeros([height, width], np.uint8)for h in range(height):for w in range(width):img_dst[h, w] = hist_cumsum[img[h, w]] # dict 用[ ]img_dst = np.clip(img_dst, 0, 255).astype(np.uint8)return img_dst, hist_cumsum

# 直方图均衡化转换函数img_1st = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(1)(top_left).tif', 0)img_2nd = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(2)(2nd_from_top).tif', 0)img_3rd = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(3)(third_from_top).tif', 0)img_4th = cv2.imread('DIP_Figures/DIP3E_Original_Images_CH03/Fig0316(4)(bottom_left).tif', 0)img_list =['img_1st', 'img_2nd', 'img_3rd', 'img_4th']fig = plt.figure(figsize=(6, 6))for i in range(len(img_list)):# plot histogram equalization transform_, map_dict = my_calhist(eval(img_list[i]))ax1 = fig.gca()ax1.plot(map_dict)plt.xlabel('Original gray value', fontsize=12)plt.ylabel('Transformed gray value', fontsize=12)plt.xticks([0, 64, 128, 192, 255])plt.yticks([0, 64, 128, 192, 255])plt.xlim([0, 255])plt.ylim([0, 255])plt.tight_layout()plt.show()

第3章 Python 数字图像处理(DIP) - 灰度变换与空间滤波8 - 直方图处理 - 直方图均衡化（全局直方图均衡化）