给出每一题的matlab代码

以下是每个任务对应的MATLAB代码示例：

### Task 1: Orthonormal Basis

#### (1) Description of Orthogonality
Orthogonality in linear algebra refers to the property of two vectors being perpendicular to each other, meaning their dot product is zero.

#### (2) Verification of Orthogonality

function is_orthogonal = verify_orthogonality(eigenfaces_blk)
    % Reshape eigenfaces to column vectors
    eigenfaces_vec = reshape(eigenfaces_blk, [], 100);
    
    % Compute the dot product matrix
    dot_product_matrix = eigenfaces_vec' * eigenfaces_vec;
    
    % Check if off-diagonal elements are close to zero
    is_orthogonal = isequal(dot_product_matrix, diag(diag(dot_product_matrix)));
end

#### (3) Orthonormal Basis and Normalization

function eigenfaces_blk_norm = normalize_eigenfaces(eigenfaces_blk)
    % Reshape eigenfaces to column vectors
    eigenfaces_vec = reshape(eigenfaces_blk, [], 100);
    
    % Normalize each eigenface
    for i = 1:100
        eigenfaces_vec(:, i) = eigenfaces_vec(:, i) / norm(eigenfaces_vec(:, i));
    end
    
    % Reshape back to original shape
    eigenfaces_blk_norm = reshape(eigenfaces_vec, 450, 300, 100);
end

### Task 2: Forward Transform

#### (1) Description of Weight Calculation
To get the weights on the orthogonal basis, we project the image onto each eigenface and compute the dot product.

#### (2) MATLAB Function to Generate Weights

function [weights_of_face] = get_face_weights(im, eigenfaces_blk)
    % Convert image to vector
    im_vec = im(:);
    
    % Reshape eigenfaces to column vectors
    eigenfaces_vec = reshape(eigenfaces_blk, [], 100);
    
    % Compute weights
    weights_of_face = eigenfaces_vec' * im_vec;
end

#### (3) Using the Function and Plotting Weights

% Load the image
im = imread('find_id.jpg');
im = rgb2gray(im);

% Get weights
weights_of_face = get_face_weights(double(im), eigenfaces_blk);

% Plot weights
figure;
plot(weights_of_face);
title('Weights of the Face');
xlabel('Eigenface Index');
ylabel('Weight');

### Task 3: Inverse Transform

#### (1) Description of Image Synthesis
To synthesize the original image from the weights, we multiply the weights by the eigenfaces and sum the results.

#### (2) MATLAB Function to Generate Face from Weights

function [im] = generate_face_from_weights(weights_of_face, eigenfaces_blk)
    % Reshape eigenfaces to column vectors
    eigenfaces_vec = reshape(eigenfaces_blk, [], 100);
    
    % Reconstruct the image
    im_vec = eigenfaces_vec * weights_of_face;
    
    % Reshape back to original shape
    im = reshape(im_vec, 450, 300);
end

#### (3) Synthesizing the Image

% Generate the face
reconstructed_im = generate_face_from_weights(weights_of_face, eigenfaces_blk);

% Display the reconstructed image
figure;
imshow(uint8(reconstructed_im));
title('Reconstructed Face');

#### (4) Comments on Number of Parameters
Using 100 parameters to describe a 450x300 image is efficient because eigenfaces capture the most significant features of the face. Fewer parameters result in lower quality reconstructions, while more parameters provide better accuracy.

#### (5) Comparison with 2D-DCT

% Perform 2D DCT
dct_coeffs = dct2(double(im));

% Zero out coefficients beyond the first 100
dct_coeffs(11:end, :) = 0;
dct_coeffs(:, 11:end) = 0;

% Reconstruct the image
reconstructed_dct_im = idct2(dct_coeffs);

% Display the reconstructed image
figure;
imshow(uint8(reconstructed_dct_im));
title('Reconstructed Face using 2D-DCT');

% Calculate PSNR
psnr_dct = psnr(double(im), double(reconstructed_dct_im));
disp(['PSNR using 2D-DCT: ', num2str(psnr_dct)]);

### Task 4: Employee Recognition

#### (1) MATLAB Function to Recognize Employee

function [ID] = get_employees_ID_from_DB(im, employees_DB, eigenfaces_blk)
    % Get weights of the input face
    weights_of_face = get_face_weights(double(im), eigenfaces_blk);
    
    % Initialize minimum distance and best match ID
    min_distance = inf;
    best_match_ID = -1;
    
    % Compare with each employee in the database
    for i = 1:size(employees_DB, 1)
        db_weights = employees_DB(i, 2:end);
        distance = norm(weights_of_face - db_weights);
        
        if distance < min_distance
            min_distance = distance;
            best_match_ID = employees_DB(i, 1);
        end
    end
    
    ID = best_match_ID;
end

#### (2) Finding the Employee's ID

% Load the image
im = imread('find_id.jpg');
im = rgb2gray(im);

% Get the employee's ID
employee_ID = get_employees_ID_from_DB(double(im), employees_DB, eigenfaces_blk);
disp(['Employee ID: ', num2str(employee_ID)]);

#### (3) Comments on Different Numbers of Eigenfaces
Using fewer eigenfaces may lead to misidentification due to insufficient feature representation. More eigenfaces generally improve accuracy but increase computational complexity.

#### (4) Improving Robustness
To improve robustness:
- **Noise Reduction**: Apply filters (e.g., Gaussian filter) to reduce noise.
- **Feature Normalization**: Normalize the input face to handle variations in lighting and pose.
- **Ensemble Methods**: Combine multiple recognition methods to enhance reliability.

#### (5) Using Your Own Face Photo
Replace `find_id.jpg` with your own face photo and repeat the steps above. Document the findings and suggest improvements based on observed issues.

These code snippets should help you complete the tasks outlined in the lab assignment. Make sure to test and validate each function thoroughly.