This paper presents a new box-constrained multiplicative iterative (BCM) algorithm for image restoration, which addresses the computational intensity and the finite pixel value range of digital images. The BCM algorithm is designed to impose box constraints on pixel values, ensuring that the restored image lies within a specified dynamic range. Unlike traditional gradient projection methods, which may suffer from slow convergence or repeated active set identification, the BCM algorithm requires only pixel-wise updates in each iteration and does not involve matrix inversion. The paper provides a convergence proof for the BCM algorithm and applies it to total variation (TV) image restoration problems under different noise models, including Poisson, Gaussian, and salt-and-pepper noises. The BCM algorithm is formulated based on the penalized likelihood (PL) principle, where the true image is estimated by minimizing a penalized negative log-likelihood function. The penalty function restricts the restored image to satisfy certain local smoothness conditions. The box-constrained problem is solved using a multiplicative iterative (MI) algorithm, which has been successfully applied to positively constrained tomographic reconstruction and TV regularized image restorations. The BCM algorithm is designed to handle general noise distributions and penalty functions, making it versatile for various image processing tasks. The paper includes a detailed description of the BCM algorithm, its implementation, and convergence properties. It also provides a comparison with other methods, such as the fast iterative shrinkage/thresholding algorithm (FISTA), projected fast total variation deconvolution (FTVD), and projected TV Minimization by Augmented Lagrangian (TVAL). The results demonstrate that the BCM algorithm converges faster and achieves higher peak signal-to-noise ratios (PSNR) compared to these methods, especially for TV regularized image restorations. Additionally, the BCM algorithm produces better PSNR values and more stable results when compared to simple projections of unconstrained or positively constrained restorations onto the box constraints. The paper concludes with a Monte-Carlo simulation that shows the BCM algorithm's ability to produce optimal solutions with better mean and variance properties of the PSNR, further validating its effectiveness in image restoration tasks.This paper presents a new box-constrained multiplicative iterative (BCM) algorithm for image restoration, which addresses the computational intensity and the finite pixel value range of digital images. The BCM algorithm is designed to impose box constraints on pixel values, ensuring that the restored image lies within a specified dynamic range. Unlike traditional gradient projection methods, which may suffer from slow convergence or repeated active set identification, the BCM algorithm requires only pixel-wise updates in each iteration and does not involve matrix inversion. The paper provides a convergence proof for the BCM algorithm and applies it to total variation (TV) image restoration problems under different noise models, including Poisson, Gaussian, and salt-and-pepper noises. The BCM algorithm is formulated based on the penalized likelihood (PL) principle, where the true image is estimated by minimizing a penalized negative log-likelihood function. The penalty function restricts the restored image to satisfy certain local smoothness conditions. The box-constrained problem is solved using a multiplicative iterative (MI) algorithm, which has been successfully applied to positively constrained tomographic reconstruction and TV regularized image restorations. The BCM algorithm is designed to handle general noise distributions and penalty functions, making it versatile for various image processing tasks. The paper includes a detailed description of the BCM algorithm, its implementation, and convergence properties. It also provides a comparison with other methods, such as the fast iterative shrinkage/thresholding algorithm (FISTA), projected fast total variation deconvolution (FTVD), and projected TV Minimization by Augmented Lagrangian (TVAL). The results demonstrate that the BCM algorithm converges faster and achieves higher peak signal-to-noise ratios (PSNR) compared to these methods, especially for TV regularized image restorations. Additionally, the BCM algorithm produces better PSNR values and more stable results when compared to simple projections of unconstrained or positively constrained restorations onto the box constraints. The paper concludes with a Monte-Carlo simulation that shows the BCM algorithm's ability to produce optimal solutions with better mean and variance properties of the PSNR, further validating its effectiveness in image restoration tasks.