---
title: "exam2"
author: "Mark Pearl"
date: "7/28/2021"
output: html_document
---

```{r setup, include=FALSE}
library(R.matlab) 
library(OpenImageR)
library(pracma)
library(Rmpfr)
```

```{r original image}
D <- rgb_2gray(readImage('image.jpg'))*255
plot(imageShow(D))
```

```{r robust pca, include=TRUE}
lambda <- 1e-2
m <- dim(D)[1]
n <- dim(D)[2]
tol <- 1e-7
maxIter <- 1000
# Initialize A,E,Y,u
Y <- D
norm_two <- norm(Y,"F")
norm_inf <- Norm(as.vector(Y), Inf) / lambda
Y <- Y / norm_inf
A_hat <- matrix(0,m, n)
E_hat <- matrix(0,m, n)
mu <- 1.25/norm_two 
mu_bar <- mu * 1e7 
rho <- 1.5         # this one can be tuned
d_norm <- norm(D, "F")
iter <- 0 
total_svd <- 0
converged <- FALSE 
stopCriterion <- 1

while(!converged){
  iter <- iter + 1
  iter
  temp_T <- D - A_hat + (1/mu)*Y
  E_hat <- pmax(temp_T - lambda/mu, 0)
  E_hat <- E_hat+pmin(temp_T + lambda/mu, 0)
  decomp <- svd(D - E_hat + (1/mu)*Y)
  U <- decomp$u
  S <- decomp$d
  V <- decomp$v
  S_t <- S- 1/mu
  S_t[S_t<0] <- 0
  A_hat <- U %*% diag(S_t) %*% t(V)
  total_svd <- total_svd+1
  Z <- D - A_hat - E_hat
  Y <- Y + mu*Z
  mu <- min(mu*rho, mu_bar)
  # stop Criterion
  stopCriterion <- norm(Z, "F") / d_norm
  if (stopCriterion < tol){
    converged <- TRUE
  }
}
```

```{r show images, include=TRUE}
plot(imageShow(A_hat))
```

```{r show images}
plot(imageShow(E_hat))
```