摘要：不同于传统的卷积，八度卷积主要针对图像的高频信号与低频信号。

本文分享自华为云社区《OctConv：八度卷积复现》，作者：李长安 。

论文解读

八度卷积于2019年在论文《Drop an Octave: Reducing Spatial Redundancy in Convolutional Neural Networks with Octave Convol》提出，在当时引起了不小的反响。八度卷积对传统的convolution进行改进，以降低空间冗余。其中“Drop an Octave”指降低八个音阶，代表频率减半。

不同于传统的卷积，八度卷积主要针对图像的高频信号与低频信号。首先，我们回忆一下数字图像处理中的高频信号与低频信号的概念。图像中的低频信号和高频信号也叫做低频分量和高频分量。图像中的高频分量，指的是图像强度（亮度/灰度）变化剧烈的像素点，例如边缘（轮廓）、图像的细节处、噪声（即噪点）（该点之所以称为噪点,是因为它与周边像素点灰度值有明显差别,也就是图像强度有剧烈的变化,所以噪声是高频部分）。图像中的低频分量，指的是图像强度（亮度/灰度）变换平缓的像素点，例如大片色块的地方。例如当我们在读书的时候，我们会聚焦于书上的文字而不是书纸本身，这里的文字就是高频分量，白纸即为低频分量。

(资料图)

下图是论文中给出的例子，左图是原图，中图表示低频信号，右图表示高频信号。

在论文中，作者提出较高的频率通常用精细的细节编码，较低的频率通常用全局结构编码。所以作者认为那么既然图像分为高低频，那么卷积产生的特征图自然也存在高低频之分。在图像处理中，模型通过高频特征图去学习图像包含的信息，因为它包含了轮廓、边缘等的信息，有助于进行显著性检测。相反，低频特征图包含的信息较少。如果我们用相同的处理方法来处理高频特征图和低频特征图，显然，前者的效益是远大于后者的。这就是特征图的冗余信息：包含信息较少的低频部分。所以在论文中作者提出了一种分而治之的方法，称之为Octave Feature Representation，对高频特征图与低频特征图分离开来进行处理。如下图所示，作者将低频特征图的分辨率降为1/2，这不仅有助于减少冗余数据，还有利于得到全局信息。

根据尺度空间理念，我们可以知道特征具有尺度不变性和旋转不变性。

• 尺度不变性：人类在识别一个物体时，不管这个物体或远或近，都能对它进行正确的辨认，这就是所谓的尺度不变性。
• 旋转不变性：当这个物体发生旋转时，我们照样可以正确地辨认它，这就是所谓的旋转不变性。当用一个机器视觉系统分析未知场景时，计算机没有办法预先知识图像中物体尺度，因此，我们需要同时考虑图像在多尺度下的描述，获知感兴趣物体的最佳尺度。例如，高分辨率的图是人近距离的观察得到的，低分辨率的图是远距离观察得到的。

2、复现详情

2.1 Oct-Conv复现

为了同时做到同一频率内的更新和不同频率之间的交流，卷积核分成四部分：

• 高频到高频的卷积核
• 高频到低频的卷积核
• 低频到高频的卷积核
• 低频到低频的卷积核

下图直观地展示了八度卷积的卷积核，可以看出四个部分共同组成了大小为 k*k 的卷积核。其中，in和out分别表示输入和输出特征图的相关属性，在这篇文章中，输入的低频占比、通道数量都和输出的一致。

在了解了卷积核之后，下面介绍输入如何进行八度卷积操作得到输出结果。如下图所示，低频和高频的输入经过八度卷积操作得到了低频和高频的输出。红色表示高频，蓝色表示低频。绿色的箭头表示同一频率内的更新，红色的箭头表示不同频率之间的交流。

H和W分别表示特征图的长宽，可以看出低频特征图的长宽都是高频特征图的一半。因为分辨率不同，所以不同频率之间交流之前需要进行分辨率的调整：高频到低频需要进行池化（降采样）操作；低频到高频需要进行上采样操作。

import paddleimport paddle.nn as nnimport mathclass OctaveConv(nn.Layer): def __init__(self, in_channels, out_channels, kernel_size, alpha_in=0.5, alpha_out=0.5, stride=1, padding=0, dilation=1,                 groups=1, bias=False): super(OctaveConv, self).__init__() self.downsample = nn.AvgPool2D(kernel_size=(2, 2), stride=2) self.upsample = nn.Upsample(scale_factor=2, mode="nearest") assert stride == 1 or stride == 2, "Stride should be 1 or 2." self.stride = stride self.is_dw = groups == in_channels assert 0 <= alpha_in <= 1 and 0 <= alpha_out <= 1, "Alphas should be in the interval from 0 to 1." self.alpha_in, self.alpha_out = alpha_in, alpha_out self.conv_l2l = None if alpha_in == 0 or alpha_out == 0 else \ nn.Conv2D(int(alpha_in * in_channels), int(alpha_out * out_channels), kernel_size, 1, padding, dilation, math.ceil(alpha_in * groups)) self.conv_l2h = None if alpha_in == 0 or alpha_out == 1 or self.is_dw else \ nn.Conv2D(int(alpha_in * in_channels), out_channels - int(alpha_out * out_channels), kernel_size, 1, padding, dilation, groups) self.conv_h2l = None if alpha_in == 1 or alpha_out == 0 or self.is_dw else \ nn.Conv2D(in_channels - int(alpha_in * in_channels), int(alpha_out * out_channels), kernel_size, 1, padding, dilation, groups) self.conv_h2h = None if alpha_in == 1 or alpha_out == 1 else \ nn.Conv2D(in_channels - int(alpha_in * in_channels), out_channels - int(alpha_out * out_channels), kernel_size, 1, padding, dilation, math.ceil(groups - alpha_in * groups)) def forward(self, x): x_h, x_l = x if type(x) is tuple else (x, None) x_h = self.downsample(x_h) if self.stride == 2 else x_h        x_h2h = self.conv_h2h(x_h)        x_h2l = self.conv_h2l(self.downsample(x_h)) if self.alpha_out > 0 and not self.is_dw else None if x_l is not None:            x_l2l = self.downsample(x_l) if self.stride == 2 else x_l            x_l2l = self.conv_l2l(x_l2l) if self.alpha_out > 0 else None if self.is_dw: return x_h2h, x_l2l else:                x_l2h = self.conv_l2h(x_l)                x_l2h = self.upsample(x_l2h) if self.stride == 1 else x_l2h x_h = x_l2h + x_h2h x_l = x_h2l + x_l2l if x_h2l is not None and x_l2l is not None else None return x_h, x_l else: return x_h2h, x_h2lclass Conv_BN(nn.Layer): def __init__(self, in_channels, out_channels, kernel_size, alpha_in=0.5, alpha_out=0.5, stride=1, padding=0, dilation=1,                 groups=1, bias=False, norm_layer=nn.BatchNorm2D): super(Conv_BN, self).__init__() self.conv = OctaveConv(in_channels, out_channels, kernel_size, alpha_in, alpha_out, stride, padding, dilation,                               groups, bias) self.bn_h = None if alpha_out == 1 else norm_layer(int(out_channels * (1 - alpha_out))) self.bn_l = None if alpha_out == 0 else norm_layer(int(out_channels * alpha_out)) def forward(self, x): x_h, x_l = self.conv(x) x_h = self.bn_h(x_h) x_l = self.bn_l(x_l) if x_l is not None else None return x_h, x_lclass Conv_BN_ACT(nn.Layer): def __init__(self, in_channels=3, out_channels=32, kernel_size=3, alpha_in=0.5, alpha_out=0.5, stride=1, padding=0, dilation=1,                 groups=1, bias=False, norm_layer=nn.BatchNorm2D, activation_layer=nn.ReLU): super(Conv_BN_ACT, self).__init__() self.conv = OctaveConv(in_channels, out_channels, kernel_size, alpha_in, alpha_out, stride, padding, dilation,                               groups, bias) self.bn_h = None if alpha_out == 1 else norm_layer(int(out_channels * (1 - alpha_out))) self.bn_l = None if alpha_out == 0 else norm_layer(int(out_channels * alpha_out)) self.act = activation_layer() def forward(self, x): x_h, x_l = self.conv(x) x_h = self.act(self.bn_h(x_h)) x_l = self.act(self.bn_l(x_l)) if x_l is not None else None return x_h, x_l

2.2 Oct-Mobilnetv1复现

Oct-Mobilnetv1的复现即将Mobilnetv1中的原始的Conv2D替换为Oct-Conv，其他均保持不变，在后面打印了Oct-Mobilnetv1的网络结构以及参数量，方便大家查看。

# Oct-Mobilnetv1import paddle.nn as nn__all__ = ["oct_mobilenet"]def conv_bn(inp, oup, stride): return nn.Sequential( nn.Conv2D(inp, oup, 3, stride, 1), nn.BatchNorm2D(oup), nn.ReLU() )def conv_dw(inp, oup, stride, alpha_in=0.5, alpha_out=0.5): return nn.Sequential( Conv_BN_ACT(inp, inp, kernel_size=3, stride=stride, padding=1, groups=inp,  \ alpha_in=alpha_in, alpha_out=alpha_in if alpha_out != alpha_in else alpha_out), Conv_BN_ACT(inp, oup, kernel_size=1, alpha_in=alpha_in, alpha_out=alpha_out) )class OctMobileNet(nn.Layer): def __init__(self, num_classes=1000): super(OctMobileNet, self).__init__() self.features = nn.Sequential( conv_bn( 3, 32, 2), conv_dw( 32, 64, 1, 0, 0.5), conv_dw( 64, 128, 2), conv_dw(128, 128, 1), conv_dw(128, 256, 2),  conv_dw(256, 256, 1), conv_dw(256, 512, 2), conv_dw(512, 512, 1), conv_dw(512, 512, 1), conv_dw(512, 512, 1), conv_dw(512, 512, 1), conv_dw(512, 512, 1, 0.5, 0), conv_dw(512, 1024, 2, 0, 0), conv_dw(1024, 1024, 1, 0, 0), ) self.avgpool = nn.AdaptiveAvgPool2D((1, 1)) self.fc = nn.Linear(1024, num_classes) def forward(self, x): x_h, x_l = self.features(x)        x = self.avgpool(x_h)        x = x.reshape([-1, 1024])        x = self.fc(x) return xdef oct_mobilenet(**kwargs): """    Constructs a Octave MobileNet V1 model    """ return OctMobileNet(**kwargs)

2.3 OctResNet的复现

Oct-ResNet的复现即将ResNet中的原始的Conv2D替换为Oct-Conv，其他均保持不变，在后面打印了Oct-ResNet的网络结构以及参数量，方便大家查看。

import paddle.nn as nn__all__ = ["OctResNet", "oct_resnet50", "oct_resnet101", "oct_resnet152", "oct_resnet200"]class Bottleneck(nn.Layer):    expansion = 4 def __init__(self, inplanes, planes, stride=1, downsample=None, groups=1, base_width=64, alpha_in=0.5, alpha_out=0.5, norm_layer=None, output=False): super(Bottleneck, self).__init__() if norm_layer is None: norm_layer = nn.BatchNorm2D        width = int(planes * (base_width / 64.)) * groups # Both self.conv2 and self.downsample layers downsample the input when stride != 1 self.conv1 = Conv_BN_ACT(inplanes, width, kernel_size=1, alpha_in=alpha_in, alpha_out=alpha_out, norm_layer=norm_layer) self.conv2 = Conv_BN_ACT(width, width, kernel_size=3, stride=stride, padding=1, groups=groups, norm_layer=norm_layer, alpha_in=0 if output else 0.5, alpha_out=0 if output else 0.5) self.conv3 = Conv_BN(width, planes * self.expansion, kernel_size=1, norm_layer=norm_layer, alpha_in=0 if output else 0.5, alpha_out=0 if output else 0.5) self.relu = nn.ReLU() self.downsample = downsample self.stride = stride def forward(self, x): identity_h = x[0] if type(x) is tuple else x identity_l = x[1] if type(x) is tuple else None x_h, x_l = self.conv1(x) x_h, x_l = self.conv2((x_h, x_l)) x_h, x_l = self.conv3((x_h, x_l)) if self.downsample is not None: identity_h, identity_l = self.downsample(x) x_h += identity_h x_l = x_l + identity_l if identity_l is not None else None x_h = self.relu(x_h) x_l = self.relu(x_l) if x_l is not None else None return x_h, x_lclass OctResNet(nn.Layer): def __init__(self, block, layers, num_classes=1000, zero_init_residual=False,                 groups=1, width_per_group=64, norm_layer=None): super(OctResNet, self).__init__() if norm_layer is None: norm_layer = nn.BatchNorm2D self.inplanes = 64 self.groups = groups self.base_width = width_per_group self.conv1 = nn.Conv2D(3, self.inplanes, kernel_size=7, stride=2, padding=3, )        self.bn1 = norm_layer(self.inplanes) self.relu = nn.ReLU() self.maxpool = nn.MaxPool2D(kernel_size=3, stride=2, padding=1) self.layer1 = self._make_layer(block, 64, layers[0], norm_layer=norm_layer, alpha_in=0) self.layer2 = self._make_layer(block, 128, layers[1], stride=2, norm_layer=norm_layer) self.layer3 = self._make_layer(block, 256, layers[2], stride=2, norm_layer=norm_layer) self.layer4 = self._make_layer(block, 512, layers[3], stride=2, norm_layer=norm_layer, alpha_out=0, output=True) self.avgpool = nn.AdaptiveAvgPool2D((1, 1)) self.fc = nn.Linear(512 * block.expansion, num_classes) def _make_layer(self, block, planes, blocks, stride=1, alpha_in=0.5, alpha_out=0.5, norm_layer=None, output=False): if norm_layer is None: norm_layer = nn.BatchNorm2D downsample = None if stride != 1 or self.inplanes != planes * block.expansion: downsample = nn.Sequential( Conv_BN(self.inplanes, planes * block.expansion, kernel_size=1, stride=stride, alpha_in=alpha_in, alpha_out=alpha_out) )        layers = [] layers.append(block(self.inplanes, planes, stride, downsample, self.groups, self.base_width, alpha_in, alpha_out, norm_layer, output)) self.inplanes = planes * block.expansion for _ in range(1, blocks): layers.append(block(self.inplanes, planes, groups=self.groups, base_width=self.base_width, norm_layer=norm_layer, alpha_in=0 if output else 0.5, alpha_out=0 if output else 0.5, output=output)) return nn.Sequential(*layers) def forward(self, x):        x = self.conv1(x)        x = self.bn1(x)        x = self.relu(x)        x = self.maxpool(x) x_h, x_l = self.layer1(x) x_h, x_l = self.layer2((x_h,x_l)) x_h, x_l = self.layer3((x_h,x_l)) x_h, x_l = self.layer4((x_h,x_l))        x = self.avgpool(x_h)        x = x.reshape([x.shape[0], -1])        x = self.fc(x) return xdef oct_resnet50(pretrained=False, **kwargs): """Constructs a Octave ResNet-50 model. Args:        pretrained (bool): If True, returns a model pre-trained on ImageNet    """    model = OctResNet(Bottleneck, [3, 4, 6, 3], **kwargs) return modeldef oct_resnet101(pretrained=False, **kwargs): """Constructs a Octave ResNet-101 model. Args:        pretrained (bool): If True, returns a model pre-trained on ImageNet    """    model = OctResNet(Bottleneck, [3, 4, 23, 3], **kwargs) return modeldef oct_resnet152(pretrained=False, **kwargs): """Constructs a Octave ResNet-152 model. Args:        pretrained (bool): If True, returns a model pre-trained on ImageNet    """    model = OctResNet(Bottleneck, [3, 8, 36, 3], **kwargs) return modeldef oct_resnet200(pretrained=False, **kwargs): """Constructs a Octave ResNet-200 model. Args:        pretrained (bool): If True, returns a model pre-trained on ImageNet    """    model = OctResNet(Bottleneck, [3, 24, 36, 3], **kwargs) return model

3、对比实验

实验数据：Cifar10

CIFAR-10 是由 Hinton 的学生 Alex Krizhevsky 和 Ilya Sutskever 整理的一个用于识别普适物体的小型数据集。一共包含 10 个类别的 RGB 彩色图 片：飞机（ a叩lane ）、汽车（ automobile ）、鸟类（ bird ）、猫（ cat ）、鹿（ deer ）、狗（ dog ）、蛙类（ frog ）、马（ horse ）、船（ ship ）和卡车（ truck ）。图片的尺寸为 32×32 ，数据集中一共有 50000 张训练圄片和 10000 张测试图片。 CIFAR-10 的图片样例如图所示。

3.1 Oct_MobilNetv1模型网络结构可视化

Octmobilnet_model = oct_mobilenet(num_classes=10)# inputs = paddle.randn((1, 2, 224, 224))# print(model(inputs))paddle.summary(Octmobilnet_model,(16,3,224,224))

3.2 Oct_MobilNetV1模型训练

import paddlefrom paddle.metric import Accuracyfrom paddle.vision.transforms import Compose, Normalize, Resize, Transpose, ToTensorcallback = paddle.callbacks.VisualDL(log_dir="visualdl_log_dir_octmobilenet")normalize = Normalize(mean=[0.5, 0.5, 0.5],                    std=[0.5, 0.5, 0.5], data_format="HWC")transform = Compose([ToTensor(), Normalize(), Resize(size=(224,224))])cifar10_train = paddle.vision.datasets.Cifar10(mode="train",                                               transform=transform)cifar10_test = paddle.vision.datasets.Cifar10(mode="test",                                              transform=transform)# 构建训练集数据加载器train_loader = paddle.io.DataLoader(cifar10_train, batch_size=768, shuffle=True, drop_last=True)# 构建测试集数据加载器test_loader = paddle.io.DataLoader(cifar10_test, batch_size=768, shuffle=True, drop_last=True)Octmobilnet_model = paddle.Model(oct_mobilenet(num_classes=10))optim = paddle.optimizer.Adam(learning_rate=0.001, parameters=Octmobilnet_model.parameters())Octmobilnet_model.prepare( optim, paddle.nn.CrossEntropyLoss(), Accuracy() )Octmobilnet_model.fit(train_data=train_loader, eval_data=test_loader,        epochs=12,        callbacks=callback,        verbose=1 )

3.3 MobileNetV1模型网络结构可视化

from paddle.vision.models import MobileNetV1mobile_model = MobileNetV1(num_classes=10)# inputs = paddle.randn((1, 2, 224, 224))# print(model(inputs))paddle.summary(mobile_model,(16,3,224,224))

3.4 MobileNetV1模型训练

import paddlefrom paddle.metric import Accuracyfrom paddle.vision.transforms import Compose, Normalize, Resize, Transpose, ToTensorcallback = paddle.callbacks.VisualDL(log_dir="visualdl_log_dir_mobilenet")normalize = Normalize(mean=[0.5, 0.5, 0.5],                    std=[0.5, 0.5, 0.5], data_format="HWC")transform = Compose([ToTensor(), Normalize(), Resize(size=(224,224))])cifar10_train = paddle.vision.datasets.Cifar10(mode="train",                                               transform=transform)cifar10_test = paddle.vision.datasets.Cifar10(mode="test",                                              transform=transform)# 构建训练集数据加载器train_loader = paddle.io.DataLoader(cifar10_train, batch_size=768, shuffle=True, drop_last=True)# 构建测试集数据加载器test_loader = paddle.io.DataLoader(cifar10_test, batch_size=768, shuffle=True, drop_last=True)mobile_model = paddle.Model(MobileNetV1(num_classes=10))optim = paddle.optimizer.Adam(learning_rate=0.001, parameters=mobile_model.parameters())mobile_model.prepare( optim, paddle.nn.CrossEntropyLoss(), Accuracy() )mobile_model.fit(train_data=train_loader, eval_data=test_loader,        epochs=12,        callbacks=callback,        verbose=1 )

3.5 Oct_ResNet50模型网络结构可视化

octresnet50_model = oct_resnet50(num_classes=10)paddle.summary(octresnet50_model,(16,3,224,224))

3.6 Oct_ResNet50模型训练

import paddlefrom paddle.metric import Accuracyfrom paddle.vision.transforms import Compose, Normalize, Resize, Transpose, ToTensorcallback = paddle.callbacks.VisualDL(log_dir="visualdl_log_dir_octresnet")normalize = Normalize(mean=[0.5, 0.5, 0.5],                    std=[0.5, 0.5, 0.5], data_format="HWC")transform = Compose([ToTensor(), Normalize(), Resize(size=(224,224))])cifar10_train = paddle.vision.datasets.Cifar10(mode="train",                                               transform=transform)cifar10_test = paddle.vision.datasets.Cifar10(mode="test",                                              transform=transform)# 构建训练集数据加载器train_loader = paddle.io.DataLoader(cifar10_train, batch_size=256, shuffle=True, drop_last=True)# 构建测试集数据加载器test_loader = paddle.io.DataLoader(cifar10_test, batch_size=256, shuffle=True, drop_last=True)oct_resnet50 = paddle.Model(oct_resnet50(num_classes=10))optim = paddle.optimizer.Adam(learning_rate=0.001, parameters=oct_resnet50.parameters())oct_resnet50.prepare( optim, paddle.nn.CrossEntropyLoss(), Accuracy() )oct_resnet50.fit(train_data=train_loader, eval_data=test_loader,        epochs=12,        callbacks=callback,        verbose=1 )

3.7 ResNet50模型网络结构可视化

import paddle# build modelresmodel = resnet50(num_classes=10)paddle.summary(resmodel,(16,3,224,224))

3.8 ResNet50模型训练

import paddlefrom paddle.metric import Accuracyfrom paddle.vision.transforms import Compose, Normalize, Resize, Transpose, ToTensorfrom paddle.vision.models import resnet50callback = paddle.callbacks.VisualDL(log_dir="visualdl_log_dir_resnet")normalize = Normalize(mean=[0.5, 0.5, 0.5],                    std=[0.5, 0.5, 0.5], data_format="HWC")transform = Compose([ToTensor(), Normalize(), Resize(size=(224,224))])cifar10_train = paddle.vision.datasets.Cifar10(mode="train",                                               transform=transform)cifar10_test = paddle.vision.datasets.Cifar10(mode="test",                                              transform=transform)# 构建训练集数据加载器train_loader = paddle.io.DataLoader(cifar10_train, batch_size=256, shuffle=True, drop_last=True)# 构建测试集数据加载器test_loader = paddle.io.DataLoader(cifar10_test, batch_size=256, shuffle=True, drop_last=True)res_model = paddle.Model(resnet50(num_classes=10))optim = paddle.optimizer.Adam(learning_rate=0.001, parameters=res_model.parameters())res_model.prepare( optim, paddle.nn.CrossEntropyLoss(), Accuracy() )res_model.fit(train_data=train_loader, eval_data=test_loader,        epochs=12,        callbacks=callback,        verbose=1 )

3.9 实验结果

本小节提供消融实验的结果以及可视化训练结果，共计包含四个实验，分别为octmobinetv1、mobinetv1、octresnet50以及resnet50在数据集Cifar10上的结果对比。

图1：Oct_MobileNetV1对比实验结果图

图2：Oct_ResNet50对比实验结果图

4、参考资料

d-li14/octconv.pytorch

神经网络学习之OctConv：八度卷积

Drop an Octave: Reducing Spatial Redundancy in Convolutional Neural Networks with Octave Convolution

5、总结

目前我们得到的结论与论文中的结论不符，论文提供的代码为MXnet框架，本复现参考了PyTorch版本的复现，不能确定是否为框架原因，或者一些训练设置原因，比如初始化方式或模型迭代次数不够，有待查证，大家感兴趣的也可以就这个问题与我在评论区进行交流。

点击关注，第一时间了解华为云新鲜技术~

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