边缘检测系列1：传统边缘检测算子

算法原理

• 传统的边缘检测大多数是通过基于方向导数掩码（梯度方向导数）求卷积的方法。

• 计算灰度变化的卷积算子包含Roberts算子、Prewitt算子、Sobel算子、Scharr算子、Kirsch算子、Robinson算子、Laplacian算子。

• 大多数边缘检测算子是基于方向差分卷积核求卷积的方法，在使用由两个或者多个卷积核组成的边缘检测算子时假设有 n 个卷积核，记 $Con{v}_1,Con{v}_2,\mathrm{.}\mathrm{.}\mathrm{.},Con{v}_n$Conv1,Conv2,...,Convn，为图像分别与个卷积核做卷积的结果，通常有四种方式来衡量最后输出的边缘强度。

  1. 取对应位置绝对值的和：${\sum }_{i=1}^n\mathrm{\mid }{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_i\mathrm{\mid }$∑i=1n∣convi∣

  2. 取对应位置平方和的开方：$\sqrt{{\sum }_{i=1}^n{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_i^2}$∑i=1nconvi2

  3. 取对应位置绝对值的最大值：$\max \{\mathrm{\mid }{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_1\mathrm{\mid },\mathrm{\mid }{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_2\mathrm{\mid },\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_i\mathrm{\mid }\}$max{∣conv1∣,∣conv2∣,...,∣convi∣}

  4. 插值法：${\sum }_{i=1}^n{a}_i\mathrm{\mid }{\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{v}}_i\mathrm{\mid }$∑i=1nai∣convi∣，其中 $a_i>=0$ai>=0，且 ${\sum }_{i=1}^n{a}_i=1$∑i=1nai=1

代码实现

构建通用的边缘检测算子

• 因为上述的这些算子在本质上都是通过卷积计算实现的，只是所使用到的卷积核参数有所不同
• 所以可以构建一个通用的计算算子，只需要传入对应的卷积核参数即可实现不同的边缘检测
• 并且在后处理时集成了上述的四种计算最终边缘强度的方式

import numpy as npimport paddleimport paddle.nn as nnclass EdgeOP(nn.Layer):
    def __init__(self, kernel):
        '''
        kernel: shape(out_channels, in_channels, h, w)
        '''
        super(EdgeOP, self).__init__()
        out_channels, in_channels, h, w = kernel.shape
        self.filter = nn.Conv2D(in_channels=in_channels, out_channels=out_channels, kernel_size=(h, w), padding='SAME', bias_attr=False)
        self.filter.weight.set_value(kernel.astype('float32'))    
    @staticmethod
    def postprocess(outputs, mode=0, weight=None):
        '''
        Input: NCHW
        Output: NHW(mode==1-3) or NCHW(mode==4)

        Params:
            mode: switch output mode(0-4)
            weight: weight when mode==3
        '''
        if mode==0:
            results = paddle.sum(paddle.abs(outputs), axis=1)        elif mode==1:
            results = paddle.sqrt(paddle.sum(paddle.pow(outputs, 2), axis=1))        elif mode==2:
            results = paddle.max(paddle.abs(outputs), axis=1)        elif mode==3:            if weight is None:
                C = outputs.shape[1]
                weight = paddle.to_tensor([1/C] * C, dtype='float32')            else:
                weight = paddle.to_tensor(weight, dtype='float32')
            results = paddle.einsum('nchw, c -> nhw', paddle.abs(outputs), weight)        elif mode==4:
            results = paddle.abs(outputs)        return paddle.clip(results, 0, 255).cast('uint8')    
    @paddle.no_grad()
    def forward(self, images, mode=0, weight=None):
        outputs = self.filter(images)        return self.postprocess(outputs, mode, weight)

图像边缘检测测试函数

• 为了方便测试就构建了如下的测试函数，测试同一张图片不同算子/不同边缘强度计算方法的边缘检测效果

import osimport cv2from PIL import Imagedef test_edge_det(kernel, img_path='test.jpg'):
    img = cv2.imread(img_path, 0)
    img_tensor = paddle.to_tensor(img, dtype='float32')[None, None, ...]
    op = EdgeOP(kernel)
    all_results = []    for mode in range(4):
        results = op(img_tensor, mode=mode)
        all_results.append(results.numpy()[0])
    
    results = op(img_tensor, mode=4)    for result in results.numpy()[0]:
        all_results.append(result)    return all_results, np.concatenate(all_results, 1)

Roberts 算子

roberts_kernel = np.array([
    [[
        [1,  0],
        [0, -1]
    ]],
    [[
        [0, -1],
        [1,  0]
    ]]
])

_, concat_res = test_edge_det(roberts_kernel)
Image.fromarray(concat_res)

Prewitt 算子

prewitt_kernel = np.array([
    [[
        [-1, -1, -1],
        [ 0,  0,  0],
        [ 1,  1,  1]
    ]],
    [[
        [-1,  0,  1],
        [-1,  0,  1],
        [-1,  0,  1]
    ]],
    [[
        [ 0,  1,  1],
        [-1,  0,  1],
        [-1, -1,  0]
    ]],
    [[
        [ -1, -1,  0],
        [ -1,  0,  1],
        [  0,  1,  1]
    ]]
])

_, concat_res = test_edge_det(prewitt_kernel)
Image.fromarray(concat_res)

Sobel 算子

sobel_kernel = np.array([
    [[
        [-1, -2, -1],
        [ 0,  0,  0],
        [ 1,  2,  1]
    ]],
    [[
        [-1,  0,  1],
        [-2,  0,  2],
        [-1,  0,  1]
    ]],
    [[
        [ 0,  1,  2],
        [-1,  0,  1],
        [-2, -1,  0]
    ]],
    [[
        [ -2, -1,  0],
        [ -1,  0,  1],
        [  0,  1,  2]
    ]]
])

_, concat_res = test_edge_det(sobel_kernel)
Image.fromarray(concat_res)

Scharr 算子

scharr_kernel = np.array([
    [[
        [-3, -10, -3],
        [ 0,   0,  0],
        [ 3,  10,  3]
    ]],
    [[
        [-3,  0,   3],
        [-10, 0,  10],
        [-3,  0,   3]
    ]],
    [[
        [ 0,  3,  10],
        [-3,  0,  3],
        [-10, -3,  0]
    ]],
    [[
        [ -10, -3, 0],
        [ -3,  0, 3],
        [ 0,  3,  10]
    ]]
])

_, concat_res = test_edge_det(scharr_kernel)
Image.fromarray(concat_res)

Krisch 算子

Krisch_kernel = np.array([
    [[
        [5, 5, 5],
        [-3,0,-3],
        [-3,-3,-3]
    ]],
    [[
        [-3, 5,5],
        [-3,0,5],
        [-3,-3,-3]
    ]],
    [[
        [-3,-3,5],
        [-3,0,5],
        [-3,-3,5]
    ]],
    [[
        [-3,-3,-3],
        [-3,0,5],
        [-3,5,5]
    ]],
    [[
        [-3, -3, -3],
        [-3,0,-3],
        [5,5,5]
    ]],
    [[
        [-3, -3, -3],
        [5,0,-3],
        [5,5,-3]
    ]],
    [[
        [5, -3, -3],
        [5,0,-3],
        [5,-3,-3]
    ]],
    [[
        [5, 5, -3],
        [5,0,-3],
        [-3,-3,-3]
    ]],
])

_, concat_res = test_edge_det(Krisch_kernel)
Image.fromarray(concat_res)

Robinson算子

robinson_kernel = np.array([
    [[
        [1, 2, 1],
        [0, 0, 0],
        [-1, -2, -1]
    ]],
    [[
        [0, 1, 2],
        [-1, 0, 1],
        [-2, -1, 0]
    ]],
    [[
        [-1, 0, 1],
        [-2, 0, 2],
        [-1, 0, 1]
    ]],
    [[
        [-2, -1, 0],
        [-1, 0, 1],
        [0, 1, 2]
    ]],
    [[
        [-1, -2, -1],
        [0, 0, 0],
        [1, 2, 1]
    ]],
    [[
        [0, -1, -2],
        [1, 0, -1],
        [2, 1, 0]
    ]],
    [[
        [1, 0, -1],
        [2, 0, -2],
        [1, 0, -1]
    ]],
    [[
        [2, 1, 0],
        [1, 0, -1],
        [0, -1, -2]
    ]],
])

_, concat_res = test_edge_det(robinson_kernel)
Image.fromarray(concat_res)

Laplacian 算子

laplacian_kernel = np.array([
    [[
        [1, 1, 1],
        [1, -8, 1],
        [1, 1, 1]
    ]],
    [[
        [0, 1, 0],
        [1, -4, 1],
        [0, 1, 0]
    ]]
])

_, concat_res = test_edge_det(laplacian_kernel)
Image.fromarray(concat_res)

总结

• 简单介绍并实现了几种常用的传统边缘检测算子