# GAN Training Acceleration Using Frechet Descriptor-Based Coreset
## 3 Methods
### BlobGAN and Blob Descriptors  
BlobGAN (Figure 1) consists of three steps: (1) 3D synthetic blobs are rendered using a 3D elliptical Gaussian function. The 3D blobs with respect to the corresponding masks and true images comprise the training input; (2) a 3D GAN is trained to denoise the images; (3) denoised 3D blobs are identified from the denoised images. The goal of 3D blob synthesis in Step 1 is to mimic the blobs (e.g., glomeruli in [30]) in real noisy blob images and provides the blob mask to fix the location of blobs. The image denoising in Step 2 adopts the domain translation architecture from CycleGAN [4], which consists of two generators and two discriminators. Given the 3D clean synthetic blob images as source domain and the 3D real noisy blob images as target domain, BlobGAN is to train a generator to learn the mapping from the source domain to the target domain and train another generator to learn the mapping from the target domain to the source domain iteratively, until the translated 3D noisy fake blob images approximate the 3D real noisy blob images and the translated 3D denoised blob images approximate the 3D clean bob images. The two discriminators are maintained to distinguish real images and translated images obtained from generators. The total loss of BlobGAN consists of adversarial loss, cycle consistency loss and identity loss adopted from CycleGAN [4] and an additional convexity consistency loss proposed by BlobGAN. Once image denoising is completed, Hessian convexity mask and blob mask generated from denoised blob images are used to derive the final blobs under a joint constraint.  
Here, we apply the elliptical Gaussian function to identify the blob descriptors. A 3D elliptical Gaussian function is in the general form as follows:  
\[F(x,y,z)=\mathcal{A}\cdot e^{-(a(x-x_{0})^{2}+b(y-y_{0})^{2}+c(z-z_{0})^{2}+d(x-x _{0})(y-y_{0})+e(y-y_{0})(z-z_{0})+f(x-x_{0})(z-z_{0}))} \tag{1}\]  
where \(\mathcal{A}\) is a normalization factor, and \(x_{0}\), \(y_{0}\) and \(z_{0}\) are the coordinates of the center of the Gaussian function \(F(x,y,z)\). The coefficients \(a,\ b,\ c,\ d,\ e\) and \(f\) control the shape and orientation of \(F(x,y,z)\) via \(\theta\), \(\varphi\), \(\alpha_{x}\), \(\varphi_{y}\) and \(\sigma_{z}\) as given:  
\[\begin{gathered} a=\frac{sin^{2}\theta cos^{2}\varphi}{\sigma_{z} ^{2}}+\frac{sin^{2}\theta sin^{2}\varphi}{\sigma_{y}^{2}}+\frac{cos^{2}\theta }{\sigma_{z}^{2}}\\ b=\frac{cos^{2}\theta cos^{2}\varphi}{\sigma_{z}^{2}}+\frac{cos^{2} \theta sin^{2}\varphi}{\sigma_{y}^{2}}+\frac{sin^{2}\theta}{\sigma_{z}^{2}} \\ c=\frac{sin^{2}\varphi}{\sigma_{x}^{2}}+\frac{cos^{2}\varphi}{ \sigma_{y}^{2}}d=\frac{sin^{2}\theta cos^{2}\varphi}{\sigma_{x}^{2}}+\frac{ sin2\theta sin^{2}\varphi}{\sigma_{y}^{2}}-\frac{sin^{2}\theta}{\sigma_{z}^{2}} \\ e=-\frac{cos\theta sin^{2}\varphi}{\sigma_{x}^{2}}+\frac{cos \theta sin^{2}\varphi}{\sigma_{y}^{2}}\\ f=-\frac{sinsin^{2}\varphi}{\sigma_{x}^{2}}+\frac{sin\theta sin ^{2}\varphi}{\sigma_{y}^{2}}\end{gathered} \tag{2}\]  
Based on the 5 descriptors \(\theta\), \(\varphi\), \(\sigma_{x}\), \(\sigma_{y}\) and \(\sigma_{z}\) from blob distribution, an FDD distance metric is proposed to measure the difference between each pair of images.