There　is　growing　interest　in　solving　linear　L1　estimation　problems　for　sparsity　of　the　solution　and　robustness　against　non-Gaussian　noise.　This　paper　proposes　a　discrete-time　neural　network　which　can　calculate　large　linear　L1　estimation　problems　fast.　The　proposed　neural　network　has　a　fixed　computational　step　length　and　is　proved　to　be　globally　convergent　to　an　optimal　solution.　Then,　the　proposed　neural　network　is　efficiently　applied　to　image　restoration.　Numerical　results　show　that　the　proposed　neural　network　is　not　only　efficient　in　solving　degenerate　problems　resulting　from　the　nonunique　solutions　of　the　linear　L1　estimation　problems　but　also　needs　much　less　computational　time　than　the　related　algorithms　in　solving　both　linear　L1　estimation　and　image　restoration　problems.　©　2012　IEEE.