Non-Uniform Object-Space Pixelation (NUOP) for Penalized Maximum-Likelihood Image Reconstruction for a Single Photon Emission Microscope System

Abstract

This paper presents a non-uniform object-space pixelation (NUOP) approach for image reconstruction using the penalized maximum likelihood methods. This method was developed for use with a single photon emission microscope (SPEM) system that offers an ultrahigh spatial resolution for a targeted local region inside mouse brain. In this approach, the object-space is divided with non-uniform pixel sizes, which are chosen adaptively based on object-dependent criteria. These include (a) some known characteristics of a target-region, (b) the associated Fisher Information that measures the weighted correlation between the responses of the system to gamma ray emissions occurred at different spatial locations, and (c) the linear distance from a given location to the target-region. In order to quantify the impact of this non-uniform pixelation approach on image quality, we used the Modified Uniform Cramer-Rao bound (MUCRB) to evaluate the local resolution-variance and bias-variance tradeoffs achievable with different pixelation strategies. As demonstrated in this paper, an efficient object-space pixelation could improve the speed of computation by 1-2 orders of magnitude, whilst maintaining an excellent reconstruction for the target-region. This improvement is crucial for making the SPEM system a practical imaging tool for mouse brain studies. The proposed method also allows rapid computation of the first and second order statistics of reconstructed images using analytical approximations, which is the key for the evaluation of several analytical system performance indices for system design and optimization.

Keywords: Non-uniform object-space pixelation (NUOP); penalized maximum-likelihood; single-photon emission microscope (SPEM).

Fig. 1

Cross-sections of several FIM images derived with different PDFs. Note that these FIMs were evaluated using the system-response functions after rebinning. The profiles shown are corresponding to the source pixels inside a 4 mm diameter target-region. The NUOP process has virtually no effect on the values of the FIM elements compared. x-axis is the axis of the SPECt system as defined in Section II-F.

Fig. 2

Comparing reconstructed images with different object-space pixelation schemes and noise-free projection data. The pixelation-density functions used are indicated in the top row and the parameters used for each PDF are indicated. U-P: Uniform-pixelation. The 2-D slices shown are perpendicular to the common axis and 1 mm from the center. The reconstructions were performed with 500 iterations a β was set to 0 for all cases.

Fig. 3

Comparing reconstructed images with different object-space pixelation schemes and noisy projection data. The entire phantom contains 250 μCi activity. All images were reconstructed using PML algorithm with 500 iterations. The penalization factor β was 10⁻¹⁴ for all reconstructions.

Fig. 4

PDFs derived for differently sized target-regions.

Fig. 5

Reconstructed images of the brain phantom with different PDFs as shown in Fig. 4 and with uniform pixelation (U-P).

Fig. 6

Comparison between resolution-variance (R-V) and bias-variance (B–V) curves achieved with different PDFs. The target-region was defined as the central spherical volume of 4 mm diameter. The pixel of interest, for which the R–V curves were evaluated, is located at the center of the object. Parameters used in PDFs are shown in the figure. Several R–V and B–V curves achieved with uniform-pixelation (U-P) with 32³ × 256 μm pixels, 64³ × 128 μm pixels and 128³ × 64 μm pixels are also compared.

Fig. 7

Comparison between resolution-variance and bias-variance curves achieved with differently sized target-region. The simulated brain phantom was used in this study.