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# Deep Generative Image Models using a Laplacian Pyramid of Adversarial Networks

• Category: Article
• Created: January 24, 2022 6:36 PM
• Status: Open
• URL: https://arxiv.org/pdf/1506.05751.pdf
• Updated: January 25, 2022 10:21 AM

# Background

Building a model capable of producing high quality samples of natural images.

# Highlights

1. Our approach uses a cascade of convolutional networks within a Laplacian pyramid framework to generate images in a coarse-to-fine fashion.
2. At each level of the pyramid, a separate generative convnet model is trained using the Generative Adversarial Nets (GAN) approach.

# Methods

$\min _{G} \max _{D} V(D, G)=\mathbb{E}_{\boldsymbol{x} \sim p_{\text {data }}(\boldsymbol{x})}[\log D(\boldsymbol{x})]+\mathbb{E}_{\boldsymbol{z} \sim p_{\boldsymbol{z}}(\boldsymbol{z})}[\log (1-D(G(\boldsymbol{z})))]$

### Conditional generative adversarial net (CGAN)

$\min _{G} \max _{D} \mathbb{E}_{h, l \sim p_{\text {Data }}(\mathbf{h}, \mathbf{l})}[\log D(h, l)]+\mathbb{E}_{z \sim p_{\text {Noise }}(\mathbf{z}), l \sim p_{l}(\mathbf{l})}[\log (1-D(G(z, l), l))]$

### Laplacian Pyramid

1. The Laplacian pyramid is built from a Gaussian pyramid using upsampling $$u(.)$$ and downsampling $$d(.)$$ functions.
2. Let $$G(I) = [I_0;I_1; ...;I_K]$$ be the Gaussian pyramid where $$I_0 = I$$ and $$I_K$$ is $$k$$ repeated applications of $$d(.)$$ to $$I$$. Then, the coefficient $$h_k$$ **at level $$k$$ of the Laplacian pyramid is given by the difference between the adjacent levels in Gaussian pyramid, upsampling the smaller one with $$u(.)$$.

$h_{k}=L_{k}(I)=G_{k}(I)-u\left(G_{k+1}(I)\right)=I_{k}-u\left(I_{k+1}\right)$

1. Reconstruction of the Laplacian pyramid coefficients $$[h_0;h_1; ...;h_K]$$ can be performed through backward recurrence as follows:

$I_k = u(I_{k+1} + h_k)$

### The sampling procedure for our LAPGAN

Following training (explained below), we have a set of generative convnet models $${G_0 , . . . , G_K }$$, each of which captures the distribution of coefficients $$h_k$$ for natural images at a different level of the Laplacian pyramid.