CPPNs and NeRF

The toy problem of memorizing a 2D image using an MLP is a nice problem to understand before going into solving the problem NeRF tackles (memorising a 3D scene given multiple view points). The task is to train an MLP to take image coordinates as input and produce the RGB value at that point as output.

What is observed is that directly feeding in the image coordinates does not produce great results. Instead, we augment the input vector by running it through a positional encoding function where the input vector v=(x, y) is run through sin functions of increasing frequency.

Why does Positional Encoding help?

Random Fourier Features lets networks learn high frequency functions in low dimensional domains. Positional Encoding is a special case of Random Fourier Features where B is a power of 2.

Under certain conditions, NNs are effectively Kernel Regression. ReLU MLPs correspond to a dot product kernel. The Random Fourier Features is nothing but a kernel trick.

Model Architecture

MODEL_L is defined using the following logic

inp_shape = (L_FACT-1)**2
model = tf.keras.models.Sequential()
model.add(tf.keras.layers.Dense(units = inp_shape, activation = 'ReLU', input_shape = (inp_shape, )))
for i in range(0, int(math.ceil(math.log2(inp_shape)))):
    units = max(3, inp_shape // 2**i)
    model.add(tf.keras.layers.Dense(units = units, activation = 'ReLU'))

model.compile(optimizer = 'adam', loss=keras.metrics.mean_squared_error, metrics = ['mse'])

Following is the Architecure of MODEL_33

Results

Model	Training	Params	Size
Original		N/A	126.80 KB
33		11,115	43.41 KB
65		44,075	172.16 KB
129		175,531	685.66 KB
257		700,587	2.67 MB

Refrences

1. NeRF Video - https://www.youtube.com/watch?v=nRyOzHpcr4Q&t=1706s
2. Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains - https://arxiv.org/abs/2006.10739
3. Neural Tangent Kernel: Convergence and Generalization in Neural Networks - https://arxiv.org/abs/1806.07572
4. NeRF - https://arxiv.org/abs/2003.08934