--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.16.4 kernelspec: display_name: Python 3 language: python name: python3 --- +++ {"id": "TVT_MVvc02AA"} # Generalized convolutions in JAX [![Open in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/jax-ml/jax/blob/main/docs/notebooks/convolutions.ipynb) [![Open in Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/jax-ml/jax/blob/main/docs/notebooks/convolutions.ipynb) JAX provides a number of interfaces to compute convolutions across data, including: - {func}`jax.numpy.convolve` (also {func}`jax.numpy.correlate`) - {func}`jax.scipy.signal.convolve` (also {func}`~jax.scipy.signal.correlate`) - {func}`jax.scipy.signal.convolve2d` (also {func}`~jax.scipy.signal.correlate2d`) - {func}`jax.lax.conv_general_dilated` For basic convolution operations, the `jax.numpy` and `jax.scipy` operations are usually sufficient. If you want to do more general batched multi-dimensional convolution, the `jax.lax` function is where you should start. +++ {"id": "ewZEn2X12-Ng"} ## Basic one-dimensional convolution Basic one-dimensional convolution is implemented by {func}`jax.numpy.convolve`, which provides a JAX interface for {func}`numpy.convolve`. Here is a simple example of 1D smoothing implemented via a convolution: ```{code-cell} ipython3 :id: 0qYLpeZO3Z9- :outputId: 4f6717ac-a062-4a85-8330-d57bf80de384 import matplotlib.pyplot as plt from jax import random import jax.numpy as jnp import numpy as np key = random.key(1701) x = jnp.linspace(0, 10, 500) y = jnp.sin(x) + 0.2 * random.normal(key, shape=(500,)) window = jnp.ones(10) / 10 y_smooth = jnp.convolve(y, window, mode='same') plt.plot(x, y, 'lightgray') plt.plot(x, y_smooth, 'black'); ``` +++ {"id": "dYX1tCVB4XOW"} The `mode` parameter controls how boundary conditions are treated; here we use `mode='same'` to ensure that the output is the same size as the input. For more information, see the {func}`jax.numpy.convolve` documentation, or the documentation associated with the original {func}`numpy.convolve` function. +++ {"id": "5ndvLDIH4rv6"} ## Basic N-dimensional convolution For *N*-dimensional convolution, {func}`jax.scipy.signal.convolve` provides a similar interface to that of {func}`jax.numpy.convolve`, generalized to *N* dimensions. For example, here is a simple approach to de-noising an image based on convolution with a Gaussian filter: ```{code-cell} ipython3 :id: Jk5qdnbv6QgT :outputId: 292205eb-aa09-446f-eec2-af8c23cfc718 from scipy import misc import jax.scipy as jsp fig, ax = plt.subplots(1, 3, figsize=(12, 5)) # Load a sample image; compute mean() to convert from RGB to grayscale. image = jnp.array(misc.face().mean(-1)) ax[0].imshow(image, cmap='binary_r') ax[0].set_title('original') # Create a noisy version by adding random Gaussian noise key = random.key(1701) noisy_image = image + 50 * random.normal(key, image.shape) ax[1].imshow(noisy_image, cmap='binary_r') ax[1].set_title('noisy') # Smooth the noisy image with a 2D Gaussian smoothing kernel. x = jnp.linspace(-3, 3, 7) window = jsp.stats.norm.pdf(x) * jsp.stats.norm.pdf(x[:, None]) smooth_image = jsp.signal.convolve(noisy_image, window, mode='same') ax[2].imshow(smooth_image, cmap='binary_r') ax[2].set_title('smoothed'); ``` +++ {"id": "Op-NhXy39z2U"} Like in the one-dimensional case, we use `mode='same'` to specify how we would like edges to be handled. For more information on available options in *N*-dimensional convolutions, see the {func}`jax.scipy.signal.convolve` documentation. +++ {"id": "bxuUjFVG-v1h"} ## General convolutions +++ {"id": "0pcn2LeS-03b"} For the more general types of batched convolutions often useful in the context of building deep neural networks, JAX and XLA offer the very general N-dimensional __conv_general_dilated__ function, but it's not very obvious how to use it. We'll give some examples of the common use-cases. A survey of the family of convolutional operators, [a guide to convolutional arithmetic](https://arxiv.org/abs/1603.07285), is highly recommended reading! Let's define a simple diagonal edge kernel: ```{code-cell} ipython3 :id: Yud1Y3ss-x1K :outputId: 3185fba5-1ad7-462f-96ba-7ed1b0c3d5a2 # 2D kernel - HWIO layout kernel = jnp.zeros((3, 3, 3, 3), dtype=jnp.float32) kernel += jnp.array([[1, 1, 0], [1, 0,-1], [0,-1,-1]])[:, :, jnp.newaxis, jnp.newaxis] print("Edge Conv kernel:") plt.imshow(kernel[:, :, 0, 0]); ``` +++ {"id": "dITPaPdh_cMI"} And we'll make a simple synthetic image: ```{code-cell} ipython3 :id: cpbGsIGa_Qyx :outputId: d7c5d21f-c3a0-42e9-c9bc-3da1a508c0e7 # NHWC layout img = jnp.zeros((1, 200, 198, 3), dtype=jnp.float32) for k in range(3): x = 30 + 60*k y = 20 + 60*k img = img.at[0, x:x+10, y:y+10, k].set(1.0) print("Original Image:") plt.imshow(img[0]); ``` +++ {"id": "_m90y74OWorG"} ### lax.conv and lax.conv_with_general_padding +++ {"id": "Pv9_QPDnWssM"} These are the simple convenience functions for convolutions ️⚠️ The convenience `lax.conv`, `lax.conv_with_general_padding` helper function assume __NCHW__ images and __OIHW__ kernels. ```{code-cell} ipython3 :id: kppxbxpZW0nb :outputId: 9fc5494c-b443-4e74-fe48-fac09e12378c from jax import lax out = lax.conv(jnp.transpose(img,[0,3,1,2]), # lhs = NCHW image tensor jnp.transpose(kernel,[3,2,0,1]), # rhs = OIHW conv kernel tensor (1, 1), # window strides 'SAME') # padding mode print("out shape: ", out.shape) print("First output channel:") plt.figure(figsize=(10,10)) plt.imshow(np.array(out)[0,0,:,:]); ``` ```{code-cell} ipython3 :id: aonr1tWvYCW9 :outputId: 3d44d494-9620-4736-e331-c9569a4888cd out = lax.conv_with_general_padding( jnp.transpose(img,[0,3,1,2]), # lhs = NCHW image tensor jnp.transpose(kernel,[2,3,0,1]), # rhs = IOHW conv kernel tensor (1, 1), # window strides ((2,2),(2,2)), # general padding 2x2 (1,1), # lhs/image dilation (1,1)) # rhs/kernel dilation print("out shape: ", out.shape) print("First output channel:") plt.figure(figsize=(10,10)) plt.imshow(np.array(out)[0,0,:,:]); ``` +++ {"id": "lyOwGRez_ycJ"} ### Dimension Numbers define dimensional layout for conv_general_dilated The important argument is the 3-tuple of axis layout arguments: (Input Layout, Kernel Layout, Output Layout) - __N__ - batch dimension - __H__ - spatial height - __W__ - spatial width - __C__ - channel dimension - __I__ - kernel _input_ channel dimension - __O__ - kernel _output_ channel dimension ⚠️ To demonstrate the flexibility of dimension numbers we choose a __NHWC__ image and __HWIO__ kernel convention for `lax.conv_general_dilated` below. ```{code-cell} ipython3 :id: oXKebfCb_i2B :outputId: d5a569b3-febc-4832-f725-1d5e8fd31b9b dn = lax.conv_dimension_numbers(img.shape, # only ndim matters, not shape kernel.shape, # only ndim matters, not shape ('NHWC', 'HWIO', 'NHWC')) # the important bit print(dn) ``` +++ {"id": "elZys_HzFVG6"} #### SAME padding, no stride, no dilation ```{code-cell} ipython3 :id: rgb2T15aFVG6 :outputId: 9b33cdb0-6959-4c88-832a-b92c4e42ae72 out = lax.conv_general_dilated(img, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,1), # window strides 'SAME', # padding mode (1,1), # lhs/image dilation (1,1), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape) print("First output channel:") plt.figure(figsize=(10,10)) plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "E4i3TI5JFVG9"} #### VALID padding, no stride, no dilation ```{code-cell} ipython3 :id: 1HQwudKVFVG- :outputId: be9d6b26-8e3e-44d9-dbd2-df2f6bbf98c2 out = lax.conv_general_dilated(img, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,1), # window strides 'VALID', # padding mode (1,1), # lhs/image dilation (1,1), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape, "DIFFERENT from above!") print("First output channel:") plt.figure(figsize=(10,10)) plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "VYKZdqLIFVHB"} #### SAME padding, 2,2 stride, no dilation ```{code-cell} ipython3 :id: mKq2-zmmFVHC :outputId: 14cc0114-e230-4555-a682-23e00b534863 out = lax.conv_general_dilated(img, # lhs = image tensor kernel, # rhs = conv kernel tensor (2,2), # window strides 'SAME', # padding mode (1,1), # lhs/image dilation (1,1), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape, " <-- half the size of above") plt.figure(figsize=(10,10)) print("First output channel:") plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "gPxttaiaFVHE"} #### VALID padding, no stride, rhs kernel dilation ~ Atrous convolution (excessive to illustrate) ```{code-cell} ipython3 :id: _pGr0x6qFVHF :outputId: 9edbccb6-d976-4b55-e0b7-e6f3b743e476 out = lax.conv_general_dilated(img, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,1), # window strides 'VALID', # padding mode (1,1), # lhs/image dilation (12,12), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape) plt.figure(figsize=(10,10)) print("First output channel:") plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "v-RhEeUfFVHI"} #### VALID padding, no stride, lhs=input dilation ~ Transposed Convolution ```{code-cell} ipython3 :id: B9Ail8ppFVHJ :outputId: 7aa19474-566f-4419-bfae-8286dd026c1c out = lax.conv_general_dilated(img, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,1), # window strides ((0, 0), (0, 0)), # padding mode (2,2), # lhs/image dilation (1,1), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape, "<-- larger than original!") plt.figure(figsize=(10,10)) print("First output channel:") plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "A-9OagtrVDyV"} We can use the last to, for instance, implement _transposed convolutions_: ```{code-cell} ipython3 :id: 5EYIj77-NdHE :outputId: f45b16f7-cc6e-4593-8aca-36b4152c3dfa # The following is equivalent to tensorflow: # N,H,W,C = img.shape # out = tf.nn.conv2d_transpose(img, kernel, (N,2*H,2*W,C), (1,2,2,1)) # transposed conv = 180deg kernel rotation plus LHS dilation # rotate kernel 180deg: kernel_rot = jnp.rot90(jnp.rot90(kernel, axes=(0,1)), axes=(0,1)) # need a custom output padding: padding = ((2, 1), (2, 1)) out = lax.conv_general_dilated(img, # lhs = image tensor kernel_rot, # rhs = conv kernel tensor (1,1), # window strides padding, # padding mode (2,2), # lhs/image dilation (1,1), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape, "<-- transposed_conv") plt.figure(figsize=(10,10)) print("First output channel:") plt.imshow(np.array(out)[0,:,:,0]); ``` +++ {"id": "v8HsE-NCmUxx"} ### 1D Convolutions +++ {"id": "WeP0rw0tm7HK"} You aren't limited to 2D convolutions, a simple 1D demo is below: ```{code-cell} ipython3 :id: jJ-jcAn3cig- :outputId: 67c46ace-6adc-4c47-c1c7-1f185be5fd4b # 1D kernel - WIO layout kernel = jnp.array([[[1, 0, -1], [-1, 0, 1]], [[1, 1, 1], [-1, -1, -1]]], dtype=jnp.float32).transpose([2,1,0]) # 1D data - NWC layout data = np.zeros((1, 200, 2), dtype=jnp.float32) for i in range(2): for k in range(2): x = 35*i + 30 + 60*k data[0, x:x+30, k] = 1.0 print("in shapes:", data.shape, kernel.shape) plt.figure(figsize=(10,5)) plt.plot(data[0]); dn = lax.conv_dimension_numbers(data.shape, kernel.shape, ('NWC', 'WIO', 'NWC')) print(dn) out = lax.conv_general_dilated(data, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,), # window strides 'SAME', # padding mode (1,), # lhs/image dilation (1,), # rhs/kernel dilation dn) # dimension_numbers = lhs, rhs, out dimension permutation print("out shape: ", out.shape) plt.figure(figsize=(10,5)) plt.plot(out[0]); ``` +++ {"id": "7XOgXqCTmaPa"} ### 3D Convolutions ```{code-cell} ipython3 :id: QNvSiq5-mcLd :outputId: c99ec88c-6d5c-4acd-c8d3-331f026f5631 import matplotlib as mpl # Random 3D kernel - HWDIO layout kernel = jnp.array([ [[0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, -1, 0], [-1, 0, -1], [0, -1, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, 0]]], dtype=jnp.float32)[:, :, :, jnp.newaxis, jnp.newaxis] # 3D data - NHWDC layout data = jnp.zeros((1, 30, 30, 30, 1), dtype=jnp.float32) x, y, z = np.mgrid[0:1:30j, 0:1:30j, 0:1:30j] data += (jnp.sin(2*x*jnp.pi)*jnp.cos(2*y*jnp.pi)*jnp.cos(2*z*jnp.pi))[None,:,:,:,None] print("in shapes:", data.shape, kernel.shape) dn = lax.conv_dimension_numbers(data.shape, kernel.shape, ('NHWDC', 'HWDIO', 'NHWDC')) print(dn) out = lax.conv_general_dilated(data, # lhs = image tensor kernel, # rhs = conv kernel tensor (1,1,1), # window strides 'SAME', # padding mode (1,1,1), # lhs/image dilation (1,1,1), # rhs/kernel dilation dn) # dimension_numbers print("out shape: ", out.shape) # Make some simple 3d density plots: def make_alpha(cmap): my_cmap = cmap(jnp.arange(cmap.N)) my_cmap[:,-1] = jnp.linspace(0, 1, cmap.N)**3 return mpl.colors.ListedColormap(my_cmap) my_cmap = make_alpha(plt.cm.viridis) fig = plt.figure() ax = fig.add_subplot(projection='3d') ax.scatter(x.ravel(), y.ravel(), z.ravel(), c=data.ravel(), cmap=my_cmap) ax.axis('off') ax.set_title('input') fig = plt.figure() ax = fig.add_subplot(projection='3d') ax.scatter(x.ravel(), y.ravel(), z.ravel(), c=out.ravel(), cmap=my_cmap) ax.axis('off') ax.set_title('3D conv output'); ```