Raw Moment (𝑀_𝑖𝑗)

𝑀_𝑖𝑗 = 𝛴_𝑥𝛴_𝑦𝑥^𝑖𝑦^𝑗Intensity(𝑥,𝑦)

Central Moment (𝑢_𝑖𝑗)

• translation

𝑢_𝑖𝑗 = 𝛴_𝑥𝛴_𝑦(𝑥 - 𝐶_𝑥)^𝑖(𝑦 - 𝐶_𝑦)^𝑗Intensity(𝑥,𝑦)

where 𝐶_𝑥 and 𝐶_𝑦 are centroids:

• 𝐶_𝑥 = 𝑀₁₀/𝑀₀₀ = 𝛴_𝑥𝛴_𝑦𝑥·Intensity(𝑥,𝑦) / 𝛴_x𝛴_𝑦Intensity(𝑥,𝑦)
• 𝐶_𝑦 = 𝑀₀₁/𝑀₀₀= 𝛴_𝑥𝛴_𝑦𝑦·Intensity(𝑥,𝑦) / 𝛴_𝑥𝛴_𝑦Intensity(𝑥,𝑦)

Normalized Central Moment (𝑛_𝑖𝑗)

• translation
• scale

• 𝑛_𝑖𝑗 = [𝑢_𝑖𝑗] / [𝑢₀₀^{(𝑖+𝑗)/2+1}]
• 𝑛_𝑖𝑗 = [𝛴_𝑥𝛴_𝑦(𝑥 - 𝐶_𝑥)^𝑖(𝑦 - 𝐶_𝑦)^𝑗Intensity(𝑥,𝑦)] / [𝛴_𝑥𝛴_𝑦Intensity(𝑥,𝑦)]^{(𝑖+𝑗)/2+1}

Hu Moments (ℎ₁, .., ℎ₇)

• translation
• scale
• rotation
• reflection

• ℎ₁ = 𝑛₂₀ + 𝑛₀₂
• ℎ₂ = (𝑛₂₀ - 𝑛₀₂)² + 4𝑛₁₁²
• ℎ₃ = (𝑛₃₀ - 3𝑛₁₂)² + (3𝑛₂₁ + 𝑛₀₃)²
• ℎ₄ = (𝑛₃₀ - 𝑛₁₂)² + (𝑛₂₁ + 𝑛₀₃)²
• ℎ₅ = (𝑛₃₀ - 3𝑛₁₂)(𝑛₃₀ + 𝑛₁₂)[(𝑛₃₀ + 𝑛₁₂)² - 3(𝑛₂₁ + 𝑛₀₃)²] + (3𝑛₂₁ - 𝑛₀₃)(𝑛₂₁ + 𝑛₀₃)[3(𝑛₃₀ + 𝑛₁₂)² - (𝑛₂₁ + 𝑛₀₃)²]
• ℎ₆ = (𝑛₂₀ + 𝑛₀₂)[(𝑛₃₀ + 𝑛₁₂)² - (𝑛₂₁ + 𝑛₀₃)²] + 4𝑛₁₁(𝑛₃₀ + 𝑛₁₂)(𝑛₂₁ + 𝑛₀₃)
• ℎ₇ = (3𝑛₂₁ - 𝑛₀₃)(𝑛₃₀ + 𝑛₁₂)[(𝑛₃₀ + 𝑛₁₂)² - 3(𝑛₂₁ + 𝑛₀₃)²] - (𝑛₃₀ - 3𝑛₁₂)(𝑛₂₁ + 𝑛₀₃)[3(𝑛₃₀ + 𝑛₁₂)² - (𝑛₂₁ + 𝑛₀₃)²]

Hu Moments is a set of 7 numbers calculated using central moments that are invariant to image transformations. The first 6 moments have been proven to be invariant to translation, scale, rotation, and reflection. While the 7th moment’s sign changes for image reflection