Electrical　impedance　tomography　(EIT)　is　a　promising　imaging　technique　that　recovers　the　conductivity　distribution　inside　a　domain　from　noninvasive　electrical　measurements　on　the　boundary.　In　this　work,　to　accelerate　the　solving　of　EIT　problems　in　arbitrarily-shaped　domains　and　with　a　large　number　of　unknowns　(N),　we　propose　a　fast　integral-equation-based　inversion　method.　First,　by　applying　the　Schwarz-Christoffel　conformal　transformation,　we　map　the　arbitrarily-shaped　domain　of　an　EIT　problem　to　a　rectangle,　on　which　the　Green’s　function　can　be　derived　analytically　in　Fourier-series　representation.　Leveraging　such　a　mathematical　structure　of　the　Green’s　function,　we　then　propose　a　fast-Fourier-transform　(FFT)　based　algorithm　to　compute　the　multiplication　of　the　associated　impedance　matrix　with　vectors,　where　the　time　complexity　is　substantially　reduced　from　O(N2)　to　O(Nlog(N))　and　the　memory　complexity　is　reduced　from　O(N2)　to　O(N).　By　using　the　contrast　source　inversion　method　along　with　the　accelerated　matrix-vector　multiplications,　the　conductivity　profile　can　be　reconstructed　much　more　efficiently　in　the　rectangular　transformation　domain.　As　validated　by　numerical　and　experimental　tests,　the　proposed　FFT-accelerated　transformation-domain　EIT　image　reconstruction　method　can　offer　significantly　reduced　computational　and　memory　complexity　without　sacrificing　the　image　quality.　IEEE