The effect of this is shown as scaling the projection of an object of width $aI$ from the image plane $I$ to the object plane $O$, as shown in Fig. 3, resulting in an object width of $aO$. This results in a frequency scaling in frequency space and is described by Eq. (16) and application to 4-D DSA in Eq. (17). Display Formula
$fO=1aO=maI=m\xd7fI,$(16)
Display Formula$fFBP=m\xd7fP4D=m\xd7(min[fK,fPt]).$(17)
The last operation to be performed is that of multiplying the constraining image with the backprojection of the temporal projection. A multiplication in image space is a convolution in frequency space. The wire object in the constraining image and the weighting volume image were modeled with a pair of Jinc functions^{17} to aid in determining the effects of the multiplication in image space. Various sizes were used in the model and resulted in the maximum spatial frequency tracking with object that had the highest spatial frequency or the smaller of the two objects in image space. Therefore, it was determined that maximum spatial frequency of the multiplication of the constraining image with the weighting volume is the maximum of the two spatial frequencies of the two objects when convolved in frequency space as described by Eq. (18) and with application to 4-D DSA shown in Eq. (19). Display Formula$fC=max([fA,fB]),$(18)
Display Formula$f4D=max([fIC,fFBP]).$(19)
The resulting limiting resolution of 4-D DSA, $f4D$, is the maximum of the limiting resolution of the constraining image, $iC$, and the minimum of the limiting resolution of the blurring kernel and the temporal projection data, $pt$, scaled to the image plane by the scale factor, $m$, as shown in Eq. (20). Display Formula$f4D=max([fIC,fFBP])=max([fIC,m\xd7min([fK,fPt])]).$(20)
The 4-D DSA process includes a number of nonlinear operations such as thresholding and division of the temporal projections by reprojections of the constraining volume. Although 4-D DSA is a nonlinear system, it is treated as linear at an operating point. The standard approach^{18} of scanning a fine highly attenuating wire centered in and perpendicular to the transverse plane was performed. The effects of the 4-D DSA reconstruction on volumetric spatial resolution were investigated using both an ISPH and the scan of a physical 76 micron (um) diameter tungsten wire centered in a cylindrical supporting structure. ISPH was modeled after the PPH as cylinder centered in the transverse plane with the axis parallel to the axis of rotation extending throughout the entire volume of interest. The physical phantom is shown in Fig. 4. The constraining image inherits the reconstruction parameters from the 3-D DSA as it is a 3-D DSA where a threshold has been applied. The spatial resolution of a 4-D temporal volume can be obtained by the same means as for the 3-D DSA. The 3-D DSA is compared with the 4-D temporal frames (temporally enhanced 3-D volumes) of 4-D DSA reconstructions. This was done on a volumetric basis using a single transverse slice of a physical or digital phantom of a small wire. Resulting PSF data were radially averaged and Fourier analysis performed to generate averaged MTF data for each 3-D DSA volume and 4-D DSA temporal slice. The limiting spatial resolution was automatically determined by finding the spatial resolution when 10% of the magnitude at zero spatial frequency was reached. Numerical simulations and reconstruction of the real-world phantom were performed using a combination of MATLAB^{®} (The Mathworks Inc., Natick, Massachusetts) and CUDA (NVIDIA Corp. Santa Clara, California) based software. The spatial resolution at the center of the central slice of the C-Arm CT biplane system (Artis zee, Siemens Healthcare, Forchheim, Germany) used is determined by Eq. (21) and is based on the Nyquist sampling criteria using the detector spacing ($du$), imaging geometry source to image distance (SID), and source to object distance (SOD) to determine the minimal detectable distance achievable in the transverse plane. This equation does not account for blurring effects due to focal spot, geometric instabilities of the C-Arm, or projection filtering. The minimal resolvable distance calculation is shown in Eq. (22). Display Formula$fNyquist=12\xd7du\xd71m=m2\xd7du=SID2\xd7du\xd7SOD=1200[mm]2[delcyc/]\xd70.3080[mmdel]\xd7750[mm]=2.59[cycmm]\u22482.60[lpmm].$(21)
The scan geometry was determined using a standard 4-D DSA acquisition procedure without zoom of the detector/C-Arm with SID set to the maximum of 1200 mm and SOD held constant at 750 mm resulting in the magnification factor $m$, and collimation set to maximum field of view with $2\xd72$ binning yielding a detector resolution of $1240\xd7960$ detector elements with isotropic detector element size of 0.3080 mm ($2480\xd71920$ native nonbinned resolution with 0.154 mm isotropic detector elements). The minimal resolvable distance is calculated with Eq. (22), which was used to ensure the reconstruction grid was smaller than this dimension. Display Formula$dmin=dum=1fNyquist=0.385[mm].$(22)
The selection of the maximum allowable wire size was found to be 86 micron using the approach of Nickoloff.^{19} The reconstruction grid (slice) was 512 by 512 voxels with a 0.0376 mm isotropic voxel size. The voxel size was made considerably smaller than the minimum resolvable object size, $dmin$, to ensure proper reconstruction and 4-D DSA reconstruction artifacts could be properly investigated. Off axis and off central slice resolution was not investigated in this research and is a topic for future investigation and research. The physical wire phantom was built using a Scientific Instruments (New Jersey) 0.076 mm tungsten wire part number W91 surrounded in an air-filled thin-walled plastic cylinder, 47 mm O.D., as a supporting structure as shown in Fig. 3. The phantom was then scanned on C-Arm CT biplane system. The scan procedure was 8.2 s in duration providing 248 projections, at a frame rate of 30 fps, at 60.4 kVp and 168 mAs.