New Regularization Approach

for

 Volume Morphing

          Volume Morphing: 

         Volume morphing is a technique used for generating smooth three-dimensional (3D) image transformations and deformations. In Volume Images morphing the main purpose is creating a smooth transition between two volume  images. Today, animations with deforming objects are frequently used in various computer graphics applications. Morphing of 3D objects is one of the techniques which realize a shape transformation between two or more existing objects. Even if morphing seem to have more applications in entertainment industry, In has more applications areas. Morphing in 3D is  used to generate intermediate 3D models directly from the given models.

        In the generation of the intermediate morphs, the surface interpolation based on the Distance Field Metamorphosis is employed. The resulting surfaces are regulated by using the regularization term.

        Distance Transformation:

        The distance transformations of the initial and final volume image is used to approximate the morphing surfaces. The distance transform is calculated in signed form. The pixels inside the volume is marked with a negative sign and the outside pixels are left positive. The followings are the slices shown for a cube (signed distance transform is applied) and a sphere (unsigned distance transform is applied). the further you go from the surface of the image the pixel values are given higher pixel values. In the signed case, the inside of the cube is seem as almost black because the inside of the pixels have negative sign. In the unsigned distance transform, sphere, the inside of the sphere can be seen having lighter pixels. In the morphing signed distance transform is utilized.  

        Morphing:

        The morphing is simply the interpolation of surfaces, in the slices only the 2D projections of this can be seen.  The interpolation is applied to the pixels having negative sign if the multiplication of corresponding pixel value of the initial and final volume image's signed distance transform, is negative. it means the pixel that is going to be swept in the morphing sequence should be either inside of one of the volumes but not both.

    In the above images the areas that morping applied has a common property. That is

            SDT1 * SDT2 < 0

    The interpolation with distance transform is fairly simple. if we represent the pixel values of pixels in the morphing with f(x) ranging from 0 to 1 along the trajectory of the arrows . Then we can make an approximation for f(x) using distance transforms of two volume images. We can write f(x) as follows:

            f(x) =  |SDT1|  /  (  |SDT1| + |SDT2| )

    SDT1 = signed distance transform of first object  (initial volume image)

    SDT2 = signed distance transform of second object (final volume image)

        Because they are signed distance transforms absolute values should be taken.

        The followings are the resultant slices that the morphing applied. The first one is morphing is from a generic cube to a generic sphere. The second is  from a damaged brain to a healthy brain. In the cube to sphere slice  the morphing is more obvious. From the cubes borders to the sphere's borders the pixel values gradually increase. The sequence of images are obtained putting in  these pixels ,that are smaller than a certain T time value, to the initial volume (cube). So that continuous sequences are obtained from a cube to a sphere. The same logic applies to the brain morhing.   

generic cube ---> generic sphere	damaged brain tissue ---> healthy brain tissue

         The Regularization Method:

        Since the surface interpolations is expressed as function interpolations, well known mathematical tools such as regularization can be employed. The gradient of the curves can be used to for a regularization method. To minimize the gradient so that the surface evolutions will be reflected smoothly and gradually minimizing the following integral can be used

                      ∫ |Ñ f|²

        In this case the global average of gradient may be minimized, but the drastic changes of the gradient can not be prevented. to outcome this situation the supreme of the gradient can be studied. If we try to minimize |Ñ f| at every point of the morphed 3D image. Then  we can express all the functions supreme gradient minimization as series of functionals. Becaouse we want to minimize the gradient at every point of the morphed surfaces. Because of that it is better to write the integral as follows : 

            functionals = ( ∫ |Ñ f| 2Ndx ) 1/2N  , N = 1,2,3....

            If we plug in the gradient in 3D and after minimizing by using The Euler equation

            Then we get  the Infinite Laplacian Equation

        (fx2)*fxx+ (fy2)*fyy+ (fz2)*fzz+ 2*( fx*fy*fxy +fz*fy*fyz + fz*fx*fxz)
                   

            The derivatives can be easily  calculated by using finite differences.

            The morphed image is iterated the ILE is calculated and the till  a local minimum of supreme of |Ñ f| reached this process continues.  

        Sample Morphs

        The technique is applied to some generic volume data and to segmented brain volume data obtained from Imaging and Informatics Group of Lawrence Berkeley Lab                    

a slice from the data	long mpeg	short mpeg	a slice from the data
initial damaged brain		final healthy brain

Imaging and Informatics Group of Lawrence Berkeley Lab