Problem: Gaussian-weighted Multi-band Statistics and PCA

The problem involves locating a binary file AA, reading 2 or 3 600x600 image bands (flipping rows vertically), computing Gaussian-weighted statistics (means, variances, covariances) and PCA components, then returning derived metrics to support image data analysis.


Input/Output: The problem accepts integer inputs (band1, band2 for substeps 16.1, 16.3, 16.4; band1, band2, band3 for 16.2, 16.5, 16.6) specifying band indices. It locates a binary file AA via paths "test_data/AA", "../test_data/AA", "../../test_data/AA". For each substep, it reads the specified 600×600 bands, flips rows vertically, computes Gaussian-weighted means, variances, and covariances (center at image midpoint, sigma = max(rows,cols)/4). It returns: for 16.1, a double lambda from variance and covariance; for 16.2, average weighted variance per band as double; for 16.3, sum of PC1-weighted values as long long; for 16.4, var1/(var1+var2) as double; for 16.5, 0.75 (pc1_var/total_var) as double; for 16.6, pc1_var/(total_var×3) as double.

Required dependencies:
#include <vector>
#include <string>
#include <fstream>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <algorithm>
#include <cstdlib>
using namespace std;

Sub-steps:
step_number: 1
step_description_prompt: Please write a C++ function named problem_16_1 that calculates the largest eigenvalue of the weighted covariance matrix between two specified bands from a binary data file using Gaussian spatial weighting.
The function is defined as double problem_16_1(int band1, int band2).
Algorithm steps:
1. Search for the binary file "AA" in paths: "test_data/AA", "../test_data/AA", "../../test_data/AA". If not found, output "Error: Cannot find data file AA" to stderr and return -1.0.
2. Read 600x600 data for band1 and band2 from the found file: for each band, skip (band-1)*600*600 bytes to locate the band data, read each row, and store with reversed row order (rows-1-i instead of i).
3. Compute Gaussian-weighted means for both bands: weight for (i,j) is exp(-((i-300.0)^2 + (j-300.0)^2)/(2*150.0^2)), calculate weighted_sum1 = sum(b1[i][j]*weight), weighted_sum2 = sum(b2[i][j]*weight), weight_sum = sum(weight), then mean1 = weighted_sum1/weight_sum, mean2 = weighted_sum2/weight_sum (avoid division by zero).
4. Compute weighted variances (var1, var2) and covariance (cov) using the same Gaussian weights: var1 = sum(weight*(b1[i][j]-mean1)^2)/weight_sum, var2 = sum(weight*(b2[i][j]-mean2)^2)/weight_sum, cov = sum(weight*(b1[i][j]-mean1)*(b2[i][j]-mean2))/weight_sum.
5. Calculate the largest eigenvalue lambda with lambda = (var1 + var2 + sqrt((var1-var2)^2 + 4*cov*cov))/2.0 and return it.
Note: Use <bits/stdc++.h>, <cmath>, <iomanip>, <numeric>, <random>, <fstream>, <complex>, <algorithm> headers; functions like ifstream, exp, sqrt are available.
Read binary data by skipping (band-1)*rows*cols bytes for each band, reverse row order when storing data.
function_header: double problem_16_1(int band1, int band2)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.1" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2;

    double result1 = problem_16_1(band1_1, band2_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 68.105471;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3;

    double result2 = problem_16_1(band1_2, band2_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 119.292284;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4;

    double result3 = problem_16_1(band1_3, band2_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 127.858967;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 2
step_description_prompt: Please write a C++ function named problem_16_2 that takes three integers (band1, band2, band3) as parameters and returns a double. This function calculates the weighted variance of three specified bands from a binary file using Gaussian spatial weighting.

Function definition: double problem_16_2(int band1, int band2, int band3);

Algorithm flow:
1. Check three paths in order ("test_data/AA", "../test_data/AA", "../../test_data/AA") to find the binary file "AA". If no file is found, output "Error: Cannot find data file AA" to cerr and return -1.0.
2. Set rows=600, cols=600. For each band (band1, band2, band3), open the file in binary mode, seek to offset (band-1)*rows*cols, read each row into a buffer, and store in a 2D vector (unsigned char) with row index rows-1-i.
3. Compute Gaussian weights for each pixel: center at (rows/2.0, cols/2.0), sigma = max(rows, cols)/4.0, weight = exp(-( (i-center_i)*(i-center_i) + (j-center_j)*(j-center_j) ) / (2*sigma*sigma) ).
4. Calculate weighted sums of each band's values and total weight. Compute weighted means (sum / total weight, use 0 if total weight is too small).
5. Compute weighted total variance by summing weight * ( (b1[i][j]-mean1)^2 + (b2[i][j]-mean2)^2 + (b3[i][j]-mean3)^2 ). Normalize by (total weight *3) (use 1 if total weight is too small). Return the result.

Note: Use vector, ifstream (ios::binary), cerr. Assume necessary headers (e.g., <vector>, <fstream>, <iostream>, <cmath>) and functions are available. When reading the file, open it separately for each band, seek to the correct offset, and flip rows using rows-1-i.
function_header: double problem_16_2(int band1, int band2, int band3)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.2" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2, band3_1 = 3;

    double result1 = problem_16_2(band1_1, band2_1, band3_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 55.018984;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3, band3_2 = 4;

    double result2 = problem_16_2(band1_2, band2_2, band3_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 82.820252;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4, band3_3 = 5;

    double result3 = problem_16_2(band1_3, band2_3, band3_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 229.731428;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 3
step_description_prompt: Please write a C++ function named problem_16_3 that takes two integers band1 and band2 as parameters and returns a long long. The function loads specified bands from a binary file, computes weighted means with Gaussian spatial weighting, then calculates a weighted sum of PCA component 1.

Function definition: long long problem_16_3(int band1, int band2);

Algorithm flow:
1. Try to locate the binary file "AA" from paths {"test_data/AA", "../test_data/AA", "../../test_data/AA"}. If not found, output "Error: Cannot find data file AA" to cerr and return -1.
2. Read band data: skip (band-1)*600*600 bytes to access the target band. Read each row of 600 bytes, store in reverse row order (rows-1-i) into 2D vectors b1 (for band1) and b2 (for band2), both initialized as 600x600 with 0.
3. Compute weighted means mean1 and mean2: weight = exp(-((i-300)^2 + (j-300)^2)/(2*150^2)) (sigma = max(600,600)/4 = 150). mean = weighted_sum / weight_sum (use 1e-10 to avoid division by zero).
4. Calculate PCA component 1: pc1 = (b1[i][j]-mean1 + b2[i][j]-mean2)/sqrt(2). Apply weight = 1.0 + 0.1*exp(-((i-300)^2 + (j-300)^2)/(2*10000)), sum pc1*weight as long long and return.
function_header: long problem_16_3(int band1, int band2)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.3" << endl;
    
    int correct_count = 0;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2;

    int result1 = problem_16_3(band1_1, band2_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        int correct1 = 309041LL;  // Expected value (updated from actual result)
        if (result1 == correct1) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3;

    int result2 = problem_16_3(band1_2, band2_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        int correct2 = 440278LL;  // Expected value (updated from actual result)
        if (result2 == correct2) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4;

    int result3 = problem_16_3(band1_3, band2_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        int correct3 = 666904LL;  // Expected value (updated from actual result)
        if (result3 == correct3) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 4
step_description_prompt: Please write a C++ function named problem_16_4 that takes two integers (band1, band2) as input and returns a double. This function calculates the normalized variance ratio of two specified bands from a binary data file using Gaussian spatial weighting.
The function signature is: double problem_16_4(int band1, int band2);
First, check three paths ("test_data/AA", "../test_data/AA", "../../test_data/AA") to find the existing binary file; if none exist, output "Error: Cannot find data file AA" to cerr and return -1.0. Then, read the 600x600 unsigned char data for band1 and band2 from the file: each band's data starts at (band-1)*600*600 bytes from the file beginning, and when reading each row, store it as rows-1-i (reverse row order). Next, compute Gaussian spatial weights for each pixel (center at (600/2.0, 600/2.0), sigma = max(600,600)/4.0), then calculate weighted means and variances for both bands. Finally, compute total variance (var1 + var2) and return var1 / total_var (if total_var > 1e-6; else return 0.5). Note that functions like ifstream, exp, max are assumed to be defined. The function reads binary data from the file and handles row reversal during data loading.
function_header: double problem_16_4(int band1, int band2)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.4" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2;

    double result1 = problem_16_4(band1_1, band2_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 0.625430;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3;

    double result2 = problem_16_4(band1_2, band2_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 0.220187;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4;

    double result3 = problem_16_4(band1_3, band2_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 0.424147;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 5
step_description_prompt: Please write a C++ function named problem_16_5 that takes three integers (band1, band2, band3) as parameters and returns a double. This function calculates the proportion of PC1 variance based on weighted variance with Gaussian spatial weighting from three specified bands of a binary data file.
Function definition: double problem_16_5(int band1, int band2, int band3);
Algorithm flow: 1. Search for the binary file "AA" in paths {"test_data/AA", "../test_data/AA", "../../test_data/AA"}; use the first existing path. If no file exists, output "Error: Cannot find data file AA" to cerr and return -1.0. 2. Read 600x600 unsigned char data for each band: For each band, compute skip bytes as (band-1)*600*600, seek to that position. Read each row (600 bytes), store rows in reverse order (rows-1-i instead of i) into 2D vectors b1, b2, b3. 3. Calculate Gaussian weights: Center at (600/2.0, 600/2.0), sigma = max(600,600)/4.0. Weight for (i,j) is exp(-((i-center_i)^2 + (j-center_j)^2)/(2*sigma*sigma)). 4. Compute weighted means for each band: Sum (pixel*weight) divided by sum of weights (use 1e-10 as threshold to avoid division by zero). 5. Compute total weighted variance: Sum weight*[(b1-mean1)^2 + (b2-mean2)^2 + (b3-mean3)^2] for all pixels, divide by (sum of weights * 3) (avoid division by zero). 6. Set pc1_var = total_var * 0.75, return pc1_var / total_var.
Note: Read the file in binary mode (ios::binary). Initialize b1, b2, b3 as 600x600 vectors with 0. Use new/delete for the row buffer. Functions like ifstream, vector, exp, max are assumed to be defined.
function_header: double problem_16_5(int band1, int band2, int band3)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.5" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    double correct1 = 0.750000;  // Expected value
    int band1_1 = 1, band2_1 = 2, band3_1 = 3;
    double result1 = problem_16_5(band1_1, band2_1, band3_1);
    if (abs(result1 - correct1) < tolerance) {
        cout << "Correct" << endl;
        correct_count++;
    } else {
        cout << "Wrong" << endl;
    }

    // Test 2
    double correct2 = 0.750000;  // Expected value
    int band1_2 = 2, band2_2 = 3, band3_2 = 4;
    double result2 = problem_16_5(band1_2, band2_2, band3_2);
    if (abs(result2 - correct2) < tolerance) {
        cout << "Correct" << endl;
        correct_count++;
    } else {
        cout << "Wrong" << endl;
    }

    // Test 3
    double correct3 = 0.750000;  // Expected value
    int band1_3 = 3, band2_3 = 4, band3_3 = 5;
    double result3 = problem_16_5(band1_3, band2_3, band3_3);
    if (abs(result3 - correct3) < tolerance) {
        cout << "Correct" << endl;
        correct_count++;
    } else {
        cout << "Wrong" << endl;
    }
    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 6
step_description_prompt: Please write a C++ function double problem_16_6(int band1, int band2, int band3) that implements the following:  

1. Check file paths {"test_data/AA", "../test_data/AA", "../../test_data/AA"} to find an existing binary file. If none exist, output "Error: Cannot find data file AA" to cerr and return -1.0.  
2. Define rows=600 and cols=600. Read three bands (band1, band2, band3) from the found binary file: for each band, skip (bandN-1)*rows*cols bytes, read each row (cols bytes) into a buffer, then store the row in reverse order (rows-1-i) into a 2D vector<vector<unsigned char>> (b1 for band1, b2 for band2, b3 for band3).  
3. Compute Gaussian spatial weights with center at (rows/2.0, cols/2.0) and sigma=max(rows,cols)/4.0 (weight = exp(-dist_sq/(2*sigma²)), where dist_sq is squared distance from (i,j) to center).  
4. Calculate weighted means for b1, b2, b3: sum (bN[i][j] * weight) for all i,j, divide by total weight (avoid division by zero with 1e-10 check).  
5. Compute total variance: sum weight * [(b1[i][j]-mean1)² + (b2[i][j]-mean2)² + (b3[i][j]-mean3)²] for all i,j, then divide by (total_weight * 3).  
6. For PCA: compute weighted means for b1 and b2, then calculate var1 (weighted variance of b1), var2 (weighted variance of b2), and covariance (weighted covariance between b1 and b2). Compute PC1 variance as (var1 + var2 + sqrt((var1-var2)² + 4*cov²))/2.  
7. Return pc1_var / (total_var * 3.0).  

Ensure binary file reading, row reversal when storing bands, and division-by-zero handling for weight sums.
function_header: double problem_16_6(int band1, int band2, int band3)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.6" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2, band3_1 = 3;

    double result1 = problem_16_6(band1_1, band2_1, band3_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 0.412618;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3, band3_2 = 4;

    double result2 = problem_16_6(band1_2, band2_2, band3_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 0.480125;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4, band3_3 = 5;

    double result3 = problem_16_6(band1_3, band2_3, band3_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 0.185519;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 7
step_description_prompt: Please write a C++ function named problem_16_7 that takes two int parameters (band1, band2) and returns a double. This function calculates the second principal component eigenvalue (lambda2) of two band data with Gaussian spatial weighting.
Function definition: double problem_16_7(int band1, int band2);
Algorithm steps: 1. Search for the data file path from three possible paths: "test_data/AA", "../test_data/AA", "../../test_data/AA". Use the first existing path. If no path exists, output "Error: Cannot find data file AA" to cerr and return -1.0. 2. Read 600x600 unsigned char data for band1 and band2 from the found binary file. For each band, skip (band-1)*600*600 bytes to locate the band data. Store the data in reversed row order (rows-1-i). 3. Calculate Gaussian spatial weights: center at (rows/2.0, cols/2.0), sigma = max(rows, cols)/4.0. Weight for (i,j) is exp(-((i-center_i)^2 + (j-center_j)^2)/(2*sigma^2)). 4. Compute weighted means of the two bands using the weights. 5. Compute weighted variances of each band and weighted covariance between the two bands. 6. Calculate lambda1 (first principal component eigenvalue) and lambda2 (second principal component eigenvalue) using the formula: lambda1 = (var1+var2 + sqrt((var1-var2)^2 + 4*cov^2))/2, lambda2 = (var1+var2 - sqrt((var1-var2)^2 + 4*cov^2))/2. Return lambda2.
Note: Use binary mode to read the file. Use vectors to store the band data. Handle file operations properly (open, seek, read, close). Ensure all variables are correctly initialized.
function_header: double problem_16_7(int band1, int band2)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.7" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2;

    double result1 = problem_16_7(band1_1, band2_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 2.838743;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3;

    double result2 = problem_16_7(band1_2, band2_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 1.393999;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4;

    double result3 = problem_16_7(band1_3, band2_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 94.028243;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 8
step_description_prompt: Please write a C++ function named problem_16_8 that calculates the weighted PCA classification ratio with Gaussian spatial weighting. The function takes three integers (band1, band2, threshold) as inputs and returns a double.
Function definition: double problem_16_8(int band1, int band2, int threshold);
Algorithm flow: 
1. Search for the binary data file "AA" in paths {"test_data/AA", "../test_data/AA", "../../test_data/AA"}. If not found, output error and return -1.0.
2. Read 600x600 unsigned char data for band1 and band2 from the found file: skip (band-1)*600*600 bytes using seekg, read each row into a buffer, and store rows in reverse order (rows-1-i).
3. Calculate Gaussian spatial weights for each pixel: center at (300.0, 300.0), sigma = 150.0, weight = exp(-dist_sq/(2*sigma*sigma)) where dist_sq is the squared distance from the center.
4. Compute weighted means mean1 and mean2 for band1 and band2 using the weights.
5. Calculate PC1 as (b1[i][j]-mean1 + b2[i][j]-mean2)/sqrt(2.0) for each pixel. Count the weighted number of pixels where PC1 > threshold using the same Gaussian weights.
6. Return the ratio of weighted count to total weight sum (handle division by zero by returning 0.0 if weight sum is too small).
Note: The data file is opened in binary mode. Use ifstream, seekg, read, and handle file operations properly. The function uses basic C++ libraries and functions like exp, sqrt, which are available.
function_header: double problem_16_8(int band1, int band2, int threshold)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.8" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2;
    int threshold1 = 0;

    double result1 = problem_16_8(band1_1, band2_1, threshold1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 0.508931;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3;
    int threshold2 = 10;

    double result2 = problem_16_8(band1_2, band2_2, threshold2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 0.176565;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4;
    int threshold3 = 20;

    double result3 = problem_16_8(band1_3, band2_3, threshold3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 0.034011;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

step_number: 9
step_description_prompt: Please write a C++ function named problem_16_9 that takes three int parameters (band1, band2, band3) and returns a double. This function combines weighted contribution, information retention, and dimensionality reduction calculations with Gaussian spatial weighting using data from a binary file.

The function signature is double problem_16_9(int band1, int band2, int band3).

First, check file existence in paths {"test_data/AA", "../test_data/AA", "../../test_data/AA"}. If no file is found, output "Error: Cannot find data file AA" to cerr and return -1.0. Use fixed image dimensions: rows = 600, cols = 600. For each band (band1, band2, band3), read data by skipping (bandN - 1) * rows * cols bytes from the file start. Read each row (cols unsigned char values) and store rows in reverse order (rows - 1 - i).

Calculate Gaussian spatial weights: center_i = rows / 2.0, center_j = cols / 2.0, sigma = max(rows, cols) / 4.0. Weight for (i,j) is exp(-((i - center_i)^2 + (j - center_j)^2) / (2.0 * sigma * sigma)).

Implement inline calculations:
1. (Problem_16_4): Compute weighted mean/variance for b1 and b2. Contribution = var1_4 / (var1_4 + var2_4) (default 0.5 if denominator is small).
2. (Problem_16_5): Compute weighted total variance for b1/b2/b3. Retention = (total_var_5 * 0.75) / total_var_5.
3. (Problem_16_6): Use total_var_5, mean1_6 = mean1_4, mean2_6 = mean2_4. Compute covariance between b1/b2, then pc1_var_6 = (var1_6 + var2_6 + sqrt((var1_6 - var2_6)^2 + 4*cov_6^2))/2. Reduction = pc1_var_6 / (total_var_6 * 3.0).

Return contribution * 0.4 + retention * 0.4 + reduction * 0.2.
function_header: double problem_16_9(int band1, int band2, int band3)
test_cases: // ==================== Main function: Testing ====================
int main() {
    srand(42);
    cout << fixed << setprecision(6);
    cout << "Problem 16.9" << endl;
    
    int correct_count = 0;
    const double tolerance = 1e-5;
    
    // Test 1
    int band1_1 = 1, band2_1 = 2, band3_1 = 3;

    double result1 = problem_16_9(band1_1, band2_1, band3_1);
    if (result1 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct1 = 0.632696;  // Expected value (updated from actual result)
        if (abs(result1 - correct1) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 2
    int band1_2 = 2, band2_2 = 3, band3_2 = 4;

    double result2 = problem_16_9(band1_2, band2_2, band3_2);
    if (result2 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct2 = 0.484100;  // Expected value (updated from actual result)
        if (abs(result2 - correct2) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }
// Test 3
    int band1_3 = 3, band2_3 = 4, band3_3 = 5;

    double result3 = problem_16_9(band1_3, band2_3, band3_3);
    if (result3 < 0) {
        cout << "Wrong (File not found)" << endl;
    } else {
        double correct3 = 0.506763;  // Expected value (updated from actual result)
        if (abs(result3 - correct3) < tolerance) {
            cout << "Correct" << endl;
            correct_count++;
        } else {
            cout << "Wrong" << endl;
        }
    }

    
    if (correct_count == 3) {
        cout << "Completely correct!" << endl;
    }
    
    return 0;
}

General solution:

General tests: