Evaluation method for noise-induced phase error in fringe projection profilometry

For noise-induced phase error and phase unwrapping reliability in fringe projection profilometry (FPP), a new evaluation method is proposed. By introducing a Gaussian noise model, the maximum variance of the noise-induced phase error and phase unwrapping reliability are directly obtained. According to the proposed evaluation method, the variance of the noise-induced phase error is proportional to the noise variance and inversely proportional to the fringe modulation. The phase unwrapping reliability is not only inversely proportional to the noise-induced phase error, but also inversely proportional to the fringe frequency ratio. This work analyzes and compares three efficient dual-frequency phase unwrapping algorithms. The results show that the wrapped phase accuracy and phase unwrapping reliability based on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> are the best, followed by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> . However, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:mrow> <mml:mspace width="negativethinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> has the highest measurement efficiency.