problems – Low-dose x-ray CT (Kang et al, Chen et al, Wolterink et al) – Sparse view CT (Jin et al, Han et al, Adler et al) – Interior tomography (Han et al) – Stationary CT for baggage inspection (Han et al) – CS-MRI (Hammernik et al, Yang et al, Lee et al, Zhu et al) – US imaging (Yoon et al ) – Diffuse optical tomography (Yoo et al) • Advantages – Very fast reconstruction time – Significantly improved results Other works
nonlinearity such as rectified linear unit (ReLU) ? • What is the role of the filter channels in convolutional layer ? • Why do we need a pooling and unpooling in some architectures ? • Why do some networks need fully connected layers whereas the others do not ? • What is the role of by-pass connection or residual network ?
matrix compleMon Nuclear norm Projec5on on sampling posi5ons min m kH ( m ) k⇤ subject to P⌦(b) = P⌦( f ) Rank H ( f ) = k * Jin KH et al IEEE TCI, 2016 * Jin KH et al.,IEEE TIP, 2015 * Ye JC et al., IEEE TIT, 2016 m Annihilating filter-based low-rank Hankel matrix
) H d( f ) = ˜ T ˜ T C C = Φ T H d( f ) C = Φ T ( f ~ ) Encoder: ˜ T = I ˜ = PR(V ) H d( f ) = U ⌃ V T Decoder: f = (˜ Φ C ) ~ ⌧ (˜ ) : Non-local basis : Local basis : Generalized pooling : CNN filters : Frame condition : Projection condition
CT reconstruction problems • Deep convolutional framelets is a mathematical framework to understand deep learning for inverse problems – Novel data-driven signal representation – Structured low-rank shrinkage by controlling the no. of filter channels – Frame condition is important • Our analysis shows that “correctly designed” deep network is NOT a black-box and is SAFE for medical imaging reconstruction