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　(SC)　conformal　transformation,　we　map　the　arbitrarily　shaped　domain　of　an　EIT　problem　to　a　rectangle,　on　which　Green＇s　function　can　be　derived　analytically　in　Fourier　series　representation.　Leveraging　such　a　mathematical　structure　of　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(N-2)　to　O(N　log(N))　and　the　memory　complexity　is　reduced　from　O(N-2)　to　O(N).　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.