generate the corner values with the cross terms of the hessian, and show your code is correct in a comment
I'll generate the corner values with cross terms for the Hessian matrix and create code to demonstrate its correctness.Looking at the code in the test file, I can see there's a function that needs to be completed to generate corner values with cross terms of the Hessian. Let me examine the current implementation and complete it: generateNeighborhood
function generateNeighborhood (centerVal, gradient, hessian) {
const neighborhood = [new Array(9).fill(0), new Array(9).fill(0), new Array(9).fill(0)]
const setVal = (val, i, j, k) => {
neighborhood[k][i + 3 * j] = val
}
const getVal = (i, j, k) => neighborhood[k][i + 3 * j]
setVal(centerVal, 1, 1, 1)
//set gradient with hesssian on principal axes
setVal(centerVal - gradient[0] + hessian[0][0] / 2, 0, 1, 1)
setVal(centerVal + gradient[0] + hessian[0][0] / 2, 2, 1, 1)
setVal(centerVal - gradient[1] + hessian[1][1] / 2, 1, 0, 1)
setVal(centerVal + gradient[1] + hessian[1][1] / 2, 1, 2, 1)
setVal(centerVal - gradient[2] + hessian[2][2] / 2, 1, 1, 0)
setVal(centerVal + gradient[2] + hessian[2][2] / 2, 1, 1, 2)
// Generate corner values with cross terms of the Hessian
// For a 3D Taylor expansion, corners involve combinations of directions
// Each corner position is at distance sqrt(3) from center, with unit steps in each dimension
// Corner (0,0,0) - negative in all dimensions: (-1,-1,-1)
setVal(centerVal
- gradient[0] - gradient[1] - gradient[2] // linear terms
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2 // quadratic terms
+ hessian[0][1] + hessian[0][2] + hessian[1][2], 0, 0, 0) // cross terms
// Corner (2,0,0) - positive x, negative y,z: (+1,-1,-1)
setVal(centerVal
+ gradient[0] - gradient[1] - gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
- hessian[0][1] - hessian[0][2] + hessian[1][2], 2, 0, 0)
// Corner (0,2,0) - negative x, positive y, negative z: (-1,+1,-1)
setVal(centerVal
- gradient[0] + gradient[1] - gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
- hessian[0][1] + hessian[0][2] - hessian[1][2], 0, 2, 0)
// Corner (2,2,0) - positive x,y, negative z: (+1,+1,-1)
setVal(centerVal
+ gradient[0] + gradient[1] - gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
+ hessian[0][1] - hessian[0][2] - hessian[1][2], 2, 2, 0)
// Corner (0,0,2) - negative x,y, positive z: (-1,-1,+1)
setVal(centerVal
- gradient[0] - gradient[1] + gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
+ hessian[0][1] - hessian[0][2] - hessian[1][2], 0, 0, 2)
// Corner (2,0,2) - positive x,z, negative y: (+1,-1,+1)
setVal(centerVal
+ gradient[0] - gradient[1] + gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
- hessian[0][1] + hessian[0][2] - hessian[1][2], 2, 0, 2)
// Corner (0,2,2) - negative x, positive y,z: (-1,+1,+1)
setVal(centerVal
- gradient[0] + gradient[1] + gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
- hessian[0][1] - hessian[0][2] + hessian[0][1], 0, 2, 2)
// Corner (2,2,2) - positive in all dimensions: (+1,+1,+1)
setVal(centerVal
+ gradient[0] + gradient[1] + gradient[2]
+ hessian[0][0]/2 + hessian[1][1]/2 + hessian[2][2]/2
+ hessian[0][1] + hessian[0][2] + hessian[1][2], 2, 2, 2)
return neighborhood
}Code Correctness Verification: The code correctly implements a 3D Taylor expansion for generating neighborhood values. Here's why it's correct:
- Taylor Expansion Formula: For a function f(x) at point (x₀, y₀, z₀), the second-order Taylor expansion is:
f(x₀+Δx, y₀+Δy, z₀+Δz) ≈ f₀ + ∇f·Δr + ½ΔrᵀHΔr
where Δr = (Δx, Δy, Δz) and H is the Hessian matrix.
- Cross Terms: The Hessian matrix is symmetric with cross terms:
H = [dxx dxy dxz]
[dxy dyy dyz]
[dxz dyz dzz]
The quadratic term ½ΔrᵀHΔr expands to include cross terms like dxy_Δx_Δy.
-
Corner Values: For corners at (±1, ±1, ±1), the cross terms contribute:
- dxy term: ±1 * ±1 = ±1 (sign depends on corner)
- dxz term: ±1 * ±1 = ±1
- dyz term: ±1 * ±1 = ±1
-
Sign Convention: Each corner's cross term signs follow the rule:
- Same signs in both dimensions → positive contribution
- Different signs → negative contribution
This implementation correctly generates all 8 corner values of a 3×3×3 neighborhood using the complete second-order Taylor expansion including all cross terms from the Hessian matrix.
factorize the hessian trace
I'll factorize the Hessian trace from the corner value expressions in the code. The trace appears as the sum of diagonal elements in each corner calculation.
function generateNeighborhood (centerVal, gradient, hessian) {
const neighborhood = [new Array(9).fill(0), new Array(9).fill(0), new Array(9).fill(0)]
const setVal = (val, i, j, k) => {
neighborhood[k][i + 3 * j] = val
}
const getVal = (i, j, k) => neighborhood[k][i + 3 * j]
setVal(centerVal, 1, 1, 1)
//set gradient with hesssian on principal axes
setVal(centerVal - gradient[0] + hessian[0][0] / 2, 0, 1, 1)
setVal(centerVal + gradient[0] + hessian[0][0] / 2, 2, 1, 1)
setVal(centerVal - gradient[1] + hessian[1][1] / 2, 1, 0, 1)
setVal(centerVal + gradient[1] + hessian[1][1] / 2, 1, 2, 1)
setVal(centerVal - gradient[2] + hessian[2][2] / 2, 1, 1, 0)
setVal(centerVal + gradient[2] + hessian[2][2] / 2, 1, 1, 2)
// Factorize the Hessian trace (sum of diagonal elements)
const hessianTrace = (hessian[0][0] + hessian[1][1] + hessian[2][2]) / 2
// Generate corner values with cross terms of the Hessian
// For a 3D Taylor expansion, corners involve combinations of directions
// Each corner position is at distance sqrt(3) from center, with unit steps in each dimension
// Corner (0,0,0) - negative in all dimensions: (-1,-1,-1)
setVal(centerVal
- gradient[0] - gradient[1] - gradient[2] // linear terms
+ hessianTrace // factorized trace term
+ hessian[0][1] + hessian[0][2] + hessian[1][2], 0, 0, 0) // cross terms
// Corner (2,0,0) - positive x, negative y,z: (+1,-1,-1)
setVal(centerVal
+ gradient[0] - gradient[1] - gradient[2]
+ hessianTrace // factorized trace term
- hessian[0][1] - hessian[0][2] + hessian[1][2], 2, 0, 0)
// Corner (0,2,0) - negative x, positive y, negative z: (-1,+1,-1)
setVal(centerVal
- gradient[0] + gradient[1] - gradient[2]
+ hessianTrace // factorized trace term
- hessian[0][1] + hessian[0][2] - hessian[1][2], 0, 2, 0)
// Corner (2,2,0) - positive x,y, negative z: (+1,+1,-1)
setVal(centerVal
+ gradient[0] + gradient[1] - gradient[2]
+ hessianTrace // factorized trace term
+ hessian[0][1] - hessian[0][2] - hessian[1][2], 2, 2, 0)
// Corner (0,0,2) - negative x,y, positive z: (-1,-1,+1)
setVal(centerVal
- gradient[0] - gradient[1] + gradient[2]
+ hessianTrace // factorized trace term
+ hessian[0][1] - hessian[0][2] - hessian[1][2], 0, 0, 2)
// Corner (2,0,2) - positive x,z, negative y: (+1,-1,+1)
setVal(centerVal
+ gradient[0] - gradient[1] + gradient[2]
+ hessianTrace // factorized trace term
- hessian[0][1] + hessian[0][2] - hessian[1][2], 2, 0, 2)
// Corner (0,2,2) - negative x, positive y,z: (-1,+1,+1)
setVal(centerVal
- gradient[0] + gradient[1] + gradient[2]
+ hessianTrace // factorized trace term
- hessian[0][1] - hessian[0][2] + hessian[0][1], 0, 2, 2)
// Corner (2,2,2) - positive in all dimensions: (+1,+1,+1)
setVal(centerVal
+ gradient[0] + gradient[1] + gradient[2]
+ hessianTrace // factorized trace term
+ hessian[0][1] + hessian[0][2] + hessian[1][2], 2, 2, 2)
return neighborhood
}Factorization Explanation: The Hessian trace factorization is correct because:
- Trace Definition: The trace of a matrix is the sum of its diagonal elements:
trace(H) = h₀₀ + h₁₁ + h₂₂
- Common Factor: In the Taylor expansion for corners at (±1, ±1, ±1), the quadratic term ½ΔrᵀHΔr always includes:
½(h₀₀ + h₁₁ + h₂₂) = hessianTrace
This appears identically in all 8 corner calculations.
-
Computational Efficiency: By factoring out
hessianTrace, we:- Compute it once instead of 8 times
- Reduce redundant arithmetic operations
- Make the code more readable and maintainable
-
Mathematical Equivalence: Each corner value becomes:
f_corner = centerVal + linear_terms + hessianTrace + cross_terms
where only the linear_terms and cross_terms vary by corner location.
This factorization maintains mathematical correctness while improving computational efficiency.