Data 605 HW1
Angel Gallardo
2024-10-05
1. Geometric Transformation of Shapes Using Matrix Multiplication
# triangle vertices
triangle <- matrix(c(0,0,
1,0,
.5,1),
ncol = 2, byrow = TRUE)
plot(triangle, type= "n", xlim= c(-2,2), xlab = "X", ylab= "Y", asp=1)
polygon(triangle, col="red", border = "black")
points(triangle, pch = 19)
text(triangle, labels = c("A", "B", "C"), pos = 3)
Scaling
triangle <- matrix(c(0, 0,
2, 0,
1, 2),
ncol = 2, byrow = TRUE)
plot(NULL, xlim = c(-3, 3), ylim = c(-3, 3),
xlab = "X-axis", ylab = "Y-axis", main = "Incrementally Scaled Triangle")
# Loop through specified scale values
scale_values <- c(1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.25, 2.0, 1.75, 1.5, 1.25, 1.0, 0.75, 0.5, 0.75)
for (val in scale_values) {
trans_matrix <- matrix(c(val, 0,
0, val), nrow = 2, ncol = 2)
scaled_triangle <- triangle %*% t(trans_matrix)
polygon(scaled_triangle, col = rgb(0.5, 0.5, 1, alpha = 0.5), border = "black")
}
Rotate
triangle <- matrix(c(0, 0,
2, 0,
1, 2),
ncol = 2, byrow = TRUE)
plot(NULL, xlim = c(-3, 3), ylim = c(-3, 3),
xlab = "X-axis", ylab = "Y-axis", main = "Incrementally Rotated Triangle")
rotation_angles <- seq(0, 2 * pi, length.out = 12)
for (theta in rotation_angles) {
rotation_matrix <- matrix(c(cos(theta), -sin(theta),
sin(theta), cos(theta)), ncol = 2)
rotated_triangle <- triangle %*% t(rotation_matrix)
polygon(rotated_triangle, col = rgb(0.5, 0.5, 1, alpha = 0.5), border = "black")
}
Reflection
triangle <- matrix(c(0, 0,
2, 0,
1, 2),
ncol = 2, byrow = TRUE)
plot(NULL, xlim = c(-3, 3), ylim = c(-3, 3),
xlab = "X-axis", ylab = "Y-axis", main = "Incrementally Reflected Triangle")
reflection_matrices <- list(
matrix(c(1, 0, # Reflection across X-axis
0, -1), ncol = 2),
matrix(c(-1, 0, # Reflection across Y-axis
0, 1), ncol = 2),
matrix(c(0, 1, # Reflection across Y = X
1, 0), ncol = 2),
matrix(c(0, -1, # Reflection across Y = -X
-1, 0), ncol = 2)
)
for (reflection_matrix in reflection_matrices) {
reflected_triangle <- triangle %*% t(reflection_matrix)
polygon(reflected_triangle, col = rgb(1, 0.5, 0.5, alpha = 0.5), border = "black")
}
2. Matrix Properties and Decomposition
Prove AB =/ BA
## [1] "Matrix A:"## [,1] [,2]
## [1,] 1 2
## [2,] 3 4## [1] "Matrix B:"## [,1] [,2]
## [1,] 0 1
## [2,] 1 0## [1] "AB (A * B):"## [,1] [,2]
## [1,] 2 1
## [2,] 4 3## [1] "BA (B * A):"## [,1] [,2]
## [1,] 3 4
## [2,] 1 2## [1] "Are AB and BA identical? FALSE"Prove that A^TA is symmetric
##
## A^T * A:## [,1] [,2]
## [1,] 10 14
## [2,] 14 20##
## Is A^T * A symmetric? TRUEProve that A^T A is non-negative
##
## Determinant of A^T * A: 4## Is the determinant non-negative? TRUESingular Value Decomposition (SVD) and image compression
3. Matrix Rank, Properties, and eigenspace
Determine the Rank of Matrix A
A <- matrix(c(2, 4, 1, 3,
-2, -3, 4, 1,
5, 6, 2, 8,
-1, -2, 3, 7), nrow = 4, byrow = TRUE)
# Find the rank of matrix A
rank_A <- qr(A)$rank
cat("Rank of matrix A: ", rank_A, "\n")## Rank of matrix A: 4Determine the Rank of Matrix B Using Row Reduction
## Rank of matrix B: 1Compute the Eigenvalues and Eigenvectors of Matrix A
A_eigen <- matrix(c(3, 1, 2,
0, 5, 4,
0, 0, 2), nrow = 3, byrow = TRUE)
# Compute eigenvalues and eigenvectors
eigen_A <- eigen(A_eigen)
# Print eigenvalues and eigenvectors
cat("Eigenvalues of A:\n")## Eigenvalues of A:print(eigen_A$values)## [1] 5 3 2cat("\nEigenvectors of A:\n")##
## Eigenvectors of A:print(eigen_A$vectors)## [,1] [,2] [,3]
## [1,] 0.4472136 1 -0.3713907
## [2,] 0.8944272 0 -0.7427814
## [3,] 0.0000000 0 0.5570860Check Linear Independence:
eigenvectors <- eigen_A$vectors
det_eigenvectors <- det(eigenvectors)
cat("\nDeterminant of the matrix of eigenvectors: ", det_eigenvectors, "\n")##
## Determinant of the matrix of eigenvectors: -0.4982729# If determinant is non-zero, the eigenvectors are linearly independent
cat("Are the eigenvectors linearly independent? ", det_eigenvectors != 0, "\n")## Are the eigenvectors linearly independent? TRUEDiagonalization of Matrix A
D <- diag(eigen_A$values)
# Matrix of eigenvectors
P <- eigen_A$vectors
A_diag <- P %*% D %*% solve(P)
cat("\nDiagonalized matrix A:\n")##
## Diagonalized matrix A:print(A_diag)## [,1] [,2] [,3]
## [1,] 3 1 2
## [2,] 0 5 4
## [3,] 0 0 2equality_check <- all.equal(A_diag, A)
if (isTRUE(equality_check)) {
cat("\nA_diag is approximately equal to A.\n")
} else {
cat("\nA_diag is not equal to A: ", equality_check, "\n")
}##
## A_diag is not equal to A: Attributes: < Component "dim": Mean relative difference: 0.3333333 > Numeric: lengths (9, 16) differ4. Project: Eigenfaces from the LFW (Labeled Faces in the Wild) Dataset
Download the LFW Datase, Preprocess the Images:
resize <- function(img, width, height) {
img_resized <- imager::resize(img, size_x = width, size_y = height)
return(as.matrix(img_resized))
}
preprocess_images <- function(directory, target_size = c(64, 64), max_images = 100) {
image_files <- list.files(directory, pattern = "\\.jpg$", full.names = TRUE, recursive = TRUE)
image_files <- head(image_files, max_images)
images <- list()
for (file in image_files) {
img <- readJPEG(file)
if (length(dim(img)) == 3 && dim(img)[3] == 3) {
gray_img <- 0.2989 * img[,,1] + 0.5870 * img[,,2] + 0.1140 * img[,,3]
} else if (length(dim(img)) == 2) {
gray_img <- img
} else {
stop(paste("Image format not supported for file:", file))
}
resized_img <- resize(as.cimg(gray_img), target_size[1], target_size[2])
images[[file]] <- as.vector(resized_img)
}
return(do.call(rbind, images))
}
image_matrix <- preprocess_images("C:/Users/angel/OneDrive/Desktop/lfw")Apply PCA:
# Perform PCA
if (is.null(image_matrix) || nrow(image_matrix) == 0) {
stop("image_matrix is NULL or empty. Please check your preprocessing step.")
}
pca_result <- prcomp(image_matrix, center = TRUE, scale. = TRUE)
# Determine the variance explained by each principal component
explained_variance <- pca_result$sdev^2 / sum(pca_result$sdev^2)
cumulative_variance <- cumsum(explained_variance)
# Find the number of components that explain 80% of the variance
num_components <- which(cumulative_variance >= 0.80)[1]
cat("Number of components for 80% variance:", num_components, "\n")## Number of components for 80% variance: 26visualize Eigenfaces
plot_eigenfaces <- function(pca_result, num_faces = 10) {
par(mfrow = c(2, num_faces / 2))
for (i in 1:num_faces) {
eigenface <- matrix(pca_result$rotation[, i], nrow = 64)
image(eigenface, axes = FALSE, col = gray.colors(256))
}
}
plot_eigenfaces(pca_result, num_faces = 10)
Reconstruct Images Using Eigenfaces
reconstructed_images <- pca_result$x[, 1:num_components] %*% t(pca_result$rotation[, 1:num_components])
plot_reconstructed_images <- function(reconstructed_images, num_faces = 10, image_size = 64) {
par(mfrow = c(2, num_faces / 2), mar = c(1, 1, 1, 1))
for (i in 1:num_faces) {
reconstructed_image <- matrix(reconstructed_images[i, ], nrow = image_size) # Reshape to 64x64
image(reconstructed_image, axes = FALSE, col = gray.colors(256))
title(main = paste("Reconstructed Image", i))
}
}
plot_reconstructed_images(reconstructed_images, num_faces = 10, image_size = 64)