Applied Computational Intelligence and Soft Computing

Research Article

Nonnegative Matrix Factorizations Performing Object Detection and Localization

Algorithm 4

NMF with space constraints.
Input: 𝑝 π‘œ 𝑠 𝑖 𝑑 𝑖 𝑣 𝑒 𝑐 π‘œ 𝑛 𝑠 𝑑 π‘Ž 𝑛 𝑑 𝑠 ∢ πœ‡ π‘Š > 0 , πœ‡ 𝐻 > 0
  𝐢 β„Ž π‘œ π‘œ 𝑠 𝑒 π‘Ž 𝑛 π‘Ž 𝑝 𝑝 π‘Ÿ π‘œ 𝑝 π‘Ÿ 𝑖 π‘Ž 𝑑 𝑒 𝑃 π‘Ÿ π‘œ 𝑗 𝑒 𝑐 𝑑 𝑖 π‘œ 𝑛 ( β‹… ) 𝑑 π‘œ 𝑒 𝑛 𝑠 𝑒 π‘Ÿ 𝑒 the degree of sparseness
 Initialize nonnegative matrices π‘Š ( 0 ) 𝐻 ( 0 )
 while  𝑆 𝑑 π‘œ 𝑝 𝑝 𝑖 𝑛 𝑔 𝑐 π‘Ÿ 𝑖 𝑑 𝑒 π‘Ÿ 𝑖 π‘Ž π‘Ž π‘Ÿ 𝑒 𝑛 π‘œ 𝑑 𝑠 π‘Ž 𝑑 𝑖 𝑠 𝑓 𝑖 𝑒 𝑑  do
   π‘Š 𝑖 𝑗 ← π‘Š 𝑖 𝑗 βˆ’ πœ‡ π‘Š ( ( π‘Š 𝐻 𝐻 ⊀ ) 𝑖 𝑗 βˆ’ ( 𝑋 𝐻 ⊀ ) 𝑖 𝑗 )
   π‘Š ← 𝑃 π‘Ÿ π‘œ 𝑗 𝑒 𝑐 𝑑 𝑖 π‘œ 𝑛 ( π‘Š )
   𝐻 ← 𝐻 𝑖 𝑗 βˆ’ πœ‡ 𝐻 ( ( π‘Š ⊀ π‘Š 𝐻 ) 𝑖 𝑗 βˆ’ ( π‘Š ⊀ 𝑋 ) 𝑖 𝑗 )
   𝐻 ← 𝑃 π‘Ÿ π‘œ 𝑗 𝑒 𝑐 𝑑 𝑖 π‘œ 𝑛 ( 𝐻 )
 end while
  { πœ‡ π‘Š > 0 and πœ‡ 𝐻 > 0 are positive constants representing the step size
 of the algorithm and 𝑃 π‘Ÿ π‘œ 𝑗 𝑒 𝑐 𝑑 𝑖 π‘œ 𝑛 ( β‹… ) indicates the appropriate projection
 operator}