An Uncertainty-Aware Loss Function Incorporating Fuzzy Logic: Application to MRI Brain Image Segmentation

Hanuman Verma; Akshansh Gupta; Pranabesh Maji; Saurav Mandal; Vijay Kumar Pandey

cs.CVApr 13, 2026

An Uncertainty-Aware Loss Function Incorporating Fuzzy Logic: Application to MRI Brain Image Segmentation

Authors: Hanuman Verma, Akshansh Gupta, Pranabesh Maji, Saurav Mandal, Vijay Kumar Pandey

Organizations: Department of Mathematics, Bareilly College, Bareilly (MJP Rohilkhand University), Uttar Pradesh, 243005, India · CSIR-Central Electronics Engineering Research Institute, Pilani, Rajasthan, 333031, India · CICMR, Regional Medical Research Center, Dibrugarh, India · Department of Statistics, Bareilly College, Bareilly, Uttar Pradesh, 243005, India

Abstract

Accurate brain image segmentation, particularly for distinguishing various tissues from magnetic resonance imaging (MRI) images, plays a pivotal role in finding the neurological dis ease and medical image computing. In deep learning approaches, loss functions are very crucial for optimizing the model. In this study, we introduce a novel loss function integrating fuzzy logic to deals uncertainty issues in brain image segmentation into various tissues. It integrates the well-known categorical cross-entropy (CCE) loss function and fuzzy entropy based on fuzzy logic. By employing fuzzy logic, this loss function accounts for the inherent uncertainties in pixel classifications. The proposed loss function has been evaluated on two publicly available benchmark datasets, IBSR and OASIS, using two widely recognised architectures, U-Net and U-Net++. Experimental results demonstrate that the trained model with proposed loss function provided better results in comparison to the CCE optimisation function in terms of various performance metrics. Additionally, it effectively enhances segmentation performance while handling meaningful uncer tainty during training. The findings suggest that this approach not only improves segmentation outcomes but also contributes to the reliability of model predictions.

Figures & tables

Fig. 1 : Deep learning architecture with proposed fuzzy categorial cross entropy for brain image segmentation.

Fig. 2 : U-Net architecture featuring various neurons and 2D convolution with a kernel size of $3\times 3$ , followed by max pooling with a kernel size of $2\times 2$ . The ReLU activation function is used, followed by Batch normalization.

Fig. 3 : U-Net++ architecture with the skip connectionsn.

Algorithm	Metrics	$L_{CCE}$	$L_{FCCE}$
U-Net	AC	0.9986	0.9991
	$AC_{val}$	0.9907	0.9910
	DC	0.9980	0.9986
	$DC_{val}$	0.9903	0.9906
	IoU	0.9960	0.9972
	$IoU_{val}$	0.9808	0.9815

TABLE I : U-Net and U-Net++ architecture segmentation performance with

L_{CCE}

and

L_{FCCE}

loss function for IBSR data. We take the number of epochs 100 and Batch size 2. And use the segmentation metrices as accuracy

(AC)

, Dice coefficient

(DC)

and Intersection over Union

(IoU)

and their corresponding validations.

Panel of Figure 5 — Fig. 4 : Plot for training versus validation performance for U-Net and U-Net++ architecture for IBSR data with categorical cross-entropy $(L_{CCE})$ and fuzzy categorical cross entropy $(L_{FCCE})$ loss optimization function along with epoch in term of performance metrices: accuracy, Dice coefficient and Intersection over Union. The performance of U-Net and U-Net++ with $L_{FCCE}$ in term of $AC$ , $DC$ , $IoU$ performance slightly better with the epoch. During training the model, the U-Net validation with $L_{FCCE}$ in term of ${AC}_{{val}}$ , ${DC}_{{val}}$ , ${IoU}_{{val}}$ providing better results with optimizing with fuzzy logic. The U-Net++ validation with $L_{FCCE}$ in term of ${AC}_{{val}}$ , ${DC}_{{val}}$ , ${IoU}_{{val}}$ providing more better results in comparison to $L_{CCE}$ . The fuzzy categorical cross entropy $(L_{FCCE})$ optimizing the objective with the quantifying the uncertainty during training the model.

Fig. 5 : Qualitative segmentation results using U-Net with $L_{CCE}$ , and U-Net with $L_{FCCE}$ (proposed loss function) on the IBSR brain data for the various image (a), (b) and (c). The most left row, comprised the raw image and corresponding predicted image comprises with various tissues such as background (BG), CSF, GM, and WM in this study. U-Net architecture with loss function is trained and designed and then predicted and depicted for same images (a), (b) and (c) respectively. The U-Net with $L_{FCCE}$ shows the better segmentation in CSF, GM and WM tissues as it handles the uncertainty at the boundary region, as shown in case of CSF tissues. The boundary pixels in CSF tissues are changing which can be promising with uncertainty.

Fig. 6 : Qualitative segmentation results using U-Net++ with $L_{CCE}$ , and U-Net++ with $L_{FCCE}$ (proposed loss function) on the IBSR brain data for the various image (a), (b) and (c). The most left row, comprised the raw image and corresponding predicted image comprises with various tissues such as background (BG), CSF, GM, and WM. The U-Net++ with $L_{FCCE}$ shows the better segmentation in CSF, GM and WM tissues as it handles the uncertainty at the boundary region, as shown in case of CSF tissues.

Algorithm	Metrics	$L_{CCE}$	$L_{FCCE}$
U-Net	AC	0.9873	0.9889
	$\text{AC}_{\text{val}}$	0.9518	0.9306
	DC	0.9801	0.9842
	$\text{DC}_{\text{val}}$	0.9488	0.9296
	IoU	0.9612	0.9691
	$\text{IoU}_{\text{val}}$	0.9034	0.8701

TABLE II : U-Net and U-Net++ architecture segmentation performance with

L_{CCE}

and

L_{FCCE}

loss function for OASIS data in term of

AC

DC

IoU

and respective validations

{AC}_{{val}}

{DC}_{{val}}

{IoU}_{{val}}

. We use EarlyStopping and Batch size 2 for the OASIS data.

Panel of Figure 9 — Fig. 7 : Plot for training versus validation performance for U-Net and U-Net++ architecture for OASIS brain data with $(L_{CCE})$ and $(L_{FCCE})$ loss optimization function along with epoch in term of performance metrices: accuracy, Dice coefficient and Intersection over Union. For OASIS data we used the early stoppage to avoid the overfitting problem. The epoch varies as the model achieve the optimization. The performance of U-Net and U-Net++ with $(L_{FCCE})$ in term of $AC$ , $DC$ , $IoU$ performance slightly better with the epoch for OASIS data. During training the model, the U-Net validation with $(L_{FCCE})$ in term of ${AC}_{{val}}$ , ${DC}_{{val}}$ , ${IoU}_{{val}}$ providing less or equal results. The U-Net++ validation with $(L_{FCCE})$ in term of ${AC}_{{val}}$ , ${DC}_{{val}}$ , ${IoU}_{{val}}$ providing more better results in comparison to $(L_{CCE})$ .

Medical Image Segmentation

Medical image segmentation delineates organs, lesions, and other structures in scans to support measurement, diagnosis, treatment planning, and research.

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A Novel Framework using Intuitionistic Fuzzy Logic with U-Net and U-Net++ Architecture: A case Study of MRI Bain Image Segmentation

Hanuman Verma, Kiho Im, Akshansh Gupta, M. Tanveer

Department of Mathematics, Bareilly College, Bareilly (MJP Rohilkhand University), Uttar Pradesh, 243005, India · Fetal Neonatal Neuroimaging and Developmental Science Center, Boston Children’s Hospital, Harvard Medical School, Boston, MA 02115, USA · Division of Newborn Medicine, Boston Children’s Hospital, Harvard Medical School, Boston, MA 02115, USA+3

Medical Image Segmentation U-Net Semantic Segmentation Fuzzy Logic Segmentation Uncertainty Estimation

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Accurate segmentation of brain images from magnetic resonance imaging (MRI) scans plays a pivotal role in brain image analysis and the diagnosis of neurological disorders. Deep learning algorithms, particularly U-Net and U-Net++, are widely used for image segmentation. However, it finds difficult to deal with uncertainty in images. To address this challenge, this work integrates intuitionistic fuzzy logic into U-Net and U-Net++, propose a novel framework, named as IFS U-Net and IFS U-Net++. These models accept input data in an intuitionistic fuzzy representation to manage uncertainty arising from vague ness and imprecise data. This approach effectively handles tissue ambiguity caused by the partial volume effect and boundary uncertainties. To evaluate the effectiveness of IFS U-Net and IFS U-Net++, experiments are conducted on two publicly available MRI brain datasets: the Internet Brain Segmentation Repository (IBSR) and the Open Access Series of Imaging Studies (OASIS). Segmentation performance is quantitatively assessed using Accuracy, Dice Coefficient, and Intersection over Union (IoU). The results demonstrate that the proposed architectures consistently improve segmentation performance by effectively addressing uncertainty

Abstract

Medical Image Segmentation

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