Convergence analysis for sequence or series of function is a well researched and known topic. Your need may only to have nice per-processed data for high accuracy via your proposed method. Recently, there is hot use of deep learning for the applications you mentioned, so I would suggest you to focus on deep learning algorithms, see e.g., http://downloads.hindawi.com/journals/cin/2018/7068349.pdf

http://papers.nips.cc/paper/7141-what-uncertainties-do-we-need-in-bayesian-deep-learning-for-computer-vision.pdf

Most of the current Breast Ultrasound (BUS) images segmentation approaches focuses on designing new feature extractors by changing the well-known approach "U-Net". Refer to this survey paper:

Automatic Breast Ultrasound Image Segmentation: A Survey

Hello. As other respondents have mentioned, deep learning convolutional neural networks are the most promising approach for semantic segmentation of image data. While it is not immediately clear what you mean by "earlier limitations associated with ultrasound images of breast cancer segmentation," I assume you mean approaches that use feature-based classifiers where the features are computed from transforms or other image processing stages. Deep learning, on the other hand, is able to automatically construct its own feature maps by repeated presentation of input output data image pairs. The challenge for ultrasound imaging is that the data are very noisy and there are numerous artefacts associated with the imaging modality (reverberation, speckle, angle dependence, ultrasound machine settings, etc.).

The U-Net approach mentioned previously has been applied to microscope cell segmentation from 2015, where it won the ISBI challenge. If you want to apply such an approach to ultrasound images, you will need a lot of annotated data. This means that each training image must be accompanied by a "mask" of the same size that specifies the class to which each pixel belongs (in the simplest case this is binary for healthy/cancerous). The performance of the classifier (accuracy, specificity, sensitivity) depends critically on the quality of the annotations and on the diversity of the training set. You can robustify againt variations by augmenting the data set by rotation/scaling, etc.

Please recommend if you find this answer useful.