Fractals are used in many fields, including image analysis. Fractal dimension is important for characterizing textures in images, but estimating it accurately can be challenging, especially for small images. This work introduces new algorithms for accurately estimating fractal dimension in both small and large digital images. The researchers also introduce fractal moments, which are parameters that can characterize textures in any digital image. Fractal dimension is a special case of fractal moments in some cases. These new techniques are applied to analyzing mammograms, MRI and CT images, and X-ray images for early detection of breast cancer, brain tumors, and osteoporosis. The results show that fractal moments can effectively characterize textures in digital images, which is a new and important finding for image analysis.