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Compressão Fractal de Imagens

Aluno: Bruno Silvério de Freitas - RA: 140225

Esse trabalho é parte do seminário de Compressão Fractal de Imagens apresentado na disciplina de IA898 - Processamento de Imagens. O objetivo é demonstrar, a partir de algoritmos de exemplo, os conceitos envolvendo a compressão fractal de imagens, como por exemplo, teorema da contração e Mapeamento contrativo.

O algoritmo abaixo é uma implementação da compressão fractal de imagem que tem como objetivo explorar o conceito de Mapeamento Contrativo usando diversas transformações afins. Existem diversas implementações disponíveis no github, sendo que o código utilizado nesse trabalho é um modelo simples para o entendimentos da técnica e foi obtido nesse link.

As várias etapas que envolvem o processo de codificação e decodificação foram divididas em métodos que serão explicados brevemente a seguir:

get_greyscale_image: Usado para obter os médias dos níveis de cinza da imagem.

reduce: Usado para reduzir (cortar) a imagem e obter a média dos valores da imagem de entrada.

rotate: Rotaciona os pedaços da imagens para obter o melhor resultado de transformação afim.

flip: Inverte o pedaço da imagem que está sendo codificada ou decodificada.

apply_transformation: Aplica as transformações afins nos pedaços que serão usados para mapear a imagem.

find_contrast_and_brightness2: Ajusta o brilho e constrate entre os pedaços de imagem que serão comparados (domain e range).

generate_all_transformed_blocks: Cria todos as transformações possíveis para serem comparadas na fase de codificação.

compress: Executa a codificação da imagem a partir da criação dos domain-blocks e range-blocks. Realiza as comparações das transformações com as range-blocks e mapeia quando encontra o melhor resultado comparado.

decompress: Executa a decodificação utilizando os vetores criados (Mapeamento Contrativo). Inicialmente cria uma matriz e a cada iteração uma imagem de saída é obtida para a próxima iteração.

plot_iterations: Método responsável por exibir cada etapa da iteração com a imagem de saída obtida.





In [5]:



    


import matplotlib.pyplot as plt
import matplotlib.image as mpimg
from scipy import ndimage
from scipy import optimize
import numpy as np
import math

# Manipulate channels

def get_greyscale_image(img):
    return np.mean(img[:,:,:2], 2)

# Transformations

def reduce(img, factor):
    result = np.zeros((img.shape[0] // factor, img.shape[1] // factor))
    for i in range(result.shape[0]):
        for j in range(result.shape[1]):
            result[i,j] = np.mean(img[i*factor:(i+1)*factor,j*factor:(j+1)*factor])
    return result

def rotate(img, angle):
    return ndimage.rotate(img, angle, reshape=False)

def flip(img, direction):
    return img[::direction,:]

def apply_transformation(img, direction, angle, contrast=1.0, brightness=0.0):
    #transf = contrast*rotate(np.flip(img, direction), angle) + brightness
    transf = contrast*rotate(flip(img, direction), angle) + brightness
    return transf

# Contrast and brightness

def find_contrast_and_brightness2(D, S):
    # Fit the contrast and the brightness
    A = np.concatenate((np.ones((S.size, 1)), np.reshape(S, (S.size, 1))), axis=1)
    b = np.reshape(D, (D.size,))
    x, _, _, _ = np.linalg.lstsq(A, b)
    #x = optimize.lsq_linear(A, b, [(-np.inf, -2.0), (np.inf, 2.0)]).x
    return x[1], x[0]

# Compression for greyscale images

def generate_all_transformed_blocks(img, source_size, destination_size, step):
    factor = source_size // destination_size
    transformed_blocks = []
    for k in range((img.shape[0] - source_size) // step + 1):
        for l in range((img.shape[1] - source_size) // step + 1):
            # Extract the source block and reduce it to the shape of a destination block
            S = reduce(img[k*step:k*step+source_size,l*step:l*step+source_size], factor)
            # Generate all possible transformed blocks
            for direction, angle in candidates:
                transformed_blocks.append((k, l, direction, angle, apply_transformation(S, direction, angle)))
        if k < 1:
            print(transformed_blocks)
    return transformed_blocks

def compress(img, source_size, destination_size, step):
    transformations = []
    transformed_blocks = generate_all_transformed_blocks(img, source_size, destination_size, step)
    for i in range(img.shape[0] // destination_size):
        transformations.append([])
        for j in range(img.shape[1] // destination_size):
            print(i, j)
            transformations[i].append(None)
            min_d = float('inf')
            # Extract the destination block
            D = img[i*destination_size:(i+1)*destination_size,j*destination_size:(j+1)*destination_size]
            # Test all possible transformations and take the best one
            for k, l, direction, angle, S in transformed_blocks:
                contrast, brightness = find_contrast_and_brightness2(D, S)
                S = contrast*S + brightness
                #d = np.sum(np.square(D - S))
                d = np.sqrt(np.sum(np.square(D - S)))
                if d < min_d:
                    min_d = d
                    transformations[i][j] = (k, l, direction, angle, contrast, brightness)
    return transformations

def decompress(transformations, source_size, destination_size, step, nb_iter=8):
    factor = source_size // destination_size
    height = len(transformations) * destination_size
    width = len(transformations[0]) * destination_size
    iterations = [np.random.randint(0, 256, (height, width))]
    print('iterations', iterations)
    cur_img = np.zeros((height, width))
    for i_iter in range(nb_iter):
        for i in range(len(transformations)):
            for j in range(len(transformations[i])):
                # Apply transform
                k, l, flip, angle, contrast, brightness = transformations[i][j]
                S = reduce(iterations[-1][k*step:k*step+source_size,l*step:l*step+source_size], factor)
                D = apply_transformation(S, flip, angle, contrast, brightness)
                cur_img[i*destination_size:(i+1)*destination_size,j*destination_size:(j+1)*destination_size] = D
        iterations.append(cur_img)
        cur_img = np.zeros((height, width))
    return iterations

# Plot

def plot_iterations(iterations, target=None):
    # Configure plot
    plt.figure(figsize=(15,15))
    nb_row = math.ceil(np.sqrt(len(iterations)))
    nb_cols = nb_row
    erro = []
    # Plot
    for i, img in enumerate(iterations):
        plt.subplot(nb_row, nb_cols, i+1)
        plt.imshow(img, cmap='gray', vmin=0, vmax=255, interpolation='none')
        if target is None:
            plt.title(str(i))
        else:
            #Display the RMSE
            erro.append(np.sqrt(np.mean(np.square(target - img))))
            plt.title(str(i) + ' (' + '{0:.2f}'.format(np.sqrt(np.mean(np.square(target - img)))) + ')')
        frame = plt.gca()
        frame.axes.get_xaxis().set_visible(False)
        frame.axes.get_yaxis().set_visible(False)
    plt.tight_layout()
    plt.savefig('temp.png', dpi=200)
    plt.show()
    return erro

# Parameters

directions = [1, -1]
#directions = [0,1]
angles = [0, 90, 180, 270]
candidates = list(zip(directions, angles))

# Tests

def test_greyscale():
    img = mpimg.imread('../data/cameraman.tif') 
    #img = get_greyscale_image(img)
    #img = reduce(img, 2)
    plt.figure()
    plt.imshow(img, cmap='gray', interpolation='none')
    plt.show()
    transformations = compress(img, 8, 4, 8)
    iterations1 = decompress(transformations, 8, 4, 8,20)
    transformations = compress(img, 16, 8, 16)
    iterations2 = decompress(transformations, 16, 8, 16,20)
    
                                
    plt.figure()     
    
    x1 = np.zeros(25)
    y1 = np.zeros(25)
    for x,y in enumerate(plot_iterations(iterations1, img)):
        x1[x] = x
        y1[x] = y
      
    x2 = np.zeros(25)
    y2 = np.zeros(25)
    for x,y in enumerate(plot_iterations(iterations2, img)):
        x2[x] = x
        y2[x] = y
    
    plt.plot(x1[0:19],y1[0:19], "-r", label='Domain 8x8 | Range 4x4')
    plt.plot(x2[0:19],y2[0:19],"-b", label='Domain 16x16 | Range 8x8')
    plt.show()
    


                    
if __name__ == '__main__':
    test_greyscale()



    


















    

















    






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       [197.  , 196.25, 195.5 , 198.25],
       [195.75, 198.25, 195.75, 194.25],
       [196.75, 194.  , 190.25, 192.  ]])), (0, 29, -1, 90, array([[192.  , 194.25, 198.25, 192.  ],
       [190.25, 195.75, 195.5 , 195.75],
       [194.  , 198.25, 196.25, 195.  ],
       [196.75, 195.75, 197.  , 198.  ]])), (0, 30, 1, 0, array([[191.25, 190.75, 188.75, 189.  ],
       [192.5 , 191.75, 192.5 , 192.5 ],
       [196.  , 193.75, 191.75, 193.  ],
       [194.25, 195.  , 190.75, 189.  ]])), (0, 30, -1, 90, array([[189.  , 193.  , 192.5 , 189.  ],
       [190.75, 191.75, 192.5 , 188.75],
       [195.  , 193.75, 191.75, 190.75],
       [194.25, 196.  , 192.5 , 191.25]])), (0, 31, 1, 0, array([[187.25, 190.  , 191.  , 189.  ],
       [189.25, 191.5 , 192.  , 188.75],
       [190.75, 187.5 , 188.25, 188.5 ],
       [188.5 , 187.25, 185.75, 188.75]])), (0, 31, -1, 90, array([[188.75, 188.5 , 188.75, 189.  ],
       [185.75, 188.25, 192.  , 191.  ],
       [187.25, 187.5 , 191.5 , 190.  ],
       [188.5 , 190.75, 189.25, 187.25]]))]
0 0










    






/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:45: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.
To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.










    






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29 49
29 50
29 51
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30 0
30 1
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30 49
30 50
30 51
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31 0
31 1
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31 49
31 50
31 51
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31 60
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31 63
32 0
32 1
32 2
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32 10
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32 13
32 14
32 15
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32 49
32 50
32 51
32 52
32 53
32 54
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32 58
32 59
32 60
32 61
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32 63
33 0
33 1
33 2
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33 4
33 5
33 6
33 7
33 8
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33 10
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33 13
33 14
33 15
33 16
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33 49
33 50
33 51
33 52
33 53
33 54
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33 58
33 59
33 60
33 61
33 62
33 63
34 0
34 1
34 2
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34 4
34 5
34 6
34 7
34 8
34 9
34 10
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34 12
34 13
34 14
34 15
34 16
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34 48
34 49
34 50
34 51
34 52
34 53
34 54
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34 57
34 58
34 59
34 60
34 61
34 62
34 63
35 0
35 1
35 2
35 3
35 4
35 5
35 6
35 7
35 8
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35 10
35 11
35 12
35 13
35 14
35 15
35 16
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35 48
35 49
35 50
35 51
35 52
35 53
35 54
35 55
35 56
35 57
35 58
35 59
35 60
35 61
35 62
35 63
36 0
36 1
36 2
36 3
36 4
36 5
36 6
36 7
36 8
36 9
36 10
36 11
36 12
36 13
36 14
36 15
36 16
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36 36
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36 48
36 49
36 50
36 51
36 52
36 53
36 54
36 55
36 56
36 57
36 58
36 59
36 60
36 61
36 62
36 63
37 0
37 1
37 2
37 3
37 4
37 5
37 6
37 7
37 8
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37 10
37 11
37 12
37 13
37 14
37 15
37 16
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37 49
37 50
37 51
37 52
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37 54
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37 59
37 60
37 61
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37 63
38 0
38 1
38 2
38 3
38 4
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38 6
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38 10
38 11
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38 13
38 14
38 15
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38 48
38 49
38 50
38 51
38 52
38 53
38 54
38 55
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38 57
38 58
38 59
38 60
38 61
38 62
38 63
39 0
39 1
39 2
39 3
39 4
39 5
39 6
39 7
39 8
39 9
39 10
39 11
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39 13
39 14
39 15
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39 49
39 50
39 51
39 52
39 53
39 54
39 55
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39 58
39 59
39 60
39 61
39 62
39 63
40 0
40 1
40 2
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40 4
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40 6
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40 15
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40 48
40 49
40 50
40 51
40 52
40 53
40 54
40 55
40 56
40 57
40 58
40 59
40 60
40 61
40 62
40 63
41 0
41 1
41 2
41 3
41 4
41 5
41 6
41 7
41 8
41 9
41 10
41 11
41 12
41 13
41 14
41 15
41 16
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41 49
41 50
41 51
41 52
41 53
41 54
41 55
41 56
41 57
41 58
41 59
41 60
41 61
41 62
41 63
42 0
42 1
42 2
42 3
42 4
42 5
42 6
42 7
42 8
42 9
42 10
42 11
42 12
42 13
42 14
42 15
42 16
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42 49
42 50
42 51
42 52
42 53
42 54
42 55
42 56
42 57
42 58
42 59
42 60
42 61
42 62
42 63
43 0
43 1
43 2
43 3
43 4
43 5
43 6
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43 8
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43 10
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43 13
43 14
43 15
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43 49
43 50
43 51
43 52
43 53
43 54
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43 58
43 59
43 60
43 61
43 62
43 63
44 0
44 1
44 2
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44 4
44 5
44 6
44 7
44 8
44 9
44 10
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44 12
44 13
44 14
44 15
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44 49
44 50
44 51
44 52
44 53
44 54
44 55
44 56
44 57
44 58
44 59
44 60
44 61
44 62
44 63
45 0
45 1
45 2
45 3
45 4
45 5
45 6
45 7
45 8
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45 10
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45 13
45 14
45 15
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45 49
45 50
45 51
45 52
45 53
45 54
45 55
45 56
45 57
45 58
45 59
45 60
45 61
45 62
45 63
46 0
46 1
46 2
46 3
46 4
46 5
46 6
46 7
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46 10
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46 12
46 13
46 14
46 15
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46 49
46 50
46 51
46 52
46 53
46 54
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46 59
46 60
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46 63
47 0
47 1
47 2
47 3
47 4
47 5
47 6
47 7
47 8
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47 10
47 11
47 12
47 13
47 14
47 15
47 16
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47 49
47 50
47 51
47 52
47 53
47 54
47 55
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47 58
47 59
47 60
47 61
47 62
47 63
48 0
48 1
48 2
48 3
48 4
48 5
48 6
48 7
48 8
48 9
48 10
48 11
48 12
48 13
48 14
48 15
48 16
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48 49
48 50
48 51
48 52
48 53
48 54
48 55
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48 57
48 58
48 59
48 60
48 61
48 62
48 63
49 0
49 1
49 2
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49 4
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49 6
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49 49
49 50
49 51
49 52
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49 54
49 55
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49 59
49 60
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49 62
49 63
50 0
50 1
50 2
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50 4
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50 6
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50 15
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50 49
50 50
50 51
50 52
50 53
50 54
50 55
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50 58
50 59
50 60
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50 62
50 63
51 0
51 1
51 2
51 3
51 4
51 5
51 6
51 7
51 8
51 9
51 10
51 11
51 12
51 13
51 14
51 15
51 16
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51 48
51 49
51 50
51 51
51 52
51 53
51 54
51 55
51 56
51 57
51 58
51 59
51 60
51 61
51 62
51 63
52 0
52 1
52 2
52 3
52 4
52 5
52 6
52 7
52 8
52 9
52 10
52 11
52 12
52 13
52 14
52 15
52 16
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52 48
52 49
52 50
52 51
52 52
52 53
52 54
52 55
52 56
52 57
52 58
52 59
52 60
52 61
52 62
52 63
53 0
53 1
53 2
53 3
53 4
53 5
53 6
53 7
53 8
53 9
53 10
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53 12
53 13
53 14
53 15
53 16
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53 49
53 50
53 51
53 52
53 53
53 54
53 55
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53 57
53 58
53 59
53 60
53 61
53 62
53 63
54 0
54 1
54 2
54 3
54 4
54 5
54 6
54 7
54 8
54 9
54 10
54 11
54 12
54 13
54 14
54 15
54 16
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54 48
54 49
54 50
54 51
54 52
54 53
54 54
54 55
54 56
54 57
54 58
54 59
54 60
54 61
54 62
54 63
55 0
55 1
55 2
55 3
55 4
55 5
55 6
55 7
55 8
55 9
55 10
55 11
55 12
55 13
55 14
55 15
55 16
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55 48
55 49
55 50
55 51
55 52
55 53
55 54
55 55
55 56
55 57
55 58
55 59
55 60
55 61
55 62
55 63
56 0
56 1
56 2
56 3
56 4
56 5
56 6
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56 10
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56 13
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56 15
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56 49
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56 60
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57 0
57 1
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58 0
58 1
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59 0
59 1
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60 0
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61 0
61 1
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61 31
61 32
61 33
61 34
61 35
61 36
61 37
61 38
61 39
61 40
61 41
61 42
61 43
61 44
61 45
61 46
61 47
61 48
61 49
61 50
61 51
61 52
61 53
61 54
61 55
61 56
61 57
61 58
61 59
61 60
61 61
61 62
61 63
62 0
62 1
62 2
62 3
62 4
62 5
62 6
62 7
62 8
62 9
62 10
62 11
62 12
62 13
62 14
62 15
62 16
62 17
62 18
62 19
62 20
62 21
62 22
62 23
62 24
62 25
62 26
62 27
62 28
62 29
62 30
62 31
62 32
62 33
62 34
62 35
62 36
62 37
62 38
62 39
62 40
62 41
62 42
62 43
62 44
62 45
62 46
62 47
62 48
62 49
62 50
62 51
62 52
62 53
62 54
62 55
62 56
62 57
62 58
62 59
62 60
62 61
62 62
62 63
63 0
63 1
63 2
63 3
63 4
63 5
63 6
63 7
63 8
63 9
63 10
63 11
63 12
63 13
63 14
63 15
63 16
63 17
63 18
63 19
63 20
63 21
63 22
63 23
63 24
63 25
63 26
63 27
63 28
63 29
63 30
63 31
63 32
63 33
63 34
63 35
63 36
63 37
63 38
63 39
63 40
63 41
63 42
63 43
63 44
63 45
63 46
63 47
63 48
63 49
63 50
63 51
63 52
63 53
63 54
63 55
63 56
63 57
63 58
63 59
63 60
63 61
63 62
63 63
iterations [array([[ 20, 252,   7, ..., 194,   3,   6],
       [176, 191, 203, ..., 183, 172, 131],
       [ 59, 232,  75, ..., 202,  95,  66],
       ...,
       [190, 171,  23, ..., 178, 245, 204],
       [183,  67, 144, ..., 140, 244, 131],
       [177, 133, 119, ..., 147, 121,  35]])]
[(0, 0, 1, 0, array([[171.5 , 174.5 , 178.25, 179.  , 181.25, 185.5 , 184.5 , 186.  ],
       [173.  , 174.25, 177.75, 181.5 , 181.25, 184.25, 183.25, 184.75],
       [175.  , 175.  , 177.  , 181.  , 183.75, 186.  , 186.75, 184.75],
       [173.  , 172.75, 177.  , 181.  , 180.25, 183.5 , 187.25, 187.75],
       [173.75, 174.75, 174.75, 180.75, 183.5 , 187.25, 188.75, 187.5 ],
       [174.75, 175.5 , 177.  , 182.75, 183.25, 187.5 , 186.25, 186.5 ],
       [174.25, 176.25, 177.  , 182.75, 184.5 , 187.5 , 189.  , 193.25],
       [176.25, 178.25, 179.25, 181.5 , 183.5 , 188.5 , 188.25, 191.5 ]])), (0, 0, -1, 90, array([[191.5 , 193.25, 186.5 , 187.5 , 187.75, 184.75, 184.75, 186.  ],
       [188.25, 189.  , 186.25, 188.75, 187.25, 186.75, 183.25, 184.5 ],
       [188.5 , 187.5 , 187.5 , 187.25, 183.5 , 186.  , 184.25, 185.5 ],
       [183.5 , 184.5 , 183.25, 183.5 , 180.25, 183.75, 181.25, 181.25],
       [181.5 , 182.75, 182.75, 180.75, 181.  , 181.  , 181.5 , 179.  ],
       [179.25, 177.  , 177.  , 174.75, 177.  , 177.  , 177.75, 178.25],
       [178.25, 176.25, 175.5 , 174.75, 172.75, 175.  , 174.25, 174.5 ],
       [176.25, 174.25, 174.75, 173.75, 173.  , 175.  , 173.  , 171.5 ]])), (0, 1, 1, 0, array([[185.75, 188.75, 188.  , 191.25, 193.  , 197.5 , 200.25, 199.5 ],
       [189.5 , 187.  , 189.  , 192.5 , 195.75, 199.  , 198.75, 200.75],
       [191.25, 191.  , 191.25, 192.  , 198.5 , 196.75, 201.  , 199.75],
       [188.  , 189.  , 190.  , 192.5 , 195.75, 199.25, 198.5 , 203.25],
       [190.  , 188.25, 191.75, 194.25, 198.25, 199.75, 202.5 , 201.25],
       [189.5 , 193.  , 193.75, 196.5 , 197.25, 201.5 , 201.25, 201.25],
       [188.75, 193.75, 194.25, 192.  , 197.75, 198.5 , 202.  , 200.5 ],
       [190.  , 194.  , 195.5 , 196.  , 195.5 , 199.  , 200.5 , 196.25]])), (0, 1, -1, 90, array([[196.25, 200.5 , 201.25, 201.25, 203.25, 199.75, 200.75, 199.5 ],
       [200.5 , 202.  , 201.25, 202.5 , 198.5 , 201.  , 198.75, 200.25],
       [199.  , 198.5 , 201.5 , 199.75, 199.25, 196.75, 199.  , 197.5 ],
       [195.5 , 197.75, 197.25, 198.25, 195.75, 198.5 , 195.75, 193.  ],
       [196.  , 192.  , 196.5 , 194.25, 192.5 , 192.  , 192.5 , 191.25],
       [195.5 , 194.25, 193.75, 191.75, 190.  , 191.25, 189.  , 188.  ],
       [194.  , 193.75, 193.  , 188.25, 189.  , 191.  , 187.  , 188.75],
       [190.  , 188.75, 189.5 , 190.  , 188.  , 191.25, 189.5 , 185.75]])), (0, 2, 1, 0, array([[202.25, 202.  , 204.25, 204.  , 202.75, 200.75, 202.5 , 203.25],
       [197.5 , 200.25, 203.5 , 203.75, 201.25, 203.75, 206.75, 205.25],
       [201.  , 202.  , 201.25, 204.5 , 205.75, 206.75, 207.75, 206.75],
       [204.5 , 201.75, 204.75, 205.5 , 206.25, 206.5 , 207.  , 207.75],
       [201.75, 202.  , 205.25, 208.75, 206.  , 205.75, 205.  , 207.25],
       [202.25, 202.25, 205.25, 206.25, 207.25, 204.75, 208.5 , 208.5 ],
       [201.75, 202.5 , 206.25, 206.5 , 206.5 , 207.75, 206.25, 209.  ],
       [206.5 , 204.5 , 203.25, 205.  , 206.5 , 207.  , 208.  , 208.  ]])), (0, 2, -1, 90, array([[208.  , 209.  , 208.5 , 207.25, 207.75, 206.75, 205.25, 203.25],
       [208.  , 206.25, 208.5 , 205.  , 207.  , 207.75, 206.75, 202.5 ],
       [207.  , 207.75, 204.75, 205.75, 206.5 , 206.75, 203.75, 200.75],
       [206.5 , 206.5 , 207.25, 206.  , 206.25, 205.75, 201.25, 202.75],
       [205.  , 206.5 , 206.25, 208.75, 205.5 , 204.5 , 203.75, 204.  ],
       [203.25, 206.25, 205.25, 205.25, 204.75, 201.25, 203.5 , 204.25],
       [204.5 , 202.5 , 202.25, 202.  , 201.75, 202.  , 200.25, 202.  ],
       [206.5 , 201.75, 202.25, 201.75, 204.5 , 201.  , 197.5 , 202.25]])), (0, 3, 1, 0, array([[205.75, 206.  , 207.5 , 210.25, 210.75, 209.75, 214.  , 215.  ],
       [206.  , 210.  , 209.5 , 212.5 , 209.5 , 213.  , 216.  , 216.  ],
       [209.25, 210.5 , 211.  , 212.  , 215.25, 211.75, 218.  , 217.  ],
       [207.75, 208.75, 212.25, 211.25, 216.  , 217.25, 220.  , 218.  ],
       [207.  , 210.75, 208.75, 211.75, 214.  , 216.75, 217.5 , 216.5 ],
       [208.5 , 210.  , 209.5 , 211.75, 213.25, 217.75, 219.  , 219.5 ],
       [209.5 , 210.5 , 210.  , 213.5 , 217.  , 216.75, 218.25, 219.5 ],
       [210.25, 211.5 , 214.  , 212.  , 216.  , 218.  , 219.  , 217.25]])), (0, 3, -1, 90, array([[217.25, 219.5 , 219.5 , 216.5 , 218.  , 217.  , 216.  , 215.  ],
       [219.  , 218.25, 219.  , 217.5 , 220.  , 218.  , 216.  , 214.  ],
       [218.  , 216.75, 217.75, 216.75, 217.25, 211.75, 213.  , 209.75],
       [216.  , 217.  , 213.25, 214.  , 216.  , 215.25, 209.5 , 210.75],
       [212.  , 213.5 , 211.75, 211.75, 211.25, 212.  , 212.5 , 210.25],
       [214.  , 210.  , 209.5 , 208.75, 212.25, 211.  , 209.5 , 207.5 ],
       [211.5 , 210.5 , 210.  , 210.75, 208.75, 210.5 , 210.  , 206.  ],
       [210.25, 209.5 , 208.5 , 207.  , 207.75, 209.25, 206.  , 205.75]])), (0, 4, 1, 0, array([[216.5 , 220.  , 221.  , 220.  , 217.5 , 219.  , 221.25, 220.75],
       [219.5 , 220.  , 220.25, 222.  , 220.5 , 222.5 , 222.25, 222.  ],
       [221.25, 222.5 , 224.5 , 221.5 , 222.5 , 224.75, 224.  , 223.75],
       [223.  , 223.25, 223.25, 223.  , 224.  , 223.75, 224.5 , 224.  ],
       [225.5 , 222.75, 219.75, 223.75, 223.25, 225.25, 222.  , 222.5 ],
       [226.5 , 222.5 , 224.75, 223.  , 223.25, 224.75, 225.  , 221.5 ],
       [220.25, 223.25, 223.75, 223.25, 225.25, 226.5 , 223.75, 223.75],
       [218.  , 223.  , 223.25, 223.75, 224.  , 223.75, 224.  , 223.25]])), (0, 4, -1, 90, array([[223.25, 223.75, 221.5 , 222.5 , 224.  , 223.75, 222.  , 220.75],
       [224.  , 223.75, 225.  , 222.  , 224.5 , 224.  , 222.25, 221.25],
       [223.75, 226.5 , 224.75, 225.25, 223.75, 224.75, 222.5 , 219.  ],
       [224.  , 225.25, 223.25, 223.25, 224.  , 222.5 , 220.5 , 217.5 ],
       [223.75, 223.25, 223.  , 223.75, 223.  , 221.5 , 222.  , 220.  ],
       [223.25, 223.75, 224.75, 219.75, 223.25, 224.5 , 220.25, 221.  ],
       [223.  , 223.25, 222.5 , 222.75, 223.25, 222.5 , 220.  , 220.  ],
       [218.  , 220.25, 226.5 , 225.5 , 223.  , 221.25, 219.5 , 216.5 ]])), (0, 5, 1, 0, array([[222.75, 224.5 , 223.5 , 227.75, 225.75, 229.5 , 232.25, 230.  ],
       [224.5 , 223.25, 225.25, 227.25, 230.  , 232.25, 232.75, 233.25],
       [225.25, 225.25, 226.75, 225.5 , 231.25, 232.5 , 238.75, 230.75],
       [222.75, 227.  , 227.  , 229.75, 233.  , 238.5 , 151.75,  39.5 ],
       [223.75, 224.5 , 228.5 , 229.75, 225.5 , 120.  ,   9.75,  13.75],
       [222.5 , 223.5 , 227.5 , 230.25, 193.  ,  26.25,  11.5 ,  13.  ],
       [224.5 , 224.  , 228.  , 233.25, 221.5 ,  24.25,   8.5 ,   5.25],
       [224.5 , 224.5 , 228.75, 235.5 , 146.75,   8.25,   4.  ,   4.5 ]])), (0, 5, -1, 90, array([[  4.5 ,   5.25,  13.  ,  13.75,  39.5 , 230.75, 233.25, 230.  ],
       [  4.  ,   8.5 ,  11.5 ,   9.75, 151.75, 238.75, 232.75, 232.25],
       [  8.25,  24.25,  26.25, 120.  , 238.5 , 232.5 , 232.25, 229.5 ],
       [146.75, 221.5 , 193.  , 225.5 , 233.  , 231.25, 230.  , 225.75],
       [235.5 , 233.25, 230.25, 229.75, 229.75, 225.5 , 227.25, 227.75],
       [228.75, 228.  , 227.5 , 228.5 , 227.  , 226.75, 225.25, 223.5 ],
       [224.5 , 224.  , 223.5 , 224.5 , 227.  , 225.25, 223.25, 224.5 ],
       [224.5 , 224.5 , 222.5 , 223.75, 222.75, 225.25, 224.5 , 222.75]])), (0, 6, 1, 0, array([[231.5 , 233.75, 233.75, 235.5 , 231.75, 234.  , 231.75, 230.75],
       [236.25, 237.75, 241.  , 242.25, 239.25, 241.75, 240.25, 234.75],
       [159.  ,  85.5 ,  14.5 ,  16.75,  46.5 , 102.5 , 125.25, 183.75],
       [ 22.5 ,  12.25,   4.  ,   4.25,   5.25,   5.5 ,   6.75,   5.5 ],
       [ 11.25,   6.25,   5.25,   4.5 ,   4.  ,   4.75,   4.5 ,   4.25],
       [ 13.  ,   8.75,   7.75,   8.75,   5.  ,   9.  ,   5.5 ,   5.  ],
       [  3.75,   4.75,   4.75,   7.25,   6.  ,   6.25,   8.5 ,   6.  ],
       [  3.25,   3.25,   3.75,  11.5 ,   7.75,   3.  ,   6.25,  10.5 ]])), (0, 6, -1, 90, array([[ 10.5 ,   6.  ,   5.  ,   4.25,   5.5 , 183.75, 234.75, 230.75],
       [  6.25,   8.5 ,   5.5 ,   4.5 ,   6.75, 125.25, 240.25, 231.75],
       [  3.  ,   6.25,   9.  ,   4.75,   5.5 , 102.5 , 241.75, 234.  ],
       [  7.75,   6.  ,   5.  ,   4.  ,   5.25,  46.5 , 239.25, 231.75],
       [ 11.5 ,   7.25,   8.75,   4.5 ,   4.25,  16.75, 242.25, 235.5 ],
       [  3.75,   4.75,   7.75,   5.25,   4.  ,  14.5 , 241.  , 233.75],
       [  3.25,   4.75,   8.75,   6.25,  12.25,  85.5 , 237.75, 233.75],
       [  3.25,   3.75,  13.  ,  11.25,  22.5 , 159.  , 236.25, 231.5 ]])), (0, 7, 1, 0, array([[228.  , 229.75, 229.  , 229.75, 230.75, 229.25, 230.  , 226.  ],
       [235.  , 232.5 , 233.25, 232.75, 231.5 , 230.5 , 230.75, 228.25],
       [236.5 , 240.75, 236.25, 236.25, 236.  , 232.75, 232.  , 231.25],
       [ 12.  , 149.25, 243.75, 238.5 , 237.  , 235.  , 234.25, 232.  ],
       [  5.  ,   5.25,  87.5 , 215.75, 236.5 , 237.25, 233.  , 233.  ],
       [  6.  ,   4.  ,   4.5 ,  62.75, 206.25, 206.25, 207.75, 234.5 ],
       [  9.5 ,   9.75,   5.25,   4.  ,  75.75, 199.75, 115.  , 229.25],
       [  6.5 ,   8.  ,   8.75,   3.5 ,   3.5 ,  52.  ,  68.25, 166.25]])), (0, 7, -1, 90, array([[166.25, 229.25, 234.5 , 233.  , 232.  , 231.25, 228.25, 226.  ],
       [ 68.25, 115.  , 207.75, 233.  , 234.25, 232.  , 230.75, 230.  ],
       [ 52.  , 199.75, 206.25, 237.25, 235.  , 232.75, 230.5 , 229.25],
       [  3.5 ,  75.75, 206.25, 236.5 , 237.  , 236.  , 231.5 , 230.75],
       [  3.5 ,   4.  ,  62.75, 215.75, 238.5 , 236.25, 232.75, 229.75],
       [  8.75,   5.25,   4.5 ,  87.5 , 243.75, 236.25, 233.25, 229.  ],
       [  8.  ,   9.75,   4.  ,   5.25, 149.25, 240.75, 232.5 , 229.75],
       [  6.5 ,   9.5 ,   6.  ,   5.  ,  12.  , 236.5 , 235.  , 228.  ]])), (0, 8, 1, 0, array([[225.5 , 225.75, 222.  , 223.75, 223.25, 224.5 , 225.5 , 224.25],
       [225.25, 226.75, 224.25, 225.75, 223.75, 226.25, 223.25, 225.  ],
       [227.  , 225.25, 227.  , 226.75, 224.5 , 225.25, 223.5 , 224.25],
       [230.75, 228.25, 225.25, 223.25, 224.25, 225.  , 223.  , 222.75],
       [229.5 , 227.75, 225.5 , 228.  , 226.  , 224.5 , 223.75, 224.25],
       [230.  , 229.75, 227.75, 226.  , 220.25, 226.  , 225.  , 222.75],
       [232.5 , 230.  , 230.25, 229.  , 227.25, 226.25, 224.  , 226.25],
       [232.  , 230.  , 229.  , 230.5 , 226.25, 224.75, 226.75, 225.  ]])), (0, 8, -1, 90, array([[225.  , 226.25, 222.75, 224.25, 222.75, 224.25, 225.  , 224.25],
       [226.75, 224.  , 225.  , 223.75, 223.  , 223.5 , 223.25, 225.5 ],
       [224.75, 226.25, 226.  , 224.5 , 225.  , 225.25, 226.25, 224.5 ],
       [226.25, 227.25, 220.25, 226.  , 224.25, 224.5 , 223.75, 223.25],
       [230.5 , 229.  , 226.  , 228.  , 223.25, 226.75, 225.75, 223.75],
       [229.  , 230.25, 227.75, 225.5 , 225.25, 227.  , 224.25, 222.  ],
       [230.  , 230.  , 229.75, 227.75, 228.25, 225.25, 226.75, 225.75],
       [232.  , 232.5 , 230.  , 229.5 , 230.75, 227.  , 225.25, 225.5 ]])), (0, 9, 1, 0, array([[225.75, 223.25, 225.  , 221.75, 222.25, 222.  , 223.  , 221.75],
       [225.  , 224.25, 223.5 , 222.25, 223.  , 220.25, 219.5 , 218.25],
       [224.  , 222.25, 220.75, 220.25, 222.  , 218.25, 221.5 , 219.75],
       [220.  , 219.  , 217.25, 223.  , 221.75, 222.75, 222.25, 219.5 ],
       [222.  , 219.  , 225.  , 222.25, 222.75, 223.5 , 222.5 , 221.25],
       [224.25, 224.75, 221.75, 224.  , 222.25, 222.5 , 221.75, 224.  ],
       [222.75, 224.5 , 226.75, 225.25, 224.25, 223.  , 224.25, 225.25],
       [226.  , 227.  , 225.5 , 224.5 , 226.25, 223.  , 226.25, 221.75]])), (0, 9, -1, 90, array([[221.75, 225.25, 224.  , 221.25, 219.5 , 219.75, 218.25, 221.75],
       [226.25, 224.25, 221.75, 222.5 , 222.25, 221.5 , 219.5 , 223.  ],
       [223.  , 223.  , 222.5 , 223.5 , 222.75, 218.25, 220.25, 222.  ],
       [226.25, 224.25, 222.25, 222.75, 221.75, 222.  , 223.  , 222.25],
       [224.5 , 225.25, 224.  , 222.25, 223.  , 220.25, 222.25, 221.75],
       [225.5 , 226.75, 221.75, 225.  , 217.25, 220.75, 223.5 , 225.  ],
       [227.  , 224.5 , 224.75, 219.  , 219.  , 222.25, 224.25, 223.25],
       [226.  , 222.75, 224.25, 222.  , 220.  , 224.  , 225.  , 225.75]])), (0, 10, 1, 0, array([[220.75, 220.5 , 221.5 , 220.75, 220.25, 219.25, 216.75, 220.5 ],
       [221.  , 223.  , 221.25, 220.75, 220.25, 217.75, 219.  , 219.25],
       [218.5 , 219.25, 219.  , 220.75, 218.5 , 218.5 , 217.75, 217.  ],
       [219.  , 217.  , 220.75, 219.25, 220.5 , 217.  , 218.5 , 220.25],
       [220.25, 221.  , 219.75, 220.5 , 220.25, 217.5 , 218.25, 219.  ],
       [221.  , 221.75, 219.  , 220.25, 220.  , 220.  , 219.75, 218.75],
       [220.25, 223.25, 223.  , 222.25, 221.  , 221.75, 220.  , 219.  ],
       [223.75, 223.  , 222.5 , 222.25, 221.25, 220.5 , 222.  , 219.75]])), (0, 10, -1, 90, array([[219.75, 219.  , 218.75, 219.  , 220.25, 217.  , 219.25, 220.5 ],
       [222.  , 220.  , 219.75, 218.25, 218.5 , 217.75, 219.  , 216.75],
       [220.5 , 221.75, 220.  , 217.5 , 217.  , 218.5 , 217.75, 219.25],
       [221.25, 221.  , 220.  , 220.25, 220.5 , 218.5 , 220.25, 220.25],
       [222.25, 222.25, 220.25, 220.5 , 219.25, 220.75, 220.75, 220.75],
       [222.5 , 223.  , 219.  , 219.75, 220.75, 219.  , 221.25, 221.5 ],
       [223.  , 223.25, 221.75, 221.  , 217.  , 219.25, 223.  , 220.5 ],
       [223.75, 220.25, 221.  , 220.25, 219.  , 218.5 , 221.  , 220.75]])), (0, 11, 1, 0, array([[218.25, 217.25, 216.5 , 220.  , 218.25, 215.25, 214.  , 215.5 ],
       [216.75, 214.75, 216.25, 218.5 , 215.75, 217.  , 214.75, 215.  ],
       [217.  , 215.25, 213.75, 216.  , 213.5 , 215.25, 213.75, 209.75],
       [221.  , 217.  , 216.  , 215.25, 215.  , 214.5 , 211.75, 212.5 ],
       [217.75, 215.25, 217.  , 216.  , 213.5 , 215.5 , 216.5 , 212.75],
       [218.25, 217.25, 217.  , 214.75, 214.75, 212.75, 215.75, 213.  ],
       [221.25, 218.25, 217.25, 218.25, 218.  , 215.25, 217.25, 217.75],
       [220.75, 221.25, 217.  , 219.5 , 217.25, 213.  , 215.5 , 216.  ]])), (0, 11, -1, 90, array([[216.  , 217.75, 213.  , 212.75, 212.5 , 209.75, 215.  , 215.5 ],
       [215.5 , 217.25, 215.75, 216.5 , 211.75, 213.75, 214.75, 214.  ],
       [213.  , 215.25, 212.75, 215.5 , 214.5 , 215.25, 217.  , 215.25],
       [217.25, 218.  , 214.75, 213.5 , 215.  , 213.5 , 215.75, 218.25],
       [219.5 , 218.25, 214.75, 216.  , 215.25, 216.  , 218.5 , 220.  ],
       [217.  , 217.25, 217.  , 217.  , 216.  , 213.75, 216.25, 216.5 ],
       [221.25, 218.25, 217.25, 215.25, 217.  , 215.25, 214.75, 217.25],
       [220.75, 221.25, 218.25, 217.75, 221.  , 217.  , 216.75, 218.25]])), (0, 12, 1, 0, array([[214.25, 215.  , 210.75, 208.75, 209.  , 209.25, 209.  , 209.  ],
       [213.5 , 215.5 , 212.25, 213.  , 213.5 , 209.5 , 209.75, 206.5 ],
       [212.75, 213.75, 211.  , 212.25, 212.25, 210.75, 210.  , 208.75],
       [209.75, 213.75, 211.5 , 212.25, 210.75, 209.75, 210.75, 208.5 ],
       [215.25, 213.75, 215.25, 210.25, 210.5 , 209.75, 209.5 , 209.  ],
       [212.75, 211.  , 212.5 , 211.25, 210.75, 210.  , 210.25, 209.  ],
       [223.75, 210.5 , 211.5 , 210.5 , 212.5 , 209.5 , 209.  , 207.  ],
       [217.  , 212.  , 213.5 , 211.  , 209.75, 209.5 , 207.5 , 208.25]])), (0, 12, -1, 90, array([[208.25, 207.  , 209.  , 209.  , 208.5 , 208.75, 206.5 , 209.  ],
       [207.5 , 209.  , 210.25, 209.5 , 210.75, 210.  , 209.75, 209.  ],
       [209.5 , 209.5 , 210.  , 209.75, 209.75, 210.75, 209.5 , 209.25],
       [209.75, 212.5 , 210.75, 210.5 , 210.75, 212.25, 213.5 , 209.  ],
       [211.  , 210.5 , 211.25, 210.25, 212.25, 212.25, 213.  , 208.75],
       [213.5 , 211.5 , 212.5 , 215.25, 211.5 , 211.  , 212.25, 210.75],
       [212.  , 210.5 , 211.  , 213.75, 213.75, 213.75, 215.5 , 215.  ],
       [217.  , 223.75, 212.75, 215.25, 209.75, 212.75, 213.5 , 214.25]])), (0, 13, 1, 0, array([[207.5 , 204.75, 204.  , 204.5 , 201.75, 200.75, 202.  , 200.75],
       [206.75, 208.  , 205.25, 204.75, 201.25, 202.5 , 199.5 , 199.75],
       [207.  , 207.5 , 209.25, 201.5 , 203.5 , 201.25, 201.  , 199.  ],
       [208.  , 205.75, 205.5 , 204.25, 202.25, 201.75, 199.75, 201.5 ],
       [206.5 , 205.  , 204.5 , 205.25, 203.25, 200.5 , 200.75, 200.5 ],
       [205.5 , 207.  , 204.25, 202.  , 205.25, 202.5 , 200.25, 200.75],
       [208.25, 206.75, 204.25, 202.  , 204.  , 201.5 , 199.75, 200.5 ],
       [206.25, 206.5 , 202.25, 206.5 , 201.25, 202.25, 202.5 , 201.25]])), (0, 13, -1, 90, array([[201.25, 200.5 , 200.75, 200.5 , 201.5 , 199.  , 199.75, 200.75],
       [202.5 , 199.75, 200.25, 200.75, 199.75, 201.  , 199.5 , 202.  ],
       [202.25, 201.5 , 202.5 , 200.5 , 201.75, 201.25, 202.5 , 200.75],
       [201.25, 204.  , 205.25, 203.25, 202.25, 203.5 , 201.25, 201.75],
       [206.5 , 202.  , 202.  , 205.25, 204.25, 201.5 , 204.75, 204.5 ],
       [202.25, 204.25, 204.25, 204.5 , 205.5 , 209.25, 205.25, 204.  ],
       [206.5 , 206.75, 207.  , 205.  , 205.75, 207.5 , 208.  , 204.75],
       [206.25, 208.25, 205.5 , 206.5 , 208.  , 207.  , 206.75, 207.5 ]])), (0, 14, 1, 0, array([[198.25, 198.25, 198.5 , 198.5 , 198.  , 195.  , 195.75, 192.  ],
       [201.75, 197.5 , 198.  , 197.25, 197.  , 196.25, 195.5 , 198.25],
       [198.5 , 196.5 , 196.  , 197.25, 195.75, 198.25, 195.75, 194.25],
       [196.5 , 196.25, 193.25, 195.  , 196.75, 194.  , 190.25, 192.  ],
       [198.25, 197.5 , 192.5 , 194.25, 195.5 , 194.5 , 193.5 , 194.  ],
       [198.75, 195.  , 195.25, 195.5 , 196.  , 195.5 , 194.75, 194.25],
       [199.  , 199.25, 196.25, 196.  , 194.5 , 194.25, 193.  , 193.25],
       [199.25, 200.75, 196.25, 195.  , 194.5 , 193.75, 192.25, 191.5 ]])), (0, 14, -1, 90, array([[191.5 , 193.25, 194.25, 194.  , 192.  , 194.25, 198.25, 192.  ],
       [192.25, 193.  , 194.75, 193.5 , 190.25, 195.75, 195.5 , 195.75],
       [193.75, 194.25, 195.5 , 194.5 , 194.  , 198.25, 196.25, 195.  ],
       [194.5 , 194.5 , 196.  , 195.5 , 196.75, 195.75, 197.  , 198.  ],
       [195.  , 196.  , 195.5 , 194.25, 195.  , 197.25, 197.25, 198.5 ],
       [196.25, 196.25, 195.25, 192.5 , 193.25, 196.  , 198.  , 198.5 ],
       [200.75, 199.25, 195.  , 197.5 , 196.25, 196.5 , 197.5 , 198.25],
       [199.25, 199.  , 198.75, 198.25, 196.5 , 198.5 , 201.75, 198.25]])), (0, 15, 1, 0, array([[191.25, 190.75, 188.75, 189.  , 187.25, 190.  , 191.  , 189.  ],
       [192.5 , 191.75, 192.5 , 192.5 , 189.25, 191.5 , 192.  , 188.75],
       [196.  , 193.75, 191.75, 193.  , 190.75, 187.5 , 188.25, 188.5 ],
       [194.25, 195.  , 190.75, 189.  , 188.5 , 187.25, 185.75, 188.75],
       [193.5 , 189.25, 193.  , 191.25, 189.5 , 190.75, 185.75, 188.5 ],
       [191.75, 191.5 , 191.75, 191.75, 191.  , 186.75, 188.25, 186.  ],
       [189.75, 191.75, 192.5 , 189.75, 188.  , 193.25, 186.  , 190.25],
       [193.  , 190.5 , 190.5 , 190.5 , 191.25, 189.25, 189.25, 189.  ]])), (0, 15, -1, 90, array([[189.  , 190.25, 186.  , 188.5 , 188.75, 188.5 , 188.75, 189.  ],
       [189.25, 186.  , 188.25, 185.75, 185.75, 188.25, 192.  , 191.  ],
       [189.25, 193.25, 186.75, 190.75, 187.25, 187.5 , 191.5 , 190.  ],
       [191.25, 188.  , 191.  , 189.5 , 188.5 , 190.75, 189.25, 187.25],
       [190.5 , 189.75, 191.75, 191.25, 189.  , 193.  , 192.5 , 189.  ],
       [190.5 , 192.5 , 191.75, 193.  , 190.75, 191.75, 192.5 , 188.75],
       [190.5 , 191.75, 191.5 , 189.25, 195.  , 193.75, 191.75, 190.75],
       [193.  , 189.75, 191.75, 193.5 , 194.25, 196.  , 192.5 , 191.25]]))]
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iterations [array([[172,  24, 135, ..., 112,  61,  62],
       [105,   4, 255, ..., 133, 209,  51],
       [141, 144,  42, ...,  44, 169,  39],
       ...,
       [ 55, 112, 182, ..., 149, 251,  16],
       [149, 111,  92, ..., 116, 127, 118],
       [112, 113,  99, ..., 239,  89, 115]])]










    








<Figure size 432x288 with 0 Axes>

IFS Fractal - Fern and Tree

O algoritmo foi implementado utilizando-se as tabelas de coeficientes presentes no artigo de BARNSLEY e SLOAN (1988) referenciado nos slides apresentados em sala de aula e também de códigos de exemplo observados nas pesquisas realizadas.





In [3]:



    


from PIL import Image
import random
import matplotlib.pyplot as plt


fern = [[0.0,0.0,0.0,0.16,0.0,0.0,0.01],
[0.85,0.04,-0.04,0.85,0.0,1.6,0.85],
[0.2,-0.26,0.23,0.22,0.0,1.6,0.07],
[-0.15,0.28,0.26,0.24,0.0,0.44,0.07]]

tree = [[0, 0, 0, 0.5, 0, 0, 0.05],
[0.1, 0, 0, 0.1, 0, 0.2, 0.15],
[0.42, -0.42, 0.42, 0.42, 0, 0.2, 0.4],
[0.42, 0.42, -0.42, 0.42, 0, 0.2, 0.4]]

x=0.0
y=0.0
A = {}
dictypes = {'tree':tree,'fern':fern}
names = {'tree': 'Tree','fern':'Fern'}
plt.figure(figsize=(7,7))

for key,t in dictypes.items():
    A[key] = []
    
    for k in range(0,10000):
        p = random.random()
        if p <= t[0][6]:
            i=0
        elif p <= t[0][6] + t[1][6]:
            i=1
        elif p <= t[0][6] + t[1][6] + t[2][6]:
            i=2
        else:
            i=3

        x0 = x * t[i][0] + y * t[i][1] + t[i][4]
        y  = x * t[i][2] + y * t[i][3] + t[i][5]
        x = x0

        pt=[x,y]

        A[key].append(pt)
    plt.title('IFS a codes for fractal {:}'.format(names[key]))
    plt.scatter( *zip(*A[key]),s=1)
    plt.axis('off')
    plt.show()

IFS Fractal - Sierpinski Triangle

A seguir, podemos visualizar a geração do triângulo de Sierpinski a partir dos conceitos de matriz e coeficientes, conforme citado por BARNSLEY e SLOAN (1988). O algoritmo utilizado para demonstrar o triângulo de Sierpinski está disponível no github no seguinte link.





In [2]:



    


# https://github.com/TaavishThaman/Barnsley-fern-generation-in-python-

import matplotlib.pyplot as plt
from random import randint

#Setting coordinates of the triangle
X1=-10
Y1=0

X2=10
Y2=0

X3=0
Y3=10

X0=0
Y0=0

x_coo = []
y_coo = []

for i in range(0,10000):
    r = randint(1,3)
    
    if r==1:
        X0 = (X0+X1)/2
        Y0 = (Y0+Y1)/2
        
    if r==2:
        X0 = (X0+X2)/2
        Y0 = (Y0+Y2)/2
        
    if r==3:
        X0 = (X0+X3)/2
        Y0 = (Y0+Y3)/2
        
    x_coo.append(X0)
    y_coo.append(Y0)
    
plt.title('Sierpinski Triangle')
plt.scatter(x_coo, y_coo, s=0.3)
plt.axis("off")
plt.savefig('Triangle.png', dpi=200)

Content source: robertoalotufo/ia898

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