.. _chapter_conv_layer:
Convolutions for Images
=======================
Now that we understand how convolutional layers work in theory, we are
ready to see how this works in practice. Since we have motivated
convolutional neural networks by their applicability to image data, we
will stick with image data in our examples, and begin by revisiting the
convolutional layer that we introduced in the previous section. We note
that strictly speaking, *convolutional* layers are a slight misnomer,
since the operations are typically expressed as cross correlations.
The Cross-Correlation Operator
------------------------------
In a convolutional layer, an input array and a correlation kernel array
are combined to produce an output array through a cross-correlation
operation. Let’s see how this works for two dimensions. In our example,
the input is a two-dimensional array with a height of 3 and width of 3.
We mark the shape of the array as :math:`3 \times 3` or (3, 3). The
height and width of the kernel array are both 2. Common names for this
array in the deep learning research community include *kernel* and
*filter*. The shape of the kernel window (also known as the convolution
window) is given precisely by the height and width of the kernel (here
it is :math:`2 \times 2`).
.. figure:: ../img/correlation.svg
Two-dimensional cross-correlation operation. The shaded portions are
the first output element and the input and kernel array elements used
in its computation: :math:`0\times0+1\times1+3\times2+4\times3=19`.
In the two-dimensional cross-correlation operation, we begin with the
convolution window positioned at the top-left corner of the input array
and slide it across the input array, both from left to right and top to
bottom. When the convolution window slides to a certain position, the
input subarray contained in that window and the kernel array are
multiplied (element-wise) and the resulting array is summed up yielding
a single scalar value. This result if precisely the value of the output
array at the corresponding location. Here, the output array has a height
of 2 and width of 2 and the four elements are derived from the
two-dimensional cross-correlation operation:
.. math::
0\times0+1\times1+3\times2+4\times3=19,\\
1\times0+2\times1+4\times2+5\times3=25,\\
3\times0+4\times1+6\times2+7\times3=37,\\
4\times0+5\times1+7\times2+8\times3=43.
Note that along each axi, the output is slightly *smaller* than the
input. Because the kernel has a width greater than one, and we can only
computer the cross-correlation for locations where the kernel fits
wholly within the image, the output size is given by the input size
:math:`H \times W` minus the size of the convolutional kernel
:math:`h \times w` via :math:`(H-h+1) \times (W-w+1)`. This is the case
since we need enough space to ‘shift’ the convolutional kernel across
the image (later we will see how to keep the size unchanged by padding
the image with zeros around its boundary such that there’s enough space
to shift the kernel). Next, we implement the above process in the
``corr2d`` function. It accepts the input array ``X`` with the kernel
array ``K`` and outputs the array ``Y``.
.. code:: python
from mxnet import autograd, nd
from mxnet.gluon import nn
# Save to the d2l package.
def corr2d(X, K):
"""Compute 2D cross-correlation."""
h, w = K.shape
Y = nd.zeros((X.shape[0] - h + 1, X.shape[1] - w + 1))
for i in range(Y.shape[0]):
for j in range(Y.shape[1]):
Y[i, j] = (X[i: i + h, j: j + w] * K).sum()
return Y
We can construct the input array ``X`` and the kernel array ``K`` from
the figure above to validate the output of the above implementations of
the two-dimensional cross-correlation operation.
.. code:: python
X = nd.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
K = nd.array([[0, 1], [2, 3]])
corr2d(X, K)
.. parsed-literal::
:class: output
[[19. 25.]
[37. 43.]]
Convolutional Layers
--------------------
A convolutional layer cross-correlates the input and kernels and adds a
scalar bias to produce an output. The parameters of the convolutional
layer are precisely the values that constitute the kernel and the scalar
bias. When training the models based on convolutional layers, we
typically initialize the kernels randomly, just as we would with a
fully-connected layer.
We are now ready to implement a two-dimensional convolutional layer
based on the ``corr2d`` function defined above. In the ``__init__``
constructor function, we declare ``weight`` and ``bias`` as the two
model parameters. The forward computation function ``forward`` calls the
``corr2d`` function and adds the bias. As with :math:`h \times w`
cross-correlation we also refer to convolutional layers as
:math:`h \times w` convolutions.
.. code:: python
class Conv2D(nn.Block):
def __init__(self, kernel_size, **kwargs):
super(Conv2D, self).__init__(**kwargs)
self.weight = self.params.get('weight', shape=kernel_size)
self.bias = self.params.get('bias', shape=(1,))
def forward(self, x):
return corr2d(x, self.weight.data()) + self.bias.data()
Object Edge Detection in Images
-------------------------------
Let’s look at a simple application of a convolutional layer: detecting
the edge of an object in an image by finding the location of the pixel
change. First, we construct an ‘image’ of :math:`6\times 8` pixels. The
middle four columns are black (0) and the rest are white (1).
.. code:: python
X = nd.ones((6, 8))
X[:, 2:6] = 0
X
.. parsed-literal::
:class: output
[[1. 1. 0. 0. 0. 0. 1. 1.]
[1. 1. 0. 0. 0. 0. 1. 1.]
[1. 1. 0. 0. 0. 0. 1. 1.]
[1. 1. 0. 0. 0. 0. 1. 1.]
[1. 1. 0. 0. 0. 0. 1. 1.]
[1. 1. 0. 0. 0. 0. 1. 1.]]
Next, we construct a kernel ``K`` with a height of 1 and width of 2.
When we perform the cross-correlation operation with the input, if the
horizontally adjacent elements are the same, the output is 0. Otherwise,
the output is non-zero.
.. code:: python
K = nd.array([[1, -1]])
Enter ``X`` and our designed kernel ``K`` to perform the
cross-correlation operations. As you can see, we will detect 1 for the
edge from white to black and -1 for the edge from black to white. The
rest of the outputs are 0.
.. code:: python
Y = corr2d(X, K)
Y
.. parsed-literal::
:class: output
[[ 0. 1. 0. 0. 0. -1. 0.]
[ 0. 1. 0. 0. 0. -1. 0.]
[ 0. 1. 0. 0. 0. -1. 0.]
[ 0. 1. 0. 0. 0. -1. 0.]
[ 0. 1. 0. 0. 0. -1. 0.]
[ 0. 1. 0. 0. 0. -1. 0.]]
Let’s apply the kernel to the transposed image. As expected, it
vanishes. The kernel ``K`` only detects vertical edges.
.. code:: python
corr2d(X.T, K)
.. parsed-literal::
:class: output
[[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]
[0. 0. 0. 0. 0.]]
Learning a Kernel
-----------------
Designing an edge detector by finite differences ``[1, -1]`` is neat if
we know this is precisely what we are looking for. However, as we look
at larger kernels, and consider successive layers of convolutions, it
might be impossible to specify precisely what each filter should be
doing manually.
Now let’s see whether we can learn the kernel that generated ``Y`` from
``X`` by looking at the (input, output) pairs only. We first construct a
convolutional layer and initialize its kernel as a random array. Next,
in each iteration, we will use the squared error to compare ``Y`` and
the output of the convolutional layer, then calculate the gradient to
update the weight. For the sake of simplicity, in this convolutional
layer, we will ignores the bias.
We previously constructed the ``Conv2D`` class. However, since we used
single-element assignments, Gluon has some trouble finding the gradient.
Instead, we use the built-in ``Conv2D`` class provided by Gluon below.
.. code:: python
# Construct a convolutional layer with 1 output channel
# (channels will be introduced in the following section)
# and a kernel array shape of (1, 2)
conv2d = nn.Conv2D(1, kernel_size=(1, 2))
conv2d.initialize()
# The two-dimensional convolutional layer uses four-dimensional input and
# output in the format of (example channel, height, width), where the batch
# size (number of examples in the batch) and the number of channels are both 1
X = X.reshape((1, 1, 6, 8))
Y = Y.reshape((1, 1, 6, 7))
for i in range(10):
with autograd.record():
Y_hat = conv2d(X)
l = (Y_hat - Y) ** 2
l.backward()
# For the sake of simplicity, we ignore the bias here
conv2d.weight.data()[:] -= 3e-2 * conv2d.weight.grad()
if (i + 1) % 2 == 0:
print('batch %d, loss %.3f' % (i + 1, l.sum().asscalar()))
.. parsed-literal::
:class: output
batch 2, loss 4.949
batch 4, loss 0.831
batch 6, loss 0.140
batch 8, loss 0.024
batch 10, loss 0.004
As you can see, the error has dropped to a small value after 10
iterations. Now we will take a look at the kernel array we learned.
.. code:: python
conv2d.weight.data().reshape((1, 2))
.. parsed-literal::
:class: output
[[ 0.9895 -0.9873705]]
Indeed, the learned kernel array is remarkably close to the kernel array
``K`` we defined earlier.
Cross-correlation and Convolution
---------------------------------
Recall the observation from the previous section that cross-correlation
and convolution are equivalent. In the figure above it is easy to see
this correspondence. Simply flip the kernel from the bottom left to the
top right. In this case the indexing in the sum is reverted, yet the
same result can be obtained. In keeping with standard terminology with
deep learning literature, we will continue to refer to the
cross-correlation operation as a convolution even though,
strictly-speaking, it is slightly different.
Summary
-------
- The core computation of a two-dimensional convolutional layer is a
two-dimensional cross-correlation operation. In its simplest form,
this performs a cross-correlation operation on the two-dimensional
input data and the kernel, and then adds a bias.
- We can design a kernel to detect edges in images.
- We can learn the kernel through data.
Exercises
---------
1. Construct an image ``X`` with diagonal edges.
- What happens if you apply the kernel ``K`` to it?
- What happens if you transpose ``X``?
- What happens if you transpose ``K``?
2. When you try to automatically find the gradient for the ``Conv2D``
class we created, what kind of error message do you see?
3. How do you represent a cross-correlation operation as a matrix
multiplication by changing the input and kernel arrays?
4. Design some kernels manually.
- What is the form of a kernel for the second derivative?
- What is the kernel for the Laplace operator?
- What is the kernel for an integral?
- What is the minimum size of a kernel to obtain a derivative of
degree :math:`d`?
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|image0|
.. |image0| image:: ../img/qr_conv-layer.svg