perlin

def perlin_noise(width, height, scale=100):
    # Generate random gradient vectors
    gradient_vectors = np.random.randn(height, width, 2)

    # Generate coordinate grid
    x = np.linspace(0, 1, width, endpoint=False)
    y = np.linspace(0, 1, height, endpoint=False)
    x_grid, y_grid = np.meshgrid(x, y)

    # Scale the grid to the desired frequency
    x_grid_scaled = x_grid * scale
    y_grid_scaled = y_grid * scale

    # Compute integer coordinates for the corners of the grid cell
    x0 = x_grid_scaled.astype(int)
    y0 = y_grid_scaled.astype(int)
    x1 = x0 + 1
    y1 = y0 + 1

    # Compute fractional part of the coordinates
    dx = x_grid_scaled - x0
    dy = y_grid_scaled - y0

    # Compute dot products between gradient vectors and offsets
    dot_top_left = np.sum(gradient_vectors[y0, x0] * np.dstack((dx, dy)), axis=2)
    dot_top_right = np.sum(gradient_vectors[y0, x1] * np.dstack((dx - 1, dy)), axis=2)
    dot_bottom_left = np.sum(gradient_vectors[y1, x0] * np.dstack((dx, dy - 1)), axis=2)
    dot_bottom_right = np.sum(gradient_vectors[y1, x1] * np.dstack((dx - 1, dy - 1)), axis=2)

    # Interpolate along x-axis
    weight_x = 6 * dx**5 - 15 * dx**4 + 10 * dx**3
    dot_top = dot_top_left + weight_x * (dot_top_right - dot_top_left)
    dot_bottom = dot_bottom_left + weight_x * (dot_bottom_right - dot_bottom_left)

    # Interpolate along y-axis
    weight_y = 6 * dy**5 - 15 * dy**4 + 10 * dy**3
    dot_interp = dot_top + weight_y * (dot_bottom - dot_top)

    return dot_interp

# Example usage
width, height = 100, 100
scale = 10
noise = perlin_noise(width, height, scale)

# Example visualization using Matplotlib
import matplotlib.pyplot as plt
plt.imshow(noise, cmap='gray', origin='lower')
plt.colorbar()
plt.show()