Accounting for the Use of Different Length Scale Factors in x, y and z Directions

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1 1 Accountng for the Use of Dfferent Length Scale Factors n x, y and z Drectons Taha Soch (taha.soch@kcl.ac.uk) Imagng Scences & Bomedcal Engneerng, Kng s College London, The Rayne Insttute, St Thomas Hosptal, London, SE1 7EH, UK Abstract: Ths short artcle presents a mathematcal formula requred for metrc correctons n mage extracton and processng when usng dfferent length scale factors n three-dmensonal space whch s normally encountered n cryomcrotome mage constructon technques. Keywords: mage extracton; correcton formula; cryomcrotome magng. In many scentfc and ndustral stuatons, the coordnates space s scaled by dfferent length factors n the three spatal drectons, x, y and z; whch affect the metrc relatons. For nstance, n the cryomcrotomc mage extracton technques, the thckness of slces may be subject to errors or varatons makng the voxel sze n one drecton larger or smaller than ts standard sze n the two other drectons. Consequently, the geometrc parameters obtaned from these mages, whch are based on the standard unts of mage space of an assumed cubc voxel unt, wll be contamnated wth errors causng a dstorton because of the mssng scale factors requred by the sotropy of the physcal space. In the followng we present a smple case based on real-lfe cryomcrotomc mage constructon algorthms n bomedcal applcatons where vasculature trees are obtaned by computng the radus of each vessel n a number of rotatonal steps through a whole crcle and the results are then averaged to obtan the fnal radus [1, 2]. As these rotatonal steps are orented dfferently n the 3D space, the contrbuton of the length scale factors wll vary from one orentaton to the other and hence a scalng correcton s requred to obtan the correct radus.

2 2 There are several possble ways for dervng a formula for ths correcton; these nclude the use of rotatonal matrces, and crcle projecton on the three standard planes (.e. xy, yz and zx) to obtan three ellpses from whch the three spatal components can be computed. However, an easer and more effcent way s to fnd a parameterzed form of an ntersecton crcle between a plane perpendcular to the vessel axs and the vessel tself, whch n essence s equvalent to a great crcle ntersecton of ths plane and a sphere havng the same radus as the vessel [3]. Ths method of dervaton s outlned below. For a regular cylndrcal straght vessel orented arbtrarly n 3D space and defned by ts two end ponts P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) and radus r, a free vector orented n ts axal drecton s gven by a = (a x, a y, a z ) = (x 1 x 2, y 1 y 2, z 1 z 2 ) (1) whle a plane perpendcular to ths vector and (for smplcty wth no loss of generalty) passng through the orgn s gven by a x x + a y y + a z z = 0 (2) Now, n 3D space a parameterzed crcle of radus r centered (wth no loss of generalty) at the orgn and lyng n a plane dentfed by two orthonormal vectors b and c s gven by the equaton r [cos(t)b + sn(t)c] 0 t < 2π (3) that s

3 3 (r [cos(t)b x + sn(t)c x ], r [cos(t)b y + sn(t)c y ], r [cos(t)b z + sn(t)c z ]) 0 t < 2π (4) To fnd b and c, a formal orthogonalzaton process, such as Gram Schmdt, wth normalzaton can be followed where random vectors non-collnear to a can be used. However a more convenent way s to fnd an arbtrary non-trval vector lyng n the plane by nsertng arbtrary values for two varables (e.g. x = 1 and y = 1 ) n the plane equaton and solvng for the other varable (z) followed by normalzng through the dvson by ts norm. If ths vector s consdered b, then vector c s found by takng the cross product a b and normalzng. If the followng length scale factors: α, β and γ are ntroduced on the x, y and z drectons respectvely, then the dstorted radus, r, at a random orentaton t = θ s gven by r = r (α [cos(θ)b x + sn(θ)c x ]) 2 + (β [cos(θ)b y + sn(θ)c y ]) 2 + (γ [cos(θ)b z + sn(θ)c z ]) 2 and hence the actual radus, r, s gven by (5) r = r (α [cos(θ)b x + sn(θ)c x ]) 2 + (β [cos(θ)b y + sn(θ)c y ]) 2 + (γ [cos(θ)b z + sn(θ)c z ]) 2 (6) As the mage constructon algorthm computes r at N rotatonal steps (e.g. 360 steps correspondng to 360 ) and averages the results, to restore the corrected radus r, ths correcton should be ntroduced at each one of these steps. In an deal stuaton where rotatonal symmetry holds, only one quarter of these steps, N, s requred, resultng n a substantal computatonal economy. However due to 4

4 4 the measurements and algorthmc errors at each step, t may be safer to mantan the N steps as the errors are expected to level out or dmnsh by applyng ths process through the whole crcle. Ths correcton can also be extended to use for post processng correcton by applyng the correcton on the fnal averaged radus followng a correcton-free extracton process. For N rotatonal steps we have N r = r N (α [cos(θ )b x + sn(θ )c x ]) 2 + (β [cos(θ )b y + sn(θ )c y ]) 2 + (γ [cos(θ )b z + sn(θ )c z ]) 2 (7) Snce the averaged post processng radus s the actual radus s then gven by R av = N N r (8) r = N NR av (α [cos(θ )b x + sn(θ )c x ]) 2 + (β [cos(θ )b y + sn(θ )c y ]) 2 + (γ [cos(θ )b z + sn(θ )c z ]) 2 (9) Although post processng correcton may not result n computatonal effcency, t may be more convenent and useful to use when the non-corrected data are already obtaned wth no requrement to repeat the extracton process. It should be remarked that ths correcton can be appled n general to correct for ths type of dstorton regardless of the number of steps (sngle or multple) and the shape of the object as long as the x, y and z components of the poston vector can be obtaned for each pont n space requred to trace the path of the dstorted shape. Ths process can also be extended from dscrete to contnuous by substtutng the summatons wth ntegratons wth some other mnor modfcatons

5 5 to account for ths correcton n analytcal contexts rather than numercal dscrete processes.

6 6 References [1] R.D. ter Wee, H. Schulten, M.J. Post, J.A.E. Spaan. Localzaton and vsualzaton of collateral vessels by means of an magng cryomcrotome. Vascular Pharmacology, 45(3): e63-e64, [2] B. Bracegrdle. A Hstory of Mcrotechnque: The Evoluton of the Mcrotome and the Development of Tssue Preparaton. Scence Hertage Ltd, 2nd edton, [3] G.B. Thomas, R.L. Fnney. Calculus and Analytc Geometry, Addson Wesley, 9th Edton, 1995.

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

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