Package gov.nih.mipav.model.algorithms

Class AlgorithmQuadraticFit

  • public class AlgorithmQuadraticFit
extends java.lang.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​(java.util.Vector<WildMagic.LibFoundation.Mathematics.Vector3f> kPoints)
      Fit a set of 3D points with a quadric surface.

    • Method Summary

      All Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      WildMagic.LibFoundation.Mathematics.Vector3f getCenter()
      Get the center of the quadric surface.
      float getCoefficient​(int i)
      Get the coefficients of the quadratic equation that represents the approximating quadric surface.
      float getConstant()
      Get the constant term for the factored quadratic equation.
      float getDiagonal​(int i)
      Get a diagonal entry for the quadric surface.
      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 Detail

      • 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 Detail

      • AlgorithmQuadraticFit

        public AlgorithmQuadraticFit​(java.util.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 Detail

      • 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 <= i < 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 <= i < 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