Feature-aware unsupervised lesion segmentation for brain tumor images using fast data density functional transform

Abstract

We demonstrate that isomorphically mapping gray-level medical image matrices onto energy spaces underlying the framework of fast data density functional transform (fDDFT) can achieve the unsupervised recognition of lesion morphology. By introducing the architecture of geometric deep learning and metrics of graph neural networks, gridized density functionals of the fDDFT establish an unsupervised feature-aware mechanism with global convolutional kernels to extract the most likely lesion boundaries and produce lesion segmentation. An AutoEncoder-assisted module reduces the computational complexity from [Formula: see text] to [Formula: see text], thus efficiently speeding up global convolutional operations. We validate their performance utilizing various open-access datasets and discuss limitations. The inference time of each object in large three-dimensional datasets is 1.76 s on average. The proposed gridized density functionals have activation capability synergized with gradient ascent operations, hence can be modularized and embedded in pipelines of modern deep neural networks. Algorithms of geometric stability and similarity convergence also raise the accuracy of unsupervised recognition and segmentation of lesion images. Their performance achieves the standard requirement for conventional deep neural networks; the median dice score is higher than 0.75. The experiment shows that the synergy of fDDFT and a naïve neural network improves the training and inference time by 58% and 51%, respectively, and the dice score raises to 0.9415. This advantage facilitates fast computational modeling in interdisciplinary applications and clinical investigation.

Figure 1

The corresponding relationship between a many-particle system, a complete graph, and an image grid space ${\mathbb{R}}^2$. Geometric deep learning establishes the formation of a set of connected pixels to a feature map in an automorphic group. We only delineated the interactions (the dashed lines) within pair nodes in the many-particle system to avoid complicated structures occurring between them.

Figure 2

The flowchart of the proposed fDDFT. In Step 1, the mapping of the FLAIR image constructs the KEDF landscape, followed by forming the reciprocal distance kernel (RDK). In Step 2, element-wise products of 2D-FFTs of the image and kernel are calculated and then inversed for PEDF estimation. In Step 3, two algorithms are introduced to deal with problems of similarity convergence and geometric instability in the PEDF and LDF landscapes, respectively. Comparing elements of the enhanced PEDF landscape and the aware feature map, indicating a specific subset group of the tumor. In the last step (Step 4), segmentation of the candidate tumor image is achieved automatically using the characteristics of pixel connectivity. We provided the relevant algorithms in Supplementary Code.

Figure 3

The performance of cases with interferences. As labeled in (a), the four sub-images of each panel indicate the FLAIR image (upper-left), the aware feature map (upper-right), normal tissue (lower-left), and the tumor candidate (lower-right). Image dimensions are next to the owning image in the lower right corner. The FLAIR images of (a) and (b) have notations added by radiologists (indicated by red arrows) and represent coronal and axial views, respectively. There is a white margin (indicated by a red arrow) in the original image of (c) and a relatively small tumor in (d). The CPU time for this small dataset ranges between 0.06 to 0.09 s. We acquired these images from an open-source dataset.

Figure 4

Demonstrations of the reinforced algorithm of similarity convergence. (a) and (b) exhibit different input figure representations and their corresponding projection area evolutions of functional landscapes. The former represents a general situation of similarity convergence, in which the PEDF projection, as expected, indicates the range with high similarity in the final state, as shown by the dashed lines. However, since the figure representation in case (b) has three discrete blocks, the biggest one attracts much attention in the PEDF estimation and products a deformed PEDF projection. Thus, we introduced the HED and LED projections to replace the original PEDF one to generate better outcomes for this kind of circumstance.

Figure 5

The accuracy distribution and structural analyses of representative cases. As illustrated in (a), the soft dice score distribution reaches the standard performance of conventional deep neural networks. The blue and pink bars in (a) represent the statistical calculations using values of the permissible smallest edge number of $3\pi \sqrt{{\Vert I\Vert }_0}/2$ and $\pi \sqrt{{\Vert I\Vert }_0}$, respectively. (b) exhibits the two-dimensional aware feature maps of three representative results. The numbers of 1, 2, and 4 represent labels of the necrotic and non-enhancing tumor core, the peritumoral edema, and the GD-enhancing tumor, respectively. (c), (d) and (e) present the representative reconstructed results of three-dimensional brain tumor images and their corresponding label structures.