宣城做网站公司,湖南智慧住建云,网络架构模拟设计报告,潼南区做网站的公司直方图均衡化 灰度映射函数#xff1a; sT(r),0≤r≤L−1(3.8)s T(r), \quad 0\leq r \leq L -1 \tag{3.8}sT(r),0≤r≤L−1(3.8) 假设#xff1a; (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} …直方图均衡化 灰度映射函数 sT(r),0≤r≤L−1(3.8)s T(r), \quad 0\leq r \leq L -1 \tag{3.8}sT(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 rT−1(s),0≤s≤L−1(3.9)r T^{-1}(s), \quad 0 \leq s \leq L-1 \tag{3.9} rT−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 sT(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} sT(r)(L−1)∫0rpr(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 drdsdT(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}dsdrdrdT(r)(L−1)drd[∫0rpr(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 skT(rk)(L−1)∑j0kpr(rj),k0,1,2,…,L−1(3.15)s_{k} T(r_{k}) (L -1) \sum_{j0}^k p_{r}(r_{j}),\quad k 0, 1, 2, \dots, L-1 \tag{3.15}skT(rk)(L−1)j0∑kpr(rj),k0,1,2,…,L−1(3.15)
def my_calhist(img):histogram equalizationparam: input img: uint8[0, 255] grayscale imagereturn uint8[0, 255] grayscale image after histogram equalizationhist, bins my_hist(img, bins256, normalizedTrue)#--------------------------------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 time
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(20, 20))
start_time time.time()
for i in range(len(img_list)):# plot orginal imageax fig.add_subplot(4, 4, i1)ax.imshow(eval(img_list[i]), cmapgray, vmin0, vmax255), ax.set_title(Original)# plot histogram equalization imageimg_dst, _ my_calhist(eval(img_list[i]))ax1 fig.add_subplot(4, 4, i5)ax1.imshow(img_dst, cmapgray, vmin0, vmax255), ax1.set_title(After Histogram Equalization)# plot orginal histax2 fig.add_subplot(4, 4, i9)hist, bins my_hist(eval(img_list[i]), bins256)ax2.bar(bins, hist), ax2.set_title(Original Histogram)# plot hist after histogram equalizationax3 fig.add_subplot(4, 4, i13)hist, bins my_hist(img_dst, bins256)ax3.bar(bins, hist), ax3.set_title(Hist After Histogram Equalization)elapse time.time() - start_time
print(fElapse - {elapse}s)
plt.tight_layout()
plt.show()Elapse - 2.3357810974121094sdef my_calhist(img):histogram equalizationparam: input img: uint8[0, 255] grayscale imagereturn uint8[0, 255] grayscale image after histogram equalizationhist, bins my_hist(img, bins256, normalizedTrue)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, fontsize12)
plt.ylabel(Transformed gray value, fontsize12)
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()
