A Pyramid Approach to Subpixel Registration Based on Intensity

P. Thévenaz, U.E. Ruttimann, M. Unser

IEEE Transactions on Image Processing, vol. 7, no. 1, pp. 27–41, January 1998.

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We present an automatic sub-pixel registration algorithm that minimizes the mean square difference of intensities between a reference and a test data set, which can be either tri-dimensional (3-D) volumes or bi-dimensional (2-D) images. It uses spline processing, is based on a coarse-to-fine strategy (pyramid approach), and performs minimization according to a new variation of the iterative Marquardt-Levenberg algorithm for non-linear least-square optimization (MLA). The geometric deformation model is a global 3-D affine transformation, which one may restrict to the case of rigid-body motion (isometric scale, rotation and translation). It may also include a parameter to adjust for image contrast differences. We obtain excellent results for the registration of intra-modality Positron Emission Tomography (PET) and functional Magnetic Resonance Imaging (fMRI) data. We conclude that the multi-resolution refinement strategy is more robust than a comparable single-scale method, being less likely to get trapped into a false local optimum. In addition, it is also faster.

Erratum

Some conventions used in the paper were not explicit. More importantly, the adherence to these untold conventions was not consistent throughout the paper, which lead to erroneous equations. The version below makes the conventions explicit and corrects the mistakes.

• Equation (3) should become

  Qb,A,γ{ƒ}(x)	= Cγ{AA{Tb{ƒ}}}(x)
  	= Cγ{AA{ƒ(· + b)}}(x)
  	= Cγ{ƒ(A · + b)}(x)
  	= {eγ ƒ(A · + b)}(x)
  	= eγ ƒ(A x + b)

• Similarly, equation (11) should become

  Qb,κ,φ,ϑ,ψ,γ{ƒ}(x)	= Cγ{Rφ,ϑ,ψ{Sκ{Tb{ƒ}}}}(x)
  	= Cγ{Rφ,ϑ,ψ{Sκ{ƒ(· + b)}}}(x)
  	= Cγ{Rφ,ϑ,ψ{ƒ(eκ · + b)}}(x)
  	= Cγ{ƒ(eκ R(φ,ϑ,ψ) · + b)}(x)
  	= {eγ ƒ(eκ R(φ,ϑ,ψ) · + b)}(x)
  	= eγ ƒ(eκ R(φ,ϑ,ψ) x + b)

• Then, Table I should be corrected as follows:

  Qp{Qq{ƒ}}	Qp	Ta	AA	Ca
  Qq	o
  Tb		Ta+b	TA-1b{AA}	Tb{Ca}
  AB		AB{TBa}	ABA	AB{Ca}
  Cb		Cb{Ta}	Cb{AA}	Ca+b

• Table II should be corrected as follows:

  Qp{Qq{ƒ}}	Qp	Ta	Sκ′	Rφ′,ϑ′,ψ′	Ca
  Qq	o
  Tb			Te-κ′b{Sκ′}
  Sκ″		Sκ″{Teκ″a}	Sκ′+κ″	Sκ″{Rφ′,ϑ′,ψ′}	Sκ″{Ca}
  Rφ″,ϑ″,ψ″			Rφ″,ϑ″,ψ″{Sκ′}	Rφ,ϑ,ψ
  Cb			Cb{Sκ′}

  with R_φ,ϑ,ψ{ƒ}(x) = (R_{φ′,ϑ′,ψ′} o R_{φ″,ϑ″,ψ″}){ƒ}(x) = ƒ(R(φ″,ϑ″,ψ″) R(φ′,ϑ′,ψ′) x).

• Equation (21) should become ε² = (e^2γ ⁄ |det(A)|) ||C_Δγ{A_I+ΔA{T_Δb{ƒ_T}}} - C_-γ{A_A^-1{T_{-((I+ΔA)A)^-1b}{ƒ_R}}}||².

• Equation (22) should become ε² = (e^2(γ+Δγ) ⁄ |det((I+ΔA)A)|) ||ƒ_T - C_-γ-Δγ{A_{((I+ΔA)A)^-1}{T_{-((I+ΔA)A)^-1(b+Δb)}{ƒ_R}}}||².

• Equation (23) should become ε² = ||C_γ+Δγ{A_(I+ΔA)A{T_b+Δb{ƒ_T}}} - ƒ_R||².

• Equation (24) should become ε² = e^2γ-3κ ||C_Δγ{R_{Δφ,Δϑ,Δψ}{S_Δκ{T_Δb{ƒ_T}}}} - C_-γ{R^-1_φ,ϑ,ψ{S_-κ{T_{-(R(Δφ,Δϑ,Δψ)R(φ,ϑ,ψ))^-1e^-κ-Δκb}{ƒ_R}}}}||².

• Equation (25) should become ε² = e^{2(γ+Δγ)-3(κ+Δκ)} ||ƒ_T - C_-γ-Δγ{(R_φ,ϑ,ψoR_{Δφ,Δϑ,Δψ})^-1{S_-κ-Δκ{T_{-(R(Δφ,Δϑ,Δψ)R(φ,ϑ,ψ))^-1e^-κ-Δκ(b+Δb)}{ƒ_R}}}}||².

• Equation (26) should become ε² = ||C_γ+Δγ{R_φ,ϑ,ψoR_{Δφ,Δϑ,Δψ}{S_κ+Δκ{T_b+Δb{ƒ_T}}}} - ƒ_R||².

• Page 32, second column, last paragraph, line 10, replace (R^-1_{Δφ,Δϑ,Δψ}oR^-1_φ,ϑ,ψ) by (R_φ,ϑ,ψoR_{Δφ,Δϑ,Δψ})^-1.

• Equation (A.1) should become χ²(p) = (e^2γ ⁄ |det(A)|) ∑_i=1…N (C_Δγ{A_I+ΔA{T_Δb{ƒ_T}}}(x_i) - C_-γ{A_A^-1{T_{-((I+ΔA)A)^-1b}{ƒ_R}}}(x_i))².

• Finally, Equation (A.2) should become

  βk	= -½ (∂χ2(p) ⁄ ∂Δpk)
  	= (e-2γ ⁄ |det(A)|) ∑i=1…N (ƒT(xi) - C-γ{AA-1{T-((I+ΔA)A)-1b{ƒR}}}(xi)) (∂QΔp{ƒT}(xi) ⁄ ∂Δpk).

@ARTICLE(http://bigwww.epfl.ch/publications/thevenaz9801.html,
AUTHOR="Th{\'{e}}venaz, P. and Ruttimann, U.E. and Unser, M.",
TITLE="A Pyramid Approach to Subpixel Registration Based on
	Intensity",
JOURNAL="{IEEE} Transactions on Image Processing",
YEAR="1998",
volume="7",
number="1",
pages="27--41",
month="January",
note="")

DOI: 10.1109/83.650848