Author: Yuzhou
Date: 2026-01-28

# BioGSP: Spectral Graph Wavelet Transform for Spatial Data (New Workflow)

#### Yuzhou

#### 2026-01-28

# Introduction

BioGSP provides Spectral Graph Wavelet Transform (SGWT) analysis for spatial biological data. This package enables multi-scale analysis of spatial patterns using graph signal processing techniques.

## Key Capabilities

- Object-oriented SGWT workflow with clear separation of concerns
- Multi-scale spatial analysis using spectral graph wavelets
- Flexible data input with custom column naming
- Energy-normalized similarity analysis with comprehensive metrics
- Multiple wavelet kernels (Mexican Hat, Meyer, Heat)
- Comprehensive visualization tools
- Detailed inverse transform results (low-pass, band-pass approximations)

## New Workflow Overview

The new workflow consists of five main steps:

1. `initSGWT()`: Initialize SGWT object with data and parameters
2. `checkKband()`: Analyze k-band limited property of signals (optional)
3. `runSpecGraph()`: Build spectral graph structure
4. `runSGWT()`: Perform forward and inverse SGWT transforms
5. `runSGCC()`: Calculate energy-normalized weighted similarity

# Setup

    # Load required packages
    library(BioGSP)
    library(ggplot2)
    library(patchwork)
    library(viridis)

    # Set random seed for reproducibility
    set.seed(123)

# New SGWT Workflow Demonstration

## Generate Example Pattern

    # Create a spatial pattern with concentric circles
    demo_pattern <- simulate_multiscale(
      grid_size = 20,        # Moderate size for fast computation
      n_centers = 1,         # Single center pattern
      Ra_seq = 5,            # Inner circle radius
      n_steps = 1,           # Single step (one pattern)
      outer_start = 10,      # Outer ring radius
      seed = 123
    )[[1]]

    ## Generating 1 patterns...
    ##   |                                                                              |                                                                      |   0%  |                                                                              |======================================================================| 100%
    ##
    ## Multicenter multiscale simulation completed!

    # Display pattern structure
    cat("Generated pattern dimensions:", nrow(demo_pattern), "x", ncol(demo_pattern), "\n")

    ## Generated pattern dimensions: 400 x 4

    cat("Column names:", paste(colnames(demo_pattern), collapse = ", "), "\n")

    ## Column names: X, Y, signal_1, signal_2

    cat("Signals available:", sum(demo_pattern$signal_1), "inner pixels,", sum(demo_pattern$signal_2), "outer pixels\n")

    ## Signals available: 81 inner pixels, 234 outer pixels

## Visualize Input Pattern

    # Single combined visualization (one plot for this section)
    demo_pattern$pattern_type <- ifelse(
      demo_pattern$signal_1 == 1,
      "Inner Circle",
      ifelse(demo_pattern$signal_2 == 1, "Outer Ring", "Background")
    )

    p_input <- ggplot(demo_pattern, aes(X, Y, fill = pattern_type)) +
      geom_tile() +
      scale_fill_manual(
        values = c("Background" = "white", "Inner Circle" = "#e31a1c", "Outer Ring" = "#1f78b4"),
        name = "Pattern"
      ) +
      coord_fixed() +
      theme_void() +
      ggtitle("Input Pattern (Signal 1 + Signal 2)")

    print(p_input)

## Step 1: Initialize SGWT Object

    # Initialize SGWT object with custom column names
    SG <- initSGWT(
      data.in = demo_pattern,
      x_col = "X",           # Custom X coordinate column name
      y_col = "Y",           # Custom Y coordinate column name
      signals = c("signal_1", "signal_2"),  # Analyze both signals
      J = 2,                 # Number of wavelet scales
      scaling_factor = 1,    # Scale progression factor
      kernel_type = "mexican_hat"
    )

    # Display initialized object
    print(SG)

    ## SGWT Object
    ## ===========
    ## Data:
    ##   Dimensions: 400 x 5
    ##   Coordinates: X , Y
    ##   Signals: signal_1, signal_2
    ##
    ## Parameters:
    ##   k (neighbors): mexican_hat
    ##   J (scales): 2
    ##   Kernel type: mexican_hat
    ##   Laplacian type:
    ##
    ## Status:
    ##   Graph computed: FALSE
    ##   Forward computed: FALSE
    ##   Inverse computed: FALSE

## Step 2: Build Spectral Graph

    # Build spectral graph structure
    SG <- runSpecGraph(SG, k = 12, laplacian_type = "normalized", length_eigenvalue = 200, verbose = TRUE)

    ## Building graph from spatial coordinates...
    ## Computing Laplacian and eigendecomposition...
    ## Auto-generated scales: 1.0188, 1.0188
    ## Graph construction completed.

    # Check updated object
    cat("Graph construction completed!\n")

    ## Graph construction completed!

    cat("Adjacency matrix dimensions:", dim(SG$Graph$adjacency_matrix), "\n")

    ## Adjacency matrix dimensions: 400 400

    cat("Number of eigenvalues computed:", length(SG$Graph$eigenvalues), "\n")

    ## Number of eigenvalues computed: 200

## Visualize Fourier Modes (Eigenvectors)

    # Visualize Fourier modes (eigenvectors) - 5 low and 5 high frequency modes
    fourier_modes <- plot_FM(SG, mode_type = "both", n_modes = 2, ncol = 2)

    cat("Fourier Modes Visualization:\n")

    ## Fourier Modes Visualization:

    cat("- Low-frequency modes (2-6): Smooth spatial patterns, excluding DC component\n")

    ## - Low-frequency modes (2-6): Smooth spatial patterns, excluding DC component

    cat("- High-frequency modes (96-100): Fine-detailed spatial patterns\n")

    ## - High-frequency modes (96-100): Fine-detailed spatial patterns

    cat("- Each mode shows its eigenvalue (λ) indicating frequency content\n")

    ## - Each mode shows its eigenvalue (λ) indicating frequency content

## Step 3: Run SGWT Analysis

    # Perform SGWT forward and inverse transforms
    SG <- runSGWT(SG, verbose = TRUE)

    ## Performing SGWT analysis for 2 signals...
    ## Using batch processing for efficiency...
    ## SGWT analysis completed.

    # Display final object with all results
    print(SG)

    ## SGWT Object
    ## ===========
    ## Data:
    ##   Dimensions: 400 x 5
    ##   Coordinates: X , Y
    ##   Signals: signal_1, signal_2
    ##
    ## Parameters:
    ##   k (neighbors): mexican_hat
    ##   J (scales): 2
    ##   Kernel type: mexican_hat
    ##   Laplacian type:
    ##   Scales: 1.0188, 1.0188
    ##
    ## Status:
    ##   Graph computed: TRUE
    ##   Forward computed: TRUE
    ##   Inverse computed: TRUE
    ##
    ## Reconstruction Errors:
    ##    signal_1 : 0.129674
    ##    signal_2 : 0.234629

## Visualize SGWT Decomposition

    # Plot SGWT decomposition results for signal_1
    decomp_plots <- plot_sgwt_decomposition(
      SG = SG,
      signal_name = "signal_1",
      plot_scales = 1,       # One plot for this section
      ncol = 1
    )

    print(decomp_plots)

    ## TableGrob (4 x 1) "arrange": 4 grobs
    ##               z     cells    name           grob
    ## original      1 (1-1,1-1) arrange gtable[layout]
    ## scaling       2 (2-2,1-1) arrange gtable[layout]
    ## wavelet_1     3 (3-3,1-1) arrange gtable[layout]
    ## reconstructed 4 (4-4,1-1) arrange gtable[layout]

## Energy Analysis

    # Analyze energy distribution across scales for signal_1
    energy_analysis <- sgwt_energy_analysis(SG, "signal_1")
    print(energy_analysis)

    ##       scale    energy energy_ratio scale_value   signal
    ## 1  low_pass 57.784192   0.95227414    1.018772 signal_1
    ## 2 wavelet_1  1.448008   0.02386293    1.018772 signal_1
    ## 3 wavelet_2  1.448008   0.02386293    1.018772 signal_1

    # Create energy distribution plot
    energy_plot <- ggplot(energy_analysis, aes(x = scale, y = energy_ratio)) +
      geom_bar(stat = "identity", fill = "steelblue", alpha = 0.7) +
      geom_text(aes(label = paste0(round(energy_ratio * 100, 1), "%")),
                vjust = -0.5, size = 3) +
      labs(title = "Energy Distribution Across SGWT Scales (Signal 1)",
           x = "Scale", y = "Energy Ratio") +
      theme_minimal() +
      theme(axis.text.x = element_text(angle = 45, hjust = 1))

    print(energy_plot)

# Advanced Features Demonstration

## Different Wavelet Kernels

    # Compare different kernel types
    kernels <- c("mexican_hat", "meyer", "heat")
    kernel_results <- list()

    for (kn in kernels) {
      # Initialize with different kernel
      SG_temp <- initSGWT(
        data.in = demo_pattern,
        x_col = "X", y_col = "Y",
        signals = "signal_1",
        J = 3,                    # Reduced for faster computation
        kernel_type = kn
      )

      # Run full workflow
      SG_temp <- runSpecGraph(SG_temp, k = 12, laplacian_type = "normalized", length_eigenvalue = 100,verbose = FALSE)
      SG_temp <- runSGWT(SG_temp, verbose = FALSE)

      kernel_results[[kn]] <- SG_temp
    }

    # Compare reconstruction errors
    comparison_df <- data.frame(
      Kernel = kernels,
      RMSE = sapply(kernel_results, function(x) x$Inverse$signal_1$reconstruction_error),
      stringsAsFactors = FALSE
    )

    print("Kernel Performance Comparison:")

    ## [1] "Kernel Performance Comparison:"

    print(comparison_df)

    ##                  Kernel      RMSE
    ## mexican_hat mexican_hat 0.2113782
    ## meyer             meyer 0.2019181
    ## heat               heat 0.1928501

    # Plot comparison
    ggplot(comparison_df, aes(x = Kernel, y = RMSE, fill = Kernel)) +
      geom_bar(stat = "identity", alpha = 0.7) +
      geom_text(aes(label = round(RMSE, 6)), vjust = -0.5) +
      scale_fill_viridis_d() +
      labs(title = "SGWT Reconstruction Error by Kernel Type",
           x = "Kernel Type", y = "RMSE") +
      theme_minimal()

## Step 4: Pattern Similarity Analysis

    # Calculate similarity between signal_1 and signal_2 in same object
    similarity_within <- runSGCC("signal_1", "signal_2", SG = SG, return_parts = TRUE)

    cat("Pattern Similarity Analysis (within same graph):\n")

    ## Pattern Similarity Analysis (within same graph):

    cat(sprintf("Overall similarity: %.4f\n", similarity_within$S))

    ## Overall similarity: -0.5930

    cat(sprintf("Low-frequency similarity: %.4f\n", similarity_within$c_low))

    ## Low-frequency similarity: -0.5996

    cat(sprintf("Non-low-frequency similarity: %.4f\n", similarity_within$c_nonlow))

    ## Non-low-frequency similarity: -0.5040

    cat(sprintf("Energy weights - Low: %.3f, Non-low: %.3f\n",
               similarity_within$w_low, similarity_within$w_NL))

    ## Energy weights - Low: 0.932, Non-low: 0.068

    # Generate a second pattern for cross-comparison
    pattern_2 <- simulate_multiscale(
      grid_size = 40,
      n_centers = 1,
      Ra_seq = 8,            # Different inner radius
      n_steps = 1,           # Single step
      outer_start = 15,      # Different outer radius
      seed = 456
    )[[1]]

    ## Generating 1 patterns...
    ##   |                                                                              |                                                                      |   0%  |                                                                              |======================================================================| 100%
    ##
    ## Multicenter multiscale simulation completed!

    # Create second SGWT object
    SG2 <- initSGWT(pattern_2, x_col = "X", y_col = "Y", signals = "signal_1",
                    J = 4)
    SG2 <- runSpecGraph(SG2, k = 12, laplacian_type = "normalized",length_eigenvalue = 100, verbose = FALSE)
    SG2 <- runSGWT(SG2, verbose = FALSE)

    # Note: Cross-object similarity comparison removed due to different eigenvalue counts
    # between SG (200 eigenvalues) and SG2 (100 eigenvalues) causing dimension mismatch

    # Visualize both patterns for comparison (single faceted plot for this section)
    pattern_compare <- rbind(
      data.frame(
        X = demo_pattern$X,
        Y = demo_pattern$Y,
        signal_1 = demo_pattern$signal_1,
        Pattern = "Pattern A (R_in=5, R_out=10)"
      ),
      data.frame(
        X = pattern_2$X,
        Y = pattern_2$Y,
        signal_1 = pattern_2$signal_1,
        Pattern = "Pattern B (R_in=8, R_out=15)"
      )
    )

    pattern_compare$Signal <- ifelse(pattern_compare$signal_1 == 1, "Signal", "Background")

    ggplot(pattern_compare, aes(X, Y, fill = Signal)) +
      geom_tile() +
      scale_fill_manual(values = c("Background" = "white", "Signal" = "#377eb8")) +
      coord_fixed() +
      facet_wrap(~ Pattern) +
      theme_void() +
      theme(legend.position = "none") +
      ggtitle("Pattern Comparison (signal_1)")

## Low-frequency Only Analysis

    # Compare using only low-frequency components
    similarity_low <- runSGCC("signal_1", "signal_2", SG = SG, low_only = TRUE, return_parts = TRUE)

    cat("Low-frequency Only Similarity Analysis:\n")

    ## Low-frequency Only Similarity Analysis:

    cat(sprintf("Low-frequency similarity: %.4f\n", similarity_low$c_low))

    ## Low-frequency similarity: -0.5996

    cat(sprintf("Overall score (same as low-freq): %.4f\n", similarity_low$S))

    ## Overall score (same as low-freq): -0.5996

    cat("Note: Non-low-frequency components are ignored (c_nonlow = NA)\n")

    ## Note: Non-low-frequency components are ignored (c_nonlow = NA)

    # Compare with full analysis
    cat("\nComparison:\n")

    ##
    ## Comparison:

    cat(sprintf("Full analysis similarity: %.4f\n", similarity_within$S))

    ## Full analysis similarity: -0.5930

    cat(sprintf("Low-only similarity: %.4f\n", similarity_low$S))

    ## Low-only similarity: -0.5996

    cat(sprintf("Difference: %.4f\n", abs(similarity_within$S - similarity_low$S)))

    ## Difference: 0.0065

## Multiple Signal Analysis

    # Analyze energy distribution for both signals
    energy_signal1 <- sgwt_energy_analysis(SG, "signal_1")
    energy_signal2 <- sgwt_energy_analysis(SG, "signal_2")

    # Combine for comparison
    energy_comparison <- rbind(energy_signal1, energy_signal2)

    # Plot comparison
    ggplot(energy_comparison, aes(x = scale, y = energy_ratio, fill = signal)) +
      geom_bar(stat = "identity", position = "dodge", alpha = 0.7) +
      geom_text(aes(label = paste0(round(energy_ratio * 100, 1), "%")),
                position = position_dodge(width = 0.9), vjust = -0.5, size = 3) +
      scale_fill_viridis_d() +
      labs(title = "Energy Distribution Comparison: Signal 1 vs Signal 2",
           x = "Scale", y = "Energy Ratio", fill = "Signal") +
      theme_minimal() +
      theme(axis.text.x = element_text(angle = 45, hjust = 1))

## Similarity Space Visualization with Multiple Patterns

    # Generate 9 paired patterns using simulate_multiscale
    cat("Generating 9 paired patterns for similarity analysis...\n")

    ## Generating 9 paired patterns for similarity analysis...

    patterns_9 <- simulate_multiscale(
      grid_size = 40,
      n_centers = 1,
      Ra_seq = c(3, 6, 9),     # Inner circle radii
      n_steps = 3,             # 3 shrinkage steps
      outer_start = 20,        # Fixed starting outer radius
      seed = 123
    )

    ## Generating 9 patterns...
    ##   |                                                                              |                                                                      |   0%  |                                                                              |========                                                              |  11%  |                                                                              |================                                                      |  22%  |                                                                              |=======================                                               |  33%  |                                                                              |===============================                                       |  44%  |                                                                              |=======================================                               |  56%  |                                                                              |===============================================                       |  67%  |                                                                              |======================================================                |  78%  |                                                                              |==============================================================        |  89%  |                                                                              |======================================================================| 100%
    ##
    ## Multicenter multiscale simulation completed!

    cat("Generated", length(patterns_9), "patterns\n")

    ## Generated 9 patterns

    cat("Pattern names:", names(patterns_9), "\n")

    ## Pattern names: Ra_3_Step_1 Ra_3_Step_2 Ra_3_Step_3 Ra_6_Step_1 Ra_6_Step_2 Ra_6_Step_3 Ra_9_Step_1 Ra_9_Step_2 Ra_9_Step_3

    # Create SGWT objects for all 9 patterns and compute similarities
    similarity_results <- list()
    sgwt_objects <- list()

    cat("\nProcessing patterns and computing SGWT analysis...\n")

    ##
    ## Processing patterns and computing SGWT analysis...

    # Process each pattern
    for (i in seq_along(patterns_9)) {
      pattern_name <- names(patterns_9)[i]
      pattern_data <- patterns_9[[i]]

      cat("Processing", pattern_name, "...\n")

      # Create SGWT object
      SG_temp <- initSGWT(
        data.in = pattern_data,
        x_col = "X",
        y_col = "Y",
        signals = c("signal_1", "signal_2"),
        J = 4,
        kernel_type = "heat"
      )

      # Build graph and run SGWT
      SG_temp <- runSpecGraph(SG_temp, k = 12, laplacian_type = "normalized",
                             length_eigenvalue = 50, verbose = FALSE)
      SG_temp <- runSGWT(SG_temp, verbose = FALSE)

      sgwt_objects[[pattern_name]] <- SG_temp

      # Compute within-pattern similarity (signal_1 vs signal_2)
      sim_within <- runSGCC("signal_1", "signal_2", SG = SG_temp, return_parts = TRUE)
      similarity_results[[pattern_name]] <- sim_within
    }

    ## Processing Ra_3_Step_1 ...
    ## Processing Ra_3_Step_2 ...
    ## Processing Ra_3_Step_3 ...
    ## Processing Ra_6_Step_1 ...
    ## Processing Ra_6_Step_2 ...
    ## Processing Ra_6_Step_3 ...
    ## Processing Ra_9_Step_1 ...
    ## Processing Ra_9_Step_2 ...
    ## Processing Ra_9_Step_3 ...

    # Print similarity results summary
    cat("\nSimilarity Results Summary (signal_1 vs signal_2 within each pattern):\n")

    ##
    ## Similarity Results Summary (signal_1 vs signal_2 within each pattern):

    similarity_df <- data.frame(
      Pattern = names(similarity_results),
      S = sapply(similarity_results, function(x) x$S),
      c_low = sapply(similarity_results, function(x) x$c_low),
      c_nonlow = sapply(similarity_results, function(x) x$c_nonlow),
      w_low = sapply(similarity_results, function(x) x$w_low),
      w_NL = sapply(similarity_results, function(x) x$w_NL),
      stringsAsFactors = FALSE
    )

    # Extract Ra and Step values for better labeling
    similarity_df$Ra <- as.numeric(gsub(".*Ra_([0-9.]+)_Step.*", "\\1", similarity_df$Pattern))
    similarity_df$Step <- as.numeric(gsub(".*Step_([0-9]+)$", "\\1", similarity_df$Pattern))
    similarity_df$Label <- paste0("Ra=", similarity_df$Ra, ",Step=", similarity_df$Step)

    # Create similarity space visualization
    similarity_plot <- visualize_similarity_xy(
      similarity_results,
      point_size = 4,
      point_color = "steelblue",
      add_diagonal = TRUE,
      add_axes_lines = TRUE,
      title = "SGWT Similarity Space: 9 Paired Patterns (signal_1 vs signal_2)",
      show_labels = TRUE
    )

    print(similarity_plot)

    # Analysis of patterns
    cat("\nPattern Analysis:\n")

    ##
    ## Pattern Analysis:

    cat("Ra (Inner Radius) Effect:\n")

    ## Ra (Inner Radius) Effect:

    for (ra in unique(similarity_df$Ra)) {
      subset_data <- similarity_df[similarity_df$Ra == ra, ]
      cat(sprintf("  Ra=%g: Mean c_low=%.3f, Mean c_nonlow=%.3f, Mean S=%.3f\n",
                  ra, mean(subset_data$c_low), mean(subset_data$c_nonlow), mean(subset_data$S)))
    }

    ##   Ra=3: Mean c_low=0.459, Mean c_nonlow=0.216, Mean S=0.382
    ##   Ra=6: Mean c_low=0.205, Mean c_nonlow=-0.153, Mean S=0.097
    ##   Ra=9: Mean c_low=-0.018, Mean c_nonlow=-0.300, Mean S=-0.102

    cat("\nStep (Shrinkage Step) Effect:\n")

    ##
    ## Step (Shrinkage Step) Effect:

    for (step in unique(similarity_df$Step)) {
      subset_data <- similarity_df[similarity_df$Step == step, ]
      cat(sprintf("  Step=%g: Mean c_low=%.3f, Mean c_nonlow=%.3f, Mean S=%.3f\n",
                  step, mean(subset_data$c_low), mean(subset_data$c_nonlow), mean(subset_data$S)))
    }

    ##   Step=1: Mean c_low=-0.266, Mean c_nonlow=-0.489, Mean S=-0.334
    ##   Step=2: Mean c_low=0.185, Mean c_nonlow=-0.256, Mean S=0.050
    ##   Step=3: Mean c_low=0.726, Mean c_nonlow=0.508, Mean S=0.661

    cat("\nSimilarity Space Interpretation:\n")

    ##
    ## Similarity Space Interpretation:

    cat("- Each point represents similarity between signal_1 (inner circle) and signal_2 (outer ring)\n")

    ## - Each point represents similarity between signal_1 (inner circle) and signal_2 (outer ring)

    cat("- Color indicates inner radius (Ra): different colors for different inner radii\n")

    ## - Color indicates inner radius (Ra): different colors for different inner radii

    cat("- Shape indicates shrinkage step: circle/triangle/square for different steps\n")

    ## - Shape indicates shrinkage step: circle/triangle/square for different steps

    cat("- Position shows frequency-domain similarity characteristics\n")

    ## - Position shows frequency-domain similarity characteristics

# Advanced Workflow Features

## Custom Parameters

    # Demonstrate advanced parameter customization
    SG_advanced <- initSGWT(
      data.in = demo_pattern,
      x_col = "X", y_col = "Y",
      signals = "signal_1",
      J = 6,                     # More scales for finer analysis
      scaling_factor = 1.5,      # Closer scales
      kernel_type = "heat"       # Heat kernel
    )

    SG_advanced <- runSpecGraph(SG_advanced, k = 15, laplacian_type = "randomwalk",length_eigenvalue = 30, verbose = FALSE)
    SG_advanced <- runSGWT(SG_advanced, verbose = FALSE)

    cat("Advanced Parameters Results:\n")

    ## Advanced Parameters Results:

    cat("Number of scales:", length(SG_advanced$Parameters$scales), "\n")

    ## Number of scales: 6

    cat("Scales:", paste(round(SG_advanced$Parameters$scales, 4), collapse = ", "), "\n")

    ## Scales: 0.5399, 0.36, 0.24, 0.16, 0.1067, 0.0711

    cat("Reconstruction error:", round(SG_advanced$Inverse$signal_1$reconstruction_error, 6), "\n")

    ## Reconstruction error: 0.538056

# Summary

## Key Takeaways

1. New Object-Oriented Workflow: Clear separation of initialization, graph building, analysis, and similarity computation
2. Flexible Input: Supports custom column naming for coordinates and signals
3. Multiple Kernels: Choose from Mexican Hat, Meyer, or Heat kernel families
4. Comprehensive Similarity: Energy-normalized weighted similarity with detailed diagnostics
5. Advanced Analysis: Low-frequency only analysis, cross-object comparison, multiple signal support

## New Workflow Benefits

    # Complete workflow in 5 clear steps:
    SG <- initSGWT(data, signals = c("signal1", "signal2"))           # 1. Initialize
    kband <- checkKband(SG, signals = c("signal1", "signal2"))       # 2. Check k-band (optional)
    SG <- runSpecGraph(SG, k = 12, laplacian_type = "normalized")    # 3. Build graph
    SG <- runSGWT(SG)                                                 # 4. Run SGWT
    similarity <- runSGCC("signal1", "signal2", SG = SG)             # 5. Compare patterns

## Parameter Guidelines

- Small datasets (<200 points): k=8-12, J=3-4
- Medium datasets (200-1000 points): k=12-20, J=4-6
- Large datasets (>1000 points): k=15-25, J=5-8

## Next Steps

- Explore your own spatial datasets using the demonstrated workflow
- Experiment with different kernel types and parameters
- Use comprehensive similarity analysis for comparative studies
- Refer to function documentation for detailed parameter specifications

BioGSP Package - Biological Graph Signal Processing for Spatial Data Analysis