Package gov.nih.mipav.model.algorithms

Class AlgorithmQuadraticFit

java.lang.Object
gov.nih.mipav.model.algorithms.AlgorithmQuadraticFit
public class AlgorithmQuadraticFit extends Object
Approximate a set of points by a quadric surface. The surface is implicitly represented as the quadratic equation 
     0 = C[0] + C[1]*X + C[2]*Y + C[3]*Z + C[4]*X^2 + C[5]*Y^2
         + C[6]*Z^2 + C[7]*X*Y + C[8]*X*Z + C[9]*Y*Z

The coefficients are estimated using a least-squares algorithm. There is one degree of freedom in the coefficients. It is removed by requiring the vector C = (C[0],...,C[9]) to be unit length. The least-square problem is to minimize E(C) = C^t M C with M = (sum_i V_i)(sum_i V_i)^t and

     V = (1, X, Y, Z, X^2, Y^2, Z^2, X*Y, X*Z, Y*Z)

Minimum value is the smallest eigenvalue of M and C is a corresponding unit length eigenvector.

The quadratic equation can be factored into P^T A P + B^T P + K = 0 where P = (X,Y,Z), K = C[0], B = (C[1],C[2],C[3]), and A is a 3x3 symmetric matrix with A00 = C[4], A11 = C[5], A22 = C[6], A01 = C[7]/2, A02 = C[8]/2, and A12 = C[9]/2. Matrix A = R^T D R where R is orthogonal and D is diagonal (using an eigendecomposition). Define V = R P = (v0,v1,v2), E = R B = (e0,e1,e2), D = diag(d0,d1,d2), and f = K to obtain

     d0 v0^2 + d1 v1^2 + d2 v^2 + e0 v0 + e1 v1 + e2 v2 + f = 0

The characterization of the surface depends on the signs of the d_i. For the MRI brain segmentation, the signs d_i are all positive, thereby producing an ellipsoid.

The quadratic equation can finally be factored into

     (P-G)^T R^T D R (P-G) = L

where G is the center of the surface, R is the orientation matrix, D is a diagonal matrix, and L is a constant.

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    protected float[]
    m_afCoeff
    the coeffficients of the approximating quadratic equation.
    protected float[]
    m_afDiagonal
    DOCUMENT ME!
    protected float
    m_fConstant
    DOCUMENT ME!
    protected WildMagic.LibFoundation.Mathematics.Vector3f
    m_kCenter
    The center G, orientation R, diagonal matrix D, and constant L in the factorization (P-G)^T R^T D R (P-G) = L.
    protected WildMagic.LibFoundation.Mathematics.Matrix3f
    m_kOrient
    DOCUMENT ME!

  • Constructor Summary

    Constructors
    Constructor
    Description
    AlgorithmQuadraticFit(Vector<WildMagic.LibFoundation.Mathematics.Vector3f> kPoints)
    Fit a set of 3D points with a quadric surface.

  • Method Summary

    Modifier and Type
    Method
    Description
    final WildMagic.LibFoundation.Mathematics.Vector3f
    getCenter()
    Get the center of the quadric surface.
    final float
    getCoefficient(int i)
    Get the coefficients of the quadratic equation that represents the approximating quadric surface.
    final float
    getConstant()
    Get the constant term for the factored quadratic equation.
    final float
    getDiagonal(int i)
    Get a diagonal entry for the quadric surface.
    final WildMagic.LibFoundation.Mathematics.Matrix3f
    getOrient()
    Get the orientation of the quadric surface.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Field Details

    • m_afCoeff

      protected float[] m_afCoeff
      the coeffficients of the approximating quadratic equation.

    • m_afDiagonal

      protected float[] m_afDiagonal
      DOCUMENT ME!

    • m_fConstant

      protected float m_fConstant
      DOCUMENT ME!

    • m_kCenter

      protected WildMagic.LibFoundation.Mathematics.Vector3f m_kCenter
      The center G, orientation R, diagonal matrix D, and constant L in the factorization (P-G)^T R^T D R (P-G) = L.

    • m_kOrient

      protected WildMagic.LibFoundation.Mathematics.Matrix3f m_kOrient
      DOCUMENT ME!

  • Constructor Details

    • AlgorithmQuadraticFit

      public AlgorithmQuadraticFit(Vector<WildMagic.LibFoundation.Mathematics.Vector3f> kPoints)
      Fit a set of 3D points with a quadric surface. The various representations of the surface can be accessed by member functions.
      Parameters:
      kPoints - a vector of points, each vector element is of type Point3f

  • Method Details

    • getCenter

      public final WildMagic.LibFoundation.Mathematics.Vector3f getCenter()
      Get the center of the quadric surface. This is the vector G in the factorization (P-G)^T R^T D R (P-G) = L.
      Returns:
      the center point

    • getCoefficient

      public final float getCoefficient(int i)
      Get the coefficients of the quadratic equation that represents the approximating quadric surface.
      Parameters:
      i - the coefficient index (0 invalid input: '<'= i invalid input: '<' 10)
      Returns:
      the value of the coefficient

    • getConstant

      public final float getConstant()
      Get the constant term for the factored quadratic equation. This is the L term in (P-G)^T R^T D R (P-G) = L.
      Returns:
      the constant term

    • getDiagonal

      public final float getDiagonal(int i)
      Get a diagonal entry for the quadric surface. This is a diagonal term for the matrix D in the factorization A = R^T D R where A is the coefficient matrix for the quadratic term in the equation.
      Parameters:
      i - the index of the diagonal (0 invalid input: '<'= i invalid input: '<' 3)
      Returns:
      the diagonal entry

    • getOrient

      public final WildMagic.LibFoundation.Mathematics.Matrix3f getOrient()
      Get the orientation of the quadric surface. This is the matrix R in the factorization A = R^T D R where A is the coefficient matrix for the quadratic term in the equation.
      Returns:
      the orientation matrix