improve default gate value

[josenimo]

double derivative of density plot

[josenimo]

idea is

import numpy as np

def kittler_illingworth_threshold(data):
    # Calculate histogram
    hist, bin_edges = np.histogram(data, bins=256, range=(0, 256))
    
    # Calculate probability of each intensity level
    P = hist / hist.sum()
    
    # Cumulative sums
    omega = np.cumsum(P)
    
    # Means
    mu = np.cumsum(P * np.arange(256))
    
    # Total mean
    mu_t = mu[-1]
    
    # Between class variance
    sigma_b_squared = (mu_t * omega - mu) ** 2 / (omega * (1 - omega))
    
    # Replace NaNs with zero
    sigma_b_squared = np.nan_to_num(sigma_b_squared)
    
    # Find threshold
    threshold = np.argmax(sigma_b_squared)
    
    return threshold

# Example usage
data = np.random.randint(0, 256, 1000)
threshold = kittler_illingworth_threshold(data)
print(f"Optimal threshold: {threshold}")