This　paper　proposes　two　homotopy　methods　for　solving　the　compressed　sensing　(CS)　problem,　which　combine　the　homotopy　technique　with　the　iterative　hard　thresholding　(IHT)　method.　The　homotopy　methods　overcome　the　difficulty　of　the　IHT　method　on　the　choice　of　the　regularization　parameter　value,　by　tracing　solutions　of　the　regularized　problem　along　a　homotopy　path.　We　prove　that　any　accumulation　point　of　the　sequences　generated　by　the　proposed　homotopy　methods　is　a　feasible　solution　of　the　problem.　We　also　show　an　upper　bound　on　the　sparsity　level　for　each　solution　of　the　proposed　methods.　Moreover,　to　improve　the　solution　quality,　we　modify　the　two　methods　into　the　corresponding　heuristic　algorithms.　Computational　experiments　demonstrate　effectiveness　of　the　two　heuristic　algorithms,　in　accurately　and　efficiently　generating　sparse　solutions　of　the　CS　problem,　whether　the　observation　is　noisy　or　not.