Patch-based regularization reconstruction requires many iterations to suppress noise and retain the fine structural features of the original image. To accelerate the algorithm convergence, reduce the number of iterations, and reduce reconstruction time, we proposed to improve this patch-based regularization reconstruction method. We did this by adding TV regularization following the MLEM image update. An FR step was then used to extract the lost fine structural features from the residual image between the TV regularized image and the fused image based on patch regularization. These features were then added back to the fused image. With these added steps, each iteration of the image should gain more structural information. Using the sparsity of image gradient magnitude to calculate TV is one of the most commonly used methods.

Here, s and t are indexes of the desired tracer distribution map location, and α is a small constant used to maintain differentiability concerning image intensity. We assigned α a value of 10−8 in this study.

The FR step mentioned above is described by the following equation:

where xFR is the feature-refined image, x is the pixel-by-pixel fused image, and ν is the residual image between the TV-based MLEM-updated image xTV and the pixel-by-pixel fused image x. The symbol ⊗ denotes pointwise multiplication. f is a feature descriptor, which is defined as follows (28):

where the constant C is introduced for numerical stability (C =1.25 × 10–6 in this study). Local statistics σp, σq, and σpq at pixel j are defined as  σp=(1N1jpj(x(j)P(j))2)1/2, σq=(1N1jpj(xd(j)Q(j))2)1/2 and σqp=1N1j(pj, qj)(x(j)P(j))(xd(j)Q(j)), where P=1Njpjx(j), Q=1Njqjxd(j), and pj and qj denote two local image patches with a size of N×N centered at pixel j. These images patches were extracted from x and degraded image xd obtained by applying a 2D Gaussian filter to x, respectively. The parameters of the Gaussian filter function were a filter size of 5×5 and a standard deviation of the Gaussian function of 10. The nature of the proposed model structure descriptor involves a contrast variation component and a structure correlation component. The former calculates the reduction of contrast variation caused by the degrading operation, and the latter is the structural correction between the original image and the degraded image. The value of each element of the feature descriptor image f falls within the interval [0, 1]; a larger value is correlated with a greater likelihood of belonging to part of the structure.

The feature descriptor, designed to distinguish structures from noise and artifacts, plays a vital role in our improved algorithm. As such, several scalar parameters in this algorithm should be carefully tuned. For example, an image patch size of 7 × 7 is a good choice to balance structure-detection capacity and computational efficiency in the FR step. Additionally, parameter C is included to avoid instability when σp2+σq2 is close to zero. C should therefore be assigned a small constant value (C =1.25 × 10–6 in our study). The added TV regularization and FR steps do not increase the calculation amount of the algorithm, but accelerate the algorithm convergence, reduce the number of iterations, and reduce reconstruction time. This improved approach is described in Supplementary file 1.

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