数字加网（二）— —调幅加网算法及实现

数字加网（一）— —概念及分色合成
数字加网（二）— —调幅加网算法及实现
数字加网（三）— —调频加网算法及实现

在上一篇中提到调幅加网的方法有有理正切、无理正切和超细胞方法。无理正切方法，与有理正切本质相同，但是因为正切值为无理数，不能保证网格的角点与输出设备记录栅格的角点重合，需要进行一个强制重合的过程，一般数字加网方法里对其讨论较少。而超细胞的方法相当于外面一个大的阈值矩阵是有理正切，里面若干小的阈值矩阵是无理正切的。下图分别是有理正切和超细胞方法。

上段提到阈值矩阵，如下图，其实也可以叫做蒙板或者窗口，即在待加网的图像上“走”一圈下来（扫描整幅图像），当前像素值比阈值矩阵中相对应的值大，则像素值置为1，反之置为0。阈值矩阵本质就是一个矩阵，通过矩阵的值控制加网的网点形状（圆形、方形、菱形等）。当然不同的加网角度也对应不同的阈值矩阵。

阈值矩阵的设计

接下来我将给出调幅加网的算法及Python代码，主要参考徐锦林的《调幅加网阈值矩阵设计方法与加网原理》，文中基于最小阈值矩阵原理，详述12×12网格的最小阈值矩阵的设计原则。网格构成为12×12时，加网角度0度，45度，15度（14.04）和75度（75.96），对应的网格大小分别为为144、128与153，加网阈值的个数远小于255。加网时需将灰度图像的像素值与加网阈值比较，因此必须建立两者的转换（映射）关系。可采用以下公式处理。

式中fix表示取最小整数（下同），Qmax为对应的网格的最大值（144,218,513）。

加网角度为0度的阈值矩阵

加网角度为45度的阈值矩阵

加网角度为14.04度的阈值矩阵

加网角度为75.96度的阈值矩阵

后两个阈值矩阵分别为网线角度14.04°和75.96°时的最小阈值矩阵，因列数太多，故将表格拆开成左右两部分。不难发现，如将这两个矩阵扩展为51×51的方阵，则它们互为转置矩阵。

加网过程及算法

采用最小阈值矩阵进行灰度图像加网的过程如图所示。设原灰度图像大小(像素数)为（W ×H ），在网格构成为12×12时，则加网的二值图像大小(像素数)必为N (12×W )×M (12×H )。加网时从上到下，从左到右扫描加网图像的每一个像素，由其坐标值如图中的(m ,n )，求出该像素在阈值矩阵中对应的阈值坐标(k ,l )，得到加网阈值Q (k ,l )，同时从原灰度图像中查到与之对应的像素坐标(i ,j )，得到原灰度图像的像素值F (i ,j )，然后将灰度图像的像素值F (i ,j )与阈值Q (k ,l )比较，即可决定加网图像中该像素的取值，即决定P (m ,n )是“1”还是“0”，从而完成一个像素的加网过程。为了清楚地表示加网过程，图中的箭头处加了序号。

0度加网算法

### Digital Screening
### Author: Sin_Geek
### Date: 2015-05-17

from PIL import Image

time = 12
K = 12
L = 12
N = 144
im = Image.open('lena0.jpg').convert('L')
im2 = Image.new("1",(im.size[0]*time,im.size[1]*time))

Mask=[144,140,132,122,107,63,54,93,106,123,133,142,
    143,137,128,104,94,41,31,65,98,116,120,139,
    135,131,114,97,61,35,24,55,80,103,113,125,
    126,117,88,83,56,29,15,51,68,90,99,111,
    109,100,81,77,48,22,8,28,47,76,85,96,
    91,44,16,12,9,3,5,21,25,33,37,73,
        59,58,30,18,10,1,2,4,11,19,34,42,
    92,64,57,52,26,6,7,14,32,46,53,74,
    101,95,70,67,38,13,20,36,50,75,82,108,
    121,110,86,78,45,17,27,39,69,79,102,119,
    134,129,112,89,49,23,43,60,71,87,115,127,
    141,138,124,118,66,40,62,72,84,105,130,136]

for m in range(im2.size[1]):
    k = m % K
    for n in range(im2.size[0]):
        l = n % L
        pix = int(im.getpixel((m/time,n/time))/255.0*N+0.5)
        if pix > Mask[k*L+l]: 
            im2.putpixel((m,n),1)
        else:
            im2.putpixel((m,n),0)
im2.save("AM0.bmp")

效果图

15度加网算法

### Digital Screening
### Author: Sin_Geek
### Date: 2015-05-17

from PIL import Image

time = 12
K = 3
L = 51
N = 153
q = 4
im = Image.open('lena0.jpg').convert('L')
im2 = Image.new("1",(im.size[0]*time,im.size[1]*time))

Mask=[  153,148,120,77,53,28,26,60,87,122,131,135,132,124,116,104,73,47,23,6,56,66,85,57,51,39, 
    19,8,15,2,7,17,55,79,83,99,102,109,112,117,105,74,54,14,24,64,84,121,137,142,150,
    145,139,101,69,48,11,34,68,100,128,138,143,147,141,125,97,71,43,13,30,62,90,107,110,96,91,
    76,52,27,20,5,4,21,25,37,45,82,92,94,95,98,63,41,1,38,67,89,127,134,140,149,
    136,126,88,59,31,12,46,75,114,130,146,151,152,144,136,86,61,40,18,49,70,103,119,123,115,111,
    108,93,80,65,36,3,22,50,35,9,16,32,44,81,78,58,29,10,42,72,106,113,118,129,133]

for m in range(im2.size[1]):
    k = m % K
    t = L-(q*K*(m/K))%L
    for n in range(im2.size[0]):
        l = (n%L+t)%L
        pix = int(im.getpixel((m/time,n/time))/255.0*N+0.5)
        if pix > Mask[k*L+l]: 
            im2.putpixel((m,n),1)
        else:
            im2.putpixel((m,n),0)
im2.save("AM15.bmp")

效果图

45度加网算法

### Digital Screening
### Author: Sin_Geek
### Date: 2015-05-17

from PIL import Image

time = 12
K = 8
L = 16
N = 128
im = Image.open('lena0.jpg').convert('L')
im2 = Image.new("1",(im.size[0]*time,im.size[1]*time))

Mask=[128,120,109,92,74,66,46,8,15,10,64,79,97,111,122,127,
        123,116,87,69,62,38,6,39,42,3,19,55,86,105,115,119,
        107,96,71,59,24,12,28,52,63,47,20,1,58,95,108,112,
        84,73,56,2,18,23,48,78,82,67,35,5,31,61,91,101,
        77,53,32,4,25,43,75,85,100,89,60,30,9,34,68,80,
        51,41,21,27,40,70,94,102,110,103,93,57,26,11,37,65,
        44,29,33,45,72,90,104,121,117,114,106,88,54,17,13,16,
        14,36,49,76,83,98,118,126,125,124,113,99,81,50,22,7]

for m in range(im2.size[1]):
    k = m % K
    if m/K%2 == 0:
        t = 0
    else:
        t = K
    for n in range(im2.size[0]):
        l = (n%L+t)%L
        pix = int(im.getpixel((m/time,n/time))/255.0*N+0.5)
        if pix > Mask[k*L+l]: 
            im2.putpixel((m,n),1)
        else:
            im2.putpixel((m,n),0)
im2.save("AM45.bmp")

效果图

75度加网算法

### Digital Screening
### Author: Sin_Geek
### Date: 2015-05-17

from PIL import Image

time = 12
K = 3
L = 51
N = 153
q = 4
im = Image.open('lena0.jpg').convert('L')
im2 = Image.new("1",(im.size[0]*time,im.size[1]*time))

Mask=[153,145,136,117,95,81,8,52,93,104,97,86,77,69,59,54,41,29,7,5,36,23,13,18,26,34,
        46,64,67,72,79,25,50,66,90,103,122,128,130,137,134,118,102,82,16,51,96,115,132,147,152,
        148,139,126,105,98,78,15,27,80,73,71,61,53,48,31,14,1,10,17,4,3,6,30,49,60,68,
        75,84,89,106,83,37,35,85,107,119,131,138,146,142,140,129,109,92,32,39,91,111,124,141,144,
        120,101,88,74,63,58,2,20,65,47,43,40,28,11,12,24,38,42,55,21,22,56,62,70,87,100,
        114,121,127,113,99,45,9,57,110,123,135,143,151,150,149,133,112,94,44,19,76,108,116,125,136]

for m in range(im2.size[1]):
    k = m % K
    t = q*K*(m/K)%L;
    for n in range(im2.size[0]):
        l = (n%L+t)%L
        pix = int(im.getpixel((m/time,n/time))/255.0*N+0.5)
        if pix > Mask[k*L+l]: 
            im2.putpixel((m,n),1)
        else:
            im2.putpixel((m,n),0)
im2.save("AM75.bmp")

效果图