Tensorflow中的padding有两种方式，其中的SAME方式比较特殊，可能产生非对称pad的情况，之前在验证一个tensorflow的网络时踩到这个坑。
Tensorflow的计算公式
二维卷积接口
tf.nn.conv2d(
input, filters, strides, padding, data_format='NHWC', dilations=None, name=None
)
padding计算公式
需要注意padding的配置，如果是字符串就有SAME和VALID两种配置，如果是数字list就明确表明padding在各个维度的数量。
首先，padding如果表示确切的数字，其维度是input维度的2倍，因为每个维度两个边需要补pad，比如宽度的左边和右边，高度的上边和下边，但是tensorflow中不会给N维度以及C维度补pad，仅仅在W和H维度补pad，因此对于NHWC，padding =[[0, 0], [pad_top, pad_bottom], [pad_left, pad_right], [0, 0]] ，对于NCHW的pad的顺序换过来。
然后，如果输入的是字符串选项，补的pad都可以映射到padding这个参数上，
-
VALID模式表示不在任何维度补pad，等价于padding =[[0, 0], [0, 0], [0, 0], [0, 0]]； -
SAME模式表示在stride的尺度下，Wo与Wi保持在stride(S)下保持一致（以宽度维度为例），需要满足如下关系[W_{o}=leftlceilfrac{W_{i}}{S} ight ceil ]我们知道如果
dilation = 1，那么在某个维度上，卷积的输出宽度(W_i)、输出宽度(W_o)和步长(S)，在没有pad的情况下，满足如下的关系式[W o=leftlfloorfrac{W i-W_{k}}{S} ight floor+1 ]我们以补最小程度的(P_a)为基准，于是有关系式
[W o=frac{W i+P_{a}-W_{k}}{S}+1 ]反推过来得到
[P_{a}=left(W_{o}-1 ight) S+W_{k}-W_{i} ]这是需要补的总的pad，tensorflow的补充pad是尽量两边对称的，
- 如果(P_a)是偶数，那么两边都补(P_l = P_a/2)；
- 如果(P_a)是奇数，那么左边补(P_l = lfloor{P_a/2} floor)，右边是(P_r = P_a-P_l)。
参考如下的代码
inputH, inputW = 7, 8
strideH, strideW = 3, 3
filterH = 4
filterW = 4
inputC = 16
outputC = 3
inputData = np.ones([1,inputH,inputW,inputC]) # format [N, H, W, C]
filterData = np.float16(np.ones([filterH, filterW, inputC, outputC]) - 0.33)
strides = (1, strideH, strideW, 1)
convOutputSame = tf.nn.conv2d(inputData, filterData, strides, padding='SAME')
convOutput = tf.nn.conv2d(inputData, filterData, strides, padding=[[0,0],[1,2],[1,1],[0,0]]) # padded input data
print("output1, ", convOutputSame)
print("output2, ", convOutput)
print("Sum of a - b is ", np.sum(np.square(convOutputSame - convOutput)))
计算结果是
output1, tf.Tensor(
[[[[ 96.46875 96.46875 96.46875]
[128.625 128.625 128.625 ]
[ 96.46875 96.46875 96.46875]]
[[128.625 128.625 128.625 ]
[171.5 171.5 171.5 ]
[128.625 128.625 128.625 ]]
[[ 64.3125 64.3125 64.3125 ]
[ 85.75 85.75 85.75 ]
[ 64.3125 64.3125 64.3125 ]]]])
output2, tf.Tensor(
[[[[ 96.46875 96.46875 96.46875]
[128.625 128.625 128.625 ]
[ 96.46875 96.46875 96.46875]]
[[128.625 128.625 128.625 ]
[171.5 171.5 171.5 ]
[128.625 128.625 128.625 ]]
[[ 64.3125 64.3125 64.3125 ]
[ 85.75 85.75 85.75 ]
[ 64.3125 64.3125 64.3125 ]]]], shape=(1, 3, 3, 3), dtype=float64)
Sum of a - b is 0.0
ONNX计算公式
onnx的接口，参考IR定义如下
std::function<void(OpSchema&)> ConvOpSchemaGenerator(const char* filter_desc) {
return [=](OpSchema& schema) {
std::string doc = R"DOC(
The convolution operator consumes an input tensor and {filter_desc}, and
computes the output.)DOC";
ReplaceAll(doc, "{filter_desc}", filter_desc);
schema.SetDoc(doc);
schema.Input(
0,
"X",
"Input data tensor from previous layer; "
"has size (N x C x H x W), where N is the batch size, "
"C is the number of channels, and H and W are the "
"height and width. Note that this is for the 2D image. "
"Otherwise the size is (N x C x D1 x D2 ... x Dn). "
"Optionally, if dimension denotation is "
"in effect, the operation expects input data tensor "
"to arrive with the dimension denotation of [DATA_BATCH, "
"DATA_CHANNEL, DATA_FEATURE, DATA_FEATURE ...].",
"T");
schema.Input(
1,
"W",
"The weight tensor that will be used in the "
"convolutions; has size (M x C/group x kH x kW), where C "
"is the number of channels, and kH and kW are the "
"height and width of the kernel, and M is the number "
"of feature maps. For more than 2 dimensions, the "
"kernel shape will be (M x C/group x k1 x k2 x ... x kn), "
"where (k1 x k2 x ... kn) is the dimension of the kernel. "
"Optionally, if dimension denotation is in effect, "
"the operation expects the weight tensor to arrive "
"with the dimension denotation of [FILTER_OUT_CHANNEL, "
"FILTER_IN_CHANNEL, FILTER_SPATIAL, FILTER_SPATIAL ...]. "
"X.shape[1] == (W.shape[1] * group) == C "
"(assuming zero based indices for the shape array). "
"Or in other words FILTER_IN_CHANNEL should be equal to DATA_CHANNEL. ",
"T");
schema.Input(
2,
"B",
"Optional 1D bias to be added to the convolution, has size of M.",
"T",
OpSchema::Optional);
schema.Output(
0,
"Y",
"Output data tensor that contains the result of the "
"convolution. The output dimensions are functions "
"of the kernel size, stride size, and pad lengths.",
"T");
schema.TypeConstraint(
"T",
{"tensor(float16)", "tensor(float)", "tensor(double)"},
"Constrain input and output types to float tensors.");
schema.Attr(
"kernel_shape",
"The shape of the convolution kernel. If not present, should be inferred from input W.",
AttributeProto::INTS,
OPTIONAL);
schema.Attr(
"dilations",
"dilation value along each spatial axis of the filter. If not present, the dilation defaults is 1 along each spatial axis.",
AttributeProto::INTS,
OPTIONAL);
schema.Attr(
"strides",
"Stride along each spatial axis. If not present, the stride defaults is 1 along each spatial axis.",
AttributeProto::INTS,
OPTIONAL);
schema.Attr(
"auto_pad",
auto_pad_doc,
AttributeProto::STRING,
std::string("NOTSET"));
schema.Attr(
"pads",
pads_doc,
AttributeProto::INTS,
OPTIONAL);
schema.Attr(
"group",
"number of groups input channels and output channels are divided into.",
AttributeProto::INT,
static_cast<int64_t>(1));
schema.TypeAndShapeInferenceFunction([](InferenceContext& ctx) {
propagateElemTypeFromInputToOutput(ctx, 0, 0);
convPoolShapeInference(ctx, true, false, 0, 1);
});
};
}
需要注意上面的auto_pad选项，与tensorflow类似有3个选项
-
NOTSET，同tensorflow的VALID -
SAME_UPPER或者SAME_LOWER，这里的内容可以参考onnx文件defs.cc中的代码std::vector<int64_t> pads; if (getRepeatedAttribute(ctx, "pads", pads)) { if (pads.size() != n_input_dims * 2) { fail_shape_inference("Attribute pads has incorrect size"); } } else { pads.assign(n_input_dims * 2, 0); // pads的size是输入维度的2倍 const auto* auto_pad_attr = ctx.getAttribute("auto_pad"); if ((nullptr != auto_pad_attr) && (auto_pad_attr->s() != "VALID")) { // 如果pad mode是SAME_UPPER或者SAME_LOWER则进入该分支 int input_dims_size = static_cast<int>(n_input_dims); for (int i = 0; i < input_dims_size; ++i) { // 计算每个axis的pads int64_t residual = 0; int64_t stride = strides[i]; if (stride > 1) { // 如果stride == 1，那么total_pad就是Wk - Stride = Wk - 1 if (!input_shape.dim(2 + i).has_dim_value()) { continue; } residual = input_shape.dim(2 + i).dim_value(); while (residual >= stride) { residual -= stride; } } int64_t total_pad = residual == 0 ? effective_kernel_shape[i] - stride : effective_kernel_shape[i] - residual; // effective_kernel_shape = Wk if (total_pad < 0) total_pad = 0; int64_t half_pad_small = total_pad >> 1; int64_t half_pad_big = total_pad - half_pad_small; if (auto_pad_attr->s() == "SAME_UPPER") { // pad mode is here pads[i] = half_pad_small; pads[i + input_dims_size] = half_pad_big; } else if (auto_pad_attr->s() == "SAME_LOWER") { pads[i] = half_pad_big; pads[i + input_dims_size] = half_pad_small; } } } }上面的代码里面最难理解14~23行，其实这里计算total_pad就是tensorflow中的(P_a)，以上面的公式，做更进一步的推导，
[egin{equation} egin{aligned} P_{a}&=left(W_{o}-1 ight) S+W_{k}-W_{i}\ &=(leftlceilfrac{W_{i}}{S} ight ceil - 1)S+W_{k}-W_{i}\ end{aligned} end{equation} ]下面的分析有两种情况，对应代码第23行，
- 如果(W_i)是(S)的整数倍，那么(W_i = nS)，带入上面的公式有(P_a = W_k - S)；
- 如果(W_i)不是(S)的整数倍，那么(W_i = nS+m, m gt 0)，带入上面的公式有(P_a = W_k - m)，这个(m)就是(W_i)被Stride相除之后的余数，即代码中的
residual。
SAME_UPPER和SAME_LOWER对应(P_a)是奇数的情况，如果是偶数，结果一样，如果是奇数，那么SAME_UPPER放小半部分(lfloor{P_a/2} floor)，SAME_LOWER放大半部分(P_a - lfloor{P_a/2} floor)。
举例
在python - What is the difference between 'SAME' and 'VALID' padding in tf.nn.max_pool of tensorflow? - Stack Overflow上有一个比较具体的例子，可以看到，使用SAME模式可以使得stride刚好完整取完所有的input而不会有剩余或者短缺。
-
VALID模式inputs: 1 2 3 4 5 6 7 8 9 10 11 (12 13) |________________| dropped |_________________| -
SAME模式pad| |pad inputs: 0 |1 2 3 4 5 6 7 8 9 10 11 12 13|0 0 |________________| |_________________| |________________|在这个例子中(W_i = 13, W_k = 6, S = 5)。
参考资料
- TensorFlow中CNN的两种padding方式“SAME”和“VALID” - wuzqChom的博客 - CSDN博客
- python - What is the difference between 'SAME' and 'VALID' padding in tf.nn.max_pool of tensorflow? - Stack Overflow
- What does the 'same' padding parameter in convolution mean in TensorFlow? - Quora