Exponential-Type Regularization with Bivariate Shrinkage for Medical Image Denoising

Noise degrades images from X-ray, ultrasound and magnetic resonance imaging (MRI) systems. Penalized least squares regression (PLSR) with a non-convex penalty suppresses it without training data. We study a base-m exponential penalty ϕ(x)=(1−m−a|x|)/(alnm) with bivariate shrinkage of neighboring wavelet coefficients. We show that this penalty equals the exponential penalty with b=alnm such that the base m adds no flexibility. We derive the convexity condition of the bivariate shrinkage explicitly and prove that its fixed-point solver converges geometrically. On X-ray, ultrasound and MRI images with additive white Gaussian noise, the method performed on par with the trigonometric and logarithmic penalties and was clearly better than soft thresholding. On 29 slices of a public 7T MRI dataset with Rician noise, it improved the PSNR of soft thresholding by 0.15 dB in all 87 paired tests. The gain is specific to PSNR, since without cycle spinning, the SSIM fell slightly from 0.835 to 0.833. BM3D and a pretrained DnCNN remained 0.8–0.9 dB ahead of its translation-invariant version.