Generating Art with Computers

In the world of deep learning, there’s never a dull moment. The breadth of interesting applications seems unbounded. It’s been applied to reach super-human performance in many areas like game playing and object recognition, and integral to exciting new technologies like self driving cars. Increasingly we find that the knowledge embodied in a trained neural network can be transferred to seemingly unrelated areas. Take for example the combination of two disparate ideas: object recognition and Picasso paintings. A net built for the sole purpose of recognizing objects in a photo has been found to be useful, completely unaltered, for the task of rendering an image in the style of any painting. This is the technology behind the scenes of apps like Prisma, but it is not hard to do yourself, and I have to say that implementing this was a lot of fun.

original image

These images were created, as described in Gatys et. al, by passing a random noise image through the VGG-16 convolutional network (with imagenet weights and top layers removed) and updating the pixels of the input image directly with gradient descent. An error function is devised, which quantifies how poorly the initial seed (random noise at first) balances between appearing like the original image, and at the same time in the style of the chosen painting. The derivative of the loss with respect to the RGB values of the noise image is calculated. The pixels are adjusted in their respective directions, and the process repeats. Amazingly, this works – the image looks more and more like a stylized version of the original image with each iteration of this optimization. This is due to the unreasonable effectiveness of gradient descent, convolutional neural networks, and well designed loss functions.

The error function combines content loss E_c and style loss E_s. This balances between preserving high level features of the original image, and with the textures of the painting. The errors compare the values of chosen convolutional layers when the input is the original image, generated image, or painting.

The content error is just the squared euclidian distance between corresponding filter activations under the stylized and original images.
\displaystyle E_c^l=\frac{1}{2}\sum _{i,j} {\left\|F_{i_j}^l-P_{i_j}^l\right\|^2} where F_{ij}^l is the activation of the i^{th} filter at position j in layer l.

The novelty that makes all this work is the style error. It can be done with a statistic on the channels of the convolutions in the higher layers of the network. This was originally designed to capture texture information in a texture synthesis algorithm. The style error of a layer is the mean squared error between the gramians, \displaystyle E_s^l=\frac{1}{4 N^2 M^2} \sum _{ij} {\left(G_{i_j}^l-A_{i_j}^l\right)^2} , where G_{ij}=\langle v_i,v_j \rangle is the inner product of the vectorized feature maps i and j in filter layer l of the style image, and A is the corresponding gramian when the input is noise. The inner product (gramian) shows the correlation between each pair of channels, and this captures texture information. The same thing in Keras code:

def gram_matrix(v):
    # In tensorflow, dim order is x,y,channel
    # Make each row a channel
    chans = K.permute_dimensions(x, (2, 0, 1))
    # vectorize the feature maps
    features = K.batch_flatten(chans)
    # gramian is just an inner product
    return, K.transpose(features)) 
             / x.get_shape().num_elements()
def style_loss(vi, vj):
    # mean squared error
    return mse(gram_matrix(vi), gram_matrix(vj))

The total error is \alpha E_c + \beta E_s with the weights on each error as hyperparameters. The results are quite astounding. Playing with weights \alpha and \beta, and with the weights on the contribution of each layer l towards the style loss allows for a large range of interesting results. This picture was created by decreasing the weight of the content loss.

One major drawback of this method can be seen in the creation above. The style loss must be tempered, or else it obstructs features of the image, like the face. The third set of images, in the style of “Woman in a Hat with Pompoms and a Printed Shirt” is another example. So portraits in general are not the best target. However, landscapes and the like come out amazing. The reason for this is that the style loss function does not take into account anything about the style image except the texture of the image. There are alternative statistics that have been tried. One successful result that I have not yet tried comes from the study of Markov Random Fields, used classically for image synthesis. The idea is to calculate the loss between patches of the filters, rather than the whole filter at once, where the loss of each patch of the generated image is calculated against the most similar patch (by cross correlation) for the painting.

Another drawback is that each generated image/painting combo must be calculated separately. This has been addressed by Johnson et. al and others by training a neural network which can turn an input image into a representation which, when passed through a loss network (such as VGG-16 as above), generates a stylized image of a particular style. The benefit is the speed – once trained, generating a stylized version of an input image is hundreds of orders of magnitude faster. The drawback is that the transformation network takes much longer to train and is only able to output images of the specific style it was trained on. However, this is the type of solution that can scale, for example to video. I have replicated exactly the network architecture as described in Johnson et. al. Here’s an example result:

I had trouble getting rid of artifacts showing up in some input images, like the blotch of white on the right of the stylized image. However, I did not train the network for very long, primarily to avoid large AWS GPU server bills, so the results are not that great. I’ll probably come back to this soon, as I am building my own GPU server! There are so many ideas to explore with this.

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