Package gov.nih.mipav.model.algorithms.registration

Class AlgorithmRegELSUNCOAR25D

java.lang.Object
java.lang.Thread
gov.nih.mipav.model.algorithms.AlgorithmBase
gov.nih.mipav.model.algorithms.registration.AlgorithmRegELSUNCOAR25D
All Implemented Interfaces:
ActionListener, WindowListener, Runnable, EventListener
public class AlgorithmRegELSUNCOAR25D extends AlgorithmBase
This is an automatic registration method based on FLIRT. FLIRT stands for FMRIB's Linear Image Registration Tool. For more information on FLIRT, visit their homepage at http://www.fmrib.ox.ac.uk/fsl/flirt/. Their main paper is:

Jenkinson, M. and Smith, S. (2001a).
A global optimisation method for robust affine registration of brain images.
Medical Image Analysis, 5(2):143-156.

Internal registration is performed within one 3D image.

In adjacent mode the first slice above the reference slice is registered to the reference slice, then the second slice above the reference slice is registered to the first slice above the reference slice, and so on until the last slice is registered the next to last slice. Then, the first slice below the reference slice is registered to the reference slice, the second slice below the reference slice is registered to first slice below the reference slice, and so on until the first slice is registered to the second slice. In reference mode every slice is simply registered to the reference slice.

Our algorithm works as follows:
1.) We find the minimum resolution of the images.
2.) We transform the images into isotropic voxels.
3.) We subsample the images by 2, 4, and 8 or 16, depending on the resolution.
We loop thru a number of iterations equal to the number of slices - 1 in adjacent mode or equal to the number of slices in reference mode with one slice as the input slice and one slice as the reference slice. For each iteration: 4.) With the slices that were subsampled by 8 or 16, we call levelEight. This function will use the coarse sampling rate and optimize translations and global scale at the given rotation. So for example, if the coarse sampling range were -30 to 30 at every 15 degrees, we would optimize at rotations of -30, -15, 0, 15, 30.
5.) Still in levelEight, we now measure the cost at the fine sampling rate. We interpolate the translations and global scale to come up with a good guess as to what the optimized translation would be at that point.
6.) We take the top 20% of the points and optimize them.
7.) We now have a large multi-array of costs. 20% of those have been optimized and placed back into their original position in the multi-array. We look at the 3 neighbors of a point: +, =, or - one fine sample. If our point has a cost greater than any of these, it is not a minima. Otherwise it is. We save it in a vector of minima.
8.) We optimize the minima over rotation as well as translations and global scale. (Previously we had not optimized over rotation.) We return two vectors, one containing the minima before optimization, one containing the minima after optimization.
9.) We now call levelFour with the slices subsampled by 4 and the vectors of minima. We measure the costs of the minima on the new slices and sort them. We take the top numMinima in each vector (pre-optimization and post-optimization) and optimize them. We put them all into one vector.
10.) We perturb the rotation by zero and by plus-minus fineDelta. If it's not a rigid transformation, we then perturb the global scaling by factors of 0.8, 0.9, 1.0, 1.1, and 1.2.
11.) We optimize the perturbations. We return a vector of the perturbed, optimized minima.
12.) We now call levelTwo with the slices subsampled by 2. We measure the costs of the minima at the new slices. We optimize the best minimum with 4 degrees of freedom, then 5, then 6. If the user has limited the degrees of freedom to 3, there will only be one optimization run, with 3 degrees of freedom. The function returns the best minimum after optimization.
13.) We call levelOne with the un-subsampled slices. At levelOne, one optimization run is performed, with the maximum allowable degrees of freedom, as specified by the user (the max is 6).
14.) The best answer is returned from levelOne. The inverse of the matrix from this answer is used to register the input slice to the reference slice. The registered input slice data is imported into inputImage.

Note that when 6 degrees of freedom is used the rotation is set equal to 0 because diffX sets (0,2), diffY sets (1,2), scaleX sets (0,0), scaleY sets (1,1), skewX sets (0,1), and skewY sets (1,0) so all 6 elements are set.

Author:
Matthew McAuliffe, Neva Cherniavsky, Benjamin Link - Added code to use less memory. (April 2003)

  • Field Details

    • allowLevel16

      private boolean allowLevel16
      DOCUMENT ME!

    • allowLevel2

      private boolean allowLevel2
      Flags are true if weighted image is not present or if weighted image subsampling occurred, false if weighted image subsampling did not occur.

    • allowLevel4

      private boolean allowLevel4
      DOCUMENT ME!

    • allowLevel8

      private boolean allowLevel8
      DOCUMENT ME!

    • answer

      private MatrixListItem answer
      Final answer after registration.

    • buffer

      private float[] buffer
      DOCUMENT ME!

    • bufferA

      private float[] bufferA
      DOCUMENT ME!

    • bufferIW

      private float[] bufferIW
      DOCUMENT ME!

    • bufferW

      private float[] bufferW
      DOCUMENT ME!

    • coarseNum

      private int coarseNum
      Number of passes that will be made in the coarse sampling and fine sampling.

    • fineNum

      private int fineNum
      Number of passes that will be made in the coarse sampling and fine sampling.

    • colorFactor

      private int colorFactor
      1 for black and white, 4 for color.

    • costChoice

      private int costChoice
      Choice of which cost function to use.

    • doColor

      private boolean doColor
      DOCUMENT ME!

    • DOF

      private int DOF
      Maximum degrees of freedom when running the optimization.

    • doGraph

      private boolean doGraph
      Produce 2 output graphs - 1 for rotation and 1 for 2 translations. Only can be used for DOF == 3 and register to reference slice

    • doSubsample

      private boolean doSubsample
      if true subsample.

    • searchAlgorithm

      private int searchAlgorithm

    • ignoreCOG

      private boolean ignoreCOG
      whether or not to use center of gravity for first translation.

    • imageWeightIso

      private ModelImage imageWeightIso
      DOCUMENT ME!

    • input1

      private ModelImage input1
      Image used to import a slice from inputImage.

    • input2

      private ModelImage input2
      Image used to import a slice from inputImage2.

    • input3

      private ModelImage input3
      Image used to import a slice from inputImage3.

    • inputImage

      private ModelImage inputImage
      This is the image in which internal registration will be performed.

    • inputImage2

      private ModelImage inputImage2
      Other images which will not determine the registration, but which will undergo the same transformations as input image. Useful if registering a color image based on only 1 of the colors

    • inputImage3

      private ModelImage inputImage3
      DOCUMENT ME!

    • inputw_1

      private ModelImage inputw_1
      Image used to import a slice from inputWeight.

    • inputWeight

      private ModelImage inputWeight
      This gives weights for the input image - higher weights mean a greater impact in that area on the registration.

    • interp

      private int interp
      Interpolation method used in transformations.

    • interp2

      private int interp2
      Interpolation method used in output.

    • iResols

      private float[] iResols
      The voxel resolutions of the input image.

    • isoImage

      private ModelImage isoImage
      new to this version: must keep isoImage around if image needed to be transformed.

    • level1Factor

      private float level1Factor
      Multiplication factor for level 1 - will be set based on subsampling.

    • level2Factor

      private float level2Factor
      Multiplication factor for level 2 - will be set based on subsampling.

    • level4Factor

      private float level4Factor
      Multiplication factor for level 4 - will be set based on subsampling.

    • maxDim

      private float maxDim
      DOCUMENT ME!

    • maxIter

      private int maxIter
      Limits number of iterations in ELSUNC, LEVENBERG_MARQUARDT, or NL2SOL optimization. maxIter in the call to elsunc will be an integer multiple of baseNumIter

    • baseNumIter

      private int baseNumIter
      Limits number of iterations in ELSUNC, LEVENBERG_MARQUARDT, or NL2SOL optimization. maxIter in the call to elsunc will be an integer multiple of baseNumIter

    • numMinima

      private int numMinima
      Number of minima from level 8 to test at level 4.

    • output_1

      private ModelImage output_1
      DOCUMENT ME!

    • output_2

      private ModelImage output_2
      DOCUMENT ME!

    • output_3

      private ModelImage output_3
      DOCUMENT ME!

    • outsidePreReferenceSlice

      private ModelImage outsidePreReferenceSlice
      DOCUMENT ME!

    • outsideReferenceSlice

      private ModelImage outsideReferenceSlice
      DOCUMENT ME!

    • refImageNo

      private int refImageNo
      Indicates the first slice used as a reference slice, if regToAdjImage is true.

    • regToAdjImage

      private boolean regToAdjImage
      If true use adjacent image for registration. If false use image slice reference number to indicate the image slice to register all images to.

    • resample

      private boolean resample
      true if resolutions unequal, false if resolutions equal.

    • resampleW

      private boolean resampleW
      true if weight image must be resampled.

    • rigidFlag

      private boolean rigidFlag
      Flag used to indicate if the registration is ridgid (rotation and translation - DOF = 3.

    • rot

      private float[] rot
      Arrays used for producing graphs for DOF == 3 and register to reference image.

    • rotateBegin

      private float rotateBegin
      Coarse and fine sampling parameters.

    • rotateEnd

      private float rotateEnd
      Coarse and fine sampling parameters.

    • coarseRate

      private float coarseRate
      Coarse and fine sampling parameters.

    • fineRate

      private float fineRate
      Coarse and fine sampling parameters.

    • simpleInput_1

      private ModelSimpleImage simpleInput_1
      Simple version of an image slice of the input image.

    • simpleInputSub16_1

      private ModelSimpleImage simpleInputSub16_1
      Simple version of an image slice of the input image, subsampled by 16.

    • simpleInputSub2_1

      private ModelSimpleImage simpleInputSub2_1
      Simple version of an image slice of the input image, subsampled by 2.

    • simpleInputSub4_1

      private ModelSimpleImage simpleInputSub4_1
      Simple version of an image slice of the input image, subsampled by 4.

    • simpleInputSub8_1

      private ModelSimpleImage simpleInputSub8_1
      Simple version of an image slice of the input image, subsampled by 8.

    • simpleRef_1

      private ModelSimpleImage simpleRef_1
      Simple version of input image.

    • simpleRefSub16_1

      private ModelSimpleImage simpleRefSub16_1
      Simple version of input image, subsampled by 16.

    • simpleRefSub2_1

      private ModelSimpleImage simpleRefSub2_1
      Simple version of input image, subsampled by 2.

    • simpleRefSub4_1

      private ModelSimpleImage simpleRefSub4_1
      Simple version of input image, subsampled by 4.

    • simpleRefSub8_1

      private ModelSimpleImage simpleRefSub8_1
      Simple version of input image, subsampled by 8.

    • simpleWeightI_1

      private ModelSimpleImage simpleWeightI_1
      DOCUMENT ME!

    • simpleWeightISub16_1

      private ModelSimpleImage simpleWeightISub16_1
      DOCUMENT ME!

    • simpleWeightISub2_1

      private ModelSimpleImage simpleWeightISub2_1
      DOCUMENT ME!

    • simpleWeightISub4_1

      private ModelSimpleImage simpleWeightISub4_1
      DOCUMENT ME!

    • simpleWeightISub8_1

      private ModelSimpleImage simpleWeightISub8_1
      DOCUMENT ME!

    • simpleWeightR_1

      private ModelSimpleImage simpleWeightR_1
      Simple version of weighted input image.

    • simpleWeightRSub16_1

      private ModelSimpleImage simpleWeightRSub16_1
      Simple version of weighted input image, subsampled by 16.

    • simpleWeightRSub2_1

      private ModelSimpleImage simpleWeightRSub2_1
      Simple version of weighted input image, subsampled by 2.

    • simpleWeightRSub4_1

      private ModelSimpleImage simpleWeightRSub4_1
      Simple version of weighted input image, subsampled by 4.

    • simpleWeightRSub8_1

      private ModelSimpleImage simpleWeightRSub8_1
      Simple version of weighted input image, subsampled by 8.

    • sliceCosts

      private double[] sliceCosts
      DOCUMENT ME!

    • trans

      private float[][] trans
      DOCUMENT ME!

    • transformVOIs

      private boolean transformVOIs
      if true transform VOIs.

    • useOutsideReferenceSlice

      private boolean useOutsideReferenceSlice
      DOCUMENT ME!

    • weighted

      private boolean weighted
      Flag to determine if there are weighted images or not.

    • weightSliceImage

      private ModelImage weightSliceImage
      DOCUMENT ME!

  • Constructor Details

    • AlgorithmRegELSUNCOAR25D

      public AlgorithmRegELSUNCOAR25D(ModelImage _image, int _costChoice, int _DOF, int _interp, int _interp2, boolean mode, int refImageNum, float _rotateBegin, float _rotateEnd, float _coarseRate, float _fineRate, boolean doGraph, boolean doSubsample, boolean transformVOIs, int _baseNumIter, int _numMinima, int searchAlgorithm)
      Creates new automatic internal registration algorithm and sets necessary variables.
      Parameters:
      _image - Input image
      _costChoice - Choice of cost functions, like correlation ratio or mutual information.
      _DOF - Degrees of freedom for registration
      _interp - Interpolation method used in transformations.
      _interp2 - Interpolation method used in output
      mode - If true, register to adjacent. If false, register to reference.
      refImageNum - If register to reference, the slice all other slices are registered to. If register to adjacent, the first slice used as a reference.
      _rotateBegin - Beginning of coarse sampling range (i.e., -60 degrees).
      _rotateEnd - End of coarse sampling range (i.e., 60 degrees).
      _coarseRate - Point at which coarse samples should be taken (i.e., every 45 degrees).
      _fineRate - Point at which fine samples should be taken (i.e., every 15 degrees).
      doGraph - If true produce 2 output graphs - one for rotation and one for 2 translations
      doSubsample - If true subsample
      transformVOIs - If true, transform VOIs
      _baseNumIter - Limits the number of iterations of elsunc algorithm. maxIter in the call to elsunc will be an integer multiple of baseNumIter
      _numMinima - Number of minima from level 8 to test at level 4
      searchAlgorithm - ESLUNC, LEVENBERG_MARQUARDT, or NL2SOL.

    • AlgorithmRegELSUNCOAR25D

      public AlgorithmRegELSUNCOAR25D(ModelImage _image, ModelImage _inputWeight, int _costChoice, int _DOF, int _interp, int _interp2, boolean mode, int refImageNum, float _rotateBegin, float _rotateEnd, float _coarseRate, float _fineRate, boolean doGraph, boolean doSubsample, boolean transformVOIs, int _baseNumIter, int _numMinima, int searchAlgorithm)
      Creates new automatic internal registration algorithm and sets necessary variables.
      Parameters:
      _image - Input image
      _inputWeight - Input weighted image, used to give certain areas of the image greater impact on the registration.
      _costChoice - Choice of cost functions, like correlation ratio or mutual information.
      _DOF - DOCUMENT ME!
      _interp - Interpolation method used in transformations.
      _interp2 - Interpolation method used in output
      mode - If true, register to adjacent. If false, register to reference.
      refImageNum - If register to reference, the slice all other slices are registered to. If register to adjacent, the first slice used as a reference.
      _rotateBegin - Beginning of coarse sampling range (i.e., -60 degrees).
      _rotateEnd - End of coarse sampling range (i.e., 60 degrees).
      _coarseRate - Point at which coarse samples should be taken (i.e., every 45 degrees).
      _fineRate - Point at which fine samples should be taken (i.e., every 15 degrees).
      doGraph - If true produce 2 output graphs - 1 for rotation and one for 2 translations
      doSubsample - If true subsample
      transformVOIs - If true, transform VOIs
      _baseNumIter - Limits the number of iterations of elsunc algorithm. maxIter in the call to elsunc will be an integer multiple of baseNumIter
      _numMinima - Number of minima from level 8 to test at level 4
      searchAlgorithm - ESLUNC, LEVENBERG_MARQUARDT, or NL2SOL.

  • Method Details

    • calculateCenterOfMass2D

      public WildMagic.LibFoundation.Mathematics.Vector2f calculateCenterOfMass2D(ModelSimpleImage image, ModelSimpleImage wgtImage, boolean isColor)
      Calculates the center of mass (gravity) of a 2D image. In image space where the upper left hand corner of the image is 0,0. The x axis goes left to right, y axis goes top to bottom. (i.e. the right hand rule). One could simply multiply by voxel resolutions.
      Parameters:
      image - DOCUMENT ME!
      wgtImage - DOCUMENT ME!
      isColor - DOCUMENT ME!
      Returns:
      the center of mass as a 2D point

    • disposeLocal

      public void disposeLocal()
      Dispose of local variables that may be taking up lots of room.

    • finalize

      public void finalize()
      Prepares this class for destruction.
      Overrides:
      finalize in class AlgorithmBase

    • getCosts

      public double[] getCosts()
      accessor for costs.
      Returns:
      double[] costs

    • getRot

      public float[] getRot()
      accessor for rot.
      Returns:
      rot

    • getTrans

      public float[][] getTrans()
      accessor for trans.
      Returns:
      trans

    • getTransformedImage

      public ModelImage getTransformedImage()
      accessor to get the internally registered image.
      Returns:
      inputImage

    • runAlgorithm

      public void runAlgorithm()
      Runs the image registration. If the resolutions are unequal, the image is transformed into isotropic pixels. The resolutions of the two images after the xy isotropic transformation will be the same in the x and y dimensions. That resolution will equal the minimum resolution. If the image is weighted, the weight image is transformed into isotropic pixels in the same manner as the original. Then the image is subsampled by 2, 4, and 8 or 16. If the image is too small it will not be subsampled down to the smallest level; if it is too big, it will be subsampled to 16. The same is done with the weight image if necessary. The program loops thru levelEight, levelFour, levelTwo, and levelOne with one slice as the reference slice and one slice as the input slice. The function levelEight is called with the slices subsampled by 8 or 16; it returns two vectors with minima. Then the function levelFour is called with slices subsampled by 4 and the two vectors; it returns one vector of minima. The function levelTwo is called with slices subsampled by 2 and the vector; it returns an "answer" in the form of a MatrixListItem, which is a convenient way of storing the point, the matrix, and the cost of the minimum. Then the function levelOne is called with the minimum; it returns a final "answer", or minimum, which is used to register the input slice to the reference slice. The registered input slice data is imported into inputImage.
      Specified by:
      runAlgorithm in class AlgorithmBase

    • setInputImage2

      public void setInputImage2(ModelImage inputImage2)
      inputImage2 is not used to determine the registration, but each slice of inputImage2 undergoes the same transformation as inputImage. This is useful for color images where you wish the registration to be determined by only 1 of the colors
      Parameters:
      inputImage2 - DOCUMENT ME!

    • setInputImage3

      public void setInputImage3(ModelImage inputImage3)
      inputImage3 is not used to determine the registration, but each slice of inputImage3 undergoes the same transformation as inputImage. This is useful for color images where you wish the registration to be determined by only 1 of the colors
      Parameters:
      inputImage3 - DOCUMENT ME!

    • setReferenceSlice

      public boolean setReferenceSlice(ModelImage refSlice)
      allows the user to pass in an OUTSIDE reference slice.
      Parameters:
      refSlice - 2-Dim image for reference
      Returns:
      DOCUMENT ME!

    • subSample2DimBy2

      private static ModelSimpleImage subSample2DimBy2(ModelSimpleImage srcImage, ModelSimpleImage resultImage, boolean isColor)
      DOCUMENT ME!
      Parameters:
      srcImage - DOCUMENT ME!
      resultImage - DOCUMENT ME!
      isColor - DOCUMENT ME!
      Returns:
      DOCUMENT ME!

    • copyFloatData

      private void copyFloatData(ModelSimpleImage srcImage, ModelSimpleImage resultImage)
      DOCUMENT ME!
      Parameters:
      srcImage - DOCUMENT ME!
      resultImage - DOCUMENT ME!

    • getConstructionInfo

      private String getConstructionInfo()
      Creates a string with the parameters that the image was constructed with.
      Returns:
      Construction info.

    • getTolerance

      private double[] getTolerance(int DOF)
      Gets the tolerance vector based on the degrees of freedom (the length of the tolerance is the degrees of freedom) and.
      Parameters:
      DOF - Degrees of freedom, will be length of vector.
      Returns:
      New tolerance vector to send to optimization.

      Based on FLIRT paper: let n=pixel dimension (in one dimension) R=brain radius, here assumed to be half of field-of-view Translation tolerance = n/2 Rotation tolerance = (180/PI)*n/(2R) (converted to degrees because AlgorithmELSUNC works in degrees) Scaling tolerance = n/(2R) Skewing tolerance = n/(2R)

    • interpolate

      private void interpolate(double x, double[] initial, double[][] tranforms, boolean scale)
      Performs a bilinear interpolation on points. Takes an initial point, a vector of values to set, and an array in which to look at neighbors of that point. Sets the appropriate values in the vector. Does not set scale if the scale parameter is false.
      Parameters:
      x - Initial index into array.
      initial - Vector to set; if scale is true, set two translations and a scale. Otherwise just set translations.
      tranforms - DOCUMENT ME!
      scale - true means set the scale in the vector.

    • levelEight

      private Vector<MatrixListItem>[] levelEight(ModelSimpleImage ref, ModelSimpleImage input)
      Takes two slices that have been subsampled by a factor of 8 or 16. Sets up the cost function with the slices and the weighted slices, if necessary. Uses the coarse sampling rate and optimize translations and global scale at the given rotation. So for example, if the coarse sampling range were -30 to 30 at every 15 degrees, we would optimize at rotations of -30, -15, 0, 15, 30. Measures the cost at the fine sampling rate. Interpolates the translations and global scale to come up with a good guess as to what the optimized translation would be at that point. Takes the top 20% of the points and optimizes them. Now have a large multi-array of costs. 20% of those have been optimized and placed back into their original position in the multi-array. Looks at the 3 neighbors of a point: +, =, or - one fine sample. If the point has a cost greater than any of these, it is not a minima. Otherwise it is. Saves it in a vector of minima. Optimizes the minima over rotation as well as translations and global scale. (Previously had not optimized over rotation.) Returns two vectors, one containing the minima before optimization, one containing the minima after optimization.
      Parameters:
      ref - Subsampled by 8 or 16 reference slice.
      input - Subsampled by 8 or 16 input slice.
      Returns:
      List of preoptimized and optimized points.

    • levelFour

      private Vector<MatrixListItem> levelFour(ModelSimpleImage ref, ModelSimpleImage input, Vector<MatrixListItem> minima, Vector<MatrixListItem> optMinima)
      Takes two slices that have been subsampled by a factor of four, and two vectors of minima. Sets up the cost function with the slices and the weighted slices, if necessary. Adds the level4Factor determined during subsampling. Measures the costs of the minima on the images and sort them. Takes the top three in each vector (pre-optimization and post-optimization) and optimizes them. Puts them all into one vector. Perturbs the rotation by zero and by plus-minus fineDelta. If it's not a rigid transformation, perturbs the scales by 0, plus-minus .1, then plus-minus .2. Optimize the perturbations. Returns a vector of the perturbed, optimized minima.
      Parameters:
      ref - Reference slice, subsampled by 4.
      input - Input slice, subsampled by 4.
      minima - Preoptimized minima.
      optMinima - DOCUMENT ME!
      Returns:
      A vector of perturbed, optimized minima.

    • levelOne

      private MatrixListItem levelOne(ModelSimpleImage ref, ModelSimpleImage input, MatrixListItem item, int frame)
      Takes the two slices, no subsampling, and the best minimum so far. Sets up the cost function with the slices and the weighted slices, if necessary. Adds the level1Factor determined during subsampling. Performs one optimization run, with the maximum allowable degrees of freedom as specified by the user (the max is 7). Returns the best minimum.
      Parameters:
      ref - Reference slice.
      input - Input slice.
      item - Best minimum so far.
      frame - DOCUMENT ME!
      Returns:
      Best minimum after optimization.

    • levelOne2D

      private MatrixListItem levelOne2D(ModelSimpleImage ref, ModelSimpleImage input)
      This routine is used only with DOF = 2, translations only. LevelEight, levelFour, and levelTwo are skipped Takes the two slices, no subsampling. Sets up the cost function with the slices and the weighted slices. Adds the level1Factor determined during subsampling. Performs one optimization run, with 2 degrees of freedom
      Parameters:
      ref - Reference slice.
      input - Input slice.
      Returns:
      Best minimum after optimization.

    • levelTwo

      private MatrixListItem levelTwo(ModelSimpleImage ref, ModelSimpleImage input, Vector<MatrixListItem> minima)
      Takes two slices that have been subsampled by a factor of 2 and a vector of minima. Sets up the cost function with the images and the weighted images, if necessary. Adds the level2Factor determined during subsampling. Measures the costs of the minima at the images. Optimizes the best minimum with 4 degrees of freedom, then 5, then 7. If the user has limited the degrees of freedom to 3, there will only be one optimization run, with 3 degrees of freedom. Returns the best minimum after optimization.
      Parameters:
      ref - Reference slice, subsampled by 2.
      input - Input slice, subsampled by 2.
      minima - Minima.
      Returns:
      The optimized minimum.