Re: A05.眼球休憩多重面向

看板optical (眼鏡)作者kramnik (marmote)時間6天前 (2025/03/27 16:21)推噓0(0推 0噓 0→)

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[回覆] 眼球休憩多重面向 [Ans.] Multiple aspects of eye relaxation ------------------------------------------------------------------- 中文字幕影片連結 https://youtu.be/DYOX2yvWGtQ

英文字幕影片連結 https://youtu.be/Ze-Y9Gm0R0s

------------------------------------------------------------------ 深度感知引理(Depth perception lemma) 根據視覺光學第三定律.. 深度感知的指向是協合的.. 其可拆解為兩項陳述.. 交叉鏈結生而協調深度感知.. couple link is born to achieve coordination between depth perceptions R*depth acc sti = k1*depth ver R is the ratio of pupillary distance between infancy and adulthood k1 = 1 overall statistically 深度感知在統計上是等強的 each depth perception is relatively equal strength overall statistically R*(ΔBlur - depth acc res) + k2*(ΔDisparity - depth ver)= 0 k2 = 1 overall statistically https://imgur.com/PBiYCUF.jpeg

令調節衰老率為 w.. 落後或外偏斜為負值..超前或內偏斜為正值.. 深度感知引理方程的解為 tonic acc res = (k1*k2/R)*[φDisparity - anatomical position]*(1-2)/(2-w) tonic ver = k1*k2*[φDisparity - anatomical position]/(2-w) + tonic ver(intrinsic) depth acc res = [(φBlur-tonic acc res) + (k1*k2/R)*(φDisparity-tonic ver)]*(1-w)/(2-w) depth ver = [R*(φBlur - tonic acc res) + k1*k2*(φDisparity - tonic ver)]/(2-w) https://imgur.com/LhoCFJA.jpeg

-------------------------------------------------------------------------- {自動驗光儀校準} https://imgur.com/4EbRWXm.jpeg

令離焦像差焦距為f..球面像差致使的像點位移為△f .. 光束在折射面交點與光學中心間距為h.. 則球面像差的形式為 w(spherical) = k2*h^4 ........k2 is Off-axis quadratic aberration coefficient 像點縱向位移 △f = 2*f^2*(1/n2)*[δW/δ(h^2)] = 2*f^2*(1/n2)*2*k2*h^2 = A*h^2 ..........A is constant ..........(1) 設R為模糊圓半徑..模糊圓擷取處與離焦像差焦距間距為ΔS.. 簡易的幾何推導可得 R = h*[f-Δf]^(-1)*(Δf-ΔS) ….when ΔS <Δf R = h*[f-Δf]^(-1)*(ΔS-Δf) ….when ΔS >Δf https://imgur.com/i8cq8Uo.jpeg

我們令ΔS為不變量..對下式進行h的微分.. R = h*[f-Δf]^(-1)*(Δf-ΔS) ….when ΔS <Δf dR/dh = [f-Δf]^(-1)*(Δf-ΔS) + h*(-1)* [f-Δf] ^(-2)*(-2*A*h)*(Δf-ΔS) + h*[f(eye)-Δf]^(-1)*(2*A*h) = [f-Δf]^(-1)*{(Δf-ΔS) + 2*A*h^2* [f-Δf] ^(-1)* (Δf-ΔS) +2*A*h^2} ≒ [f-Δf]^(-1)*{(Δf-ΔS) + 2*A*h^2* f^(-1)* (Δf-ΔS) +2*A*h^2} ...........when f >>Δf = [f-Δf]^(-1)*{(Δf-ΔS) + 2*Δf* [1/f(eye)]*(Δf-ΔS) + 2*Δf} ≒ [f-Δf]^(-1)* (3*Δf-ΔS ) ..........(2) 當dR/dh = 0 時..此時R有極大值..由(1)(2)可知.. Rmax = [ΔS/(3*A)]^(1/2)*f^(-1)*(-2/3)*ΔS ..........(a) https://imgur.com/K3hAGhk.jpeg

同理..我們令ΔS為不變量..對下式進行h的微分.. R = h*{1/[f-Δf]}*(ΔS-Δf) ….when ΔS >Δf Rmax = h*f^(-1)*( A*h^2-ΔS) ..........(b) https://imgur.com/fFFRdVX.jpeg

最小模糊圓出現在條件 (a) = (b) [ΔS/(3*A)]^(1/2)*f^(-1)*(-2/3)*ΔS = h*f^(-1)*( A*h^2-ΔS) => (4/27*A)*ΔS^3 = h^2*( A^2*h^4 - 2*A*h^2*ΔS +ΔS^2 ) 我們用可以得出最小模糊圓擷取面.. 出現在離焦像差落點與最大球面像差落點的3/4處.. 即0.866*h處的光束交匯點.. ΔS ≒ 0.75*A*h^2 = A*(0.866*h)^2 https://imgur.com/D09T9R9.jpeg

因此我們知道最清晰聚焦擷取面..也就是離焦模糊感知指向的位置.. 其並非離焦像差的落點..而是會往球面像差指向處偏移.. https://imgur.com/pzMEwHg.jpeg

理想透鏡下..影像最清晰的位置就是離焦像差為零值的所在.. 而此時焦深則是座落在離焦像差兩側.. https://imgur.com/gTM9vz4.jpeg

然而人眼存在可觀的正球面像差.. 最小模糊圓位置會往調節超前的方向位移.. https://imgur.com/nxxXInA.jpeg

人類於最大正度數最大視力處方下注視無窮遠方景物.. 此時呈現模糊超前的焦深位置剛好就落在視網膜上.. https://imgur.com/s39iyJz.jpeg

若欲使自動驗光儀讀值需吻合MPMVA處方..則自動驗光儀必需校準.. 將歸零點從預設的最小模糊圓落點修正至模糊超前的焦深位置.. https://imgur.com/stv6ZdK.jpeg

實際上為了讓調節狀態不那麼失真.. 校準歸零幅度通常會小於DoF(lead).. 因此臨床上通常自動驗光儀處方都會較MPMVA呈現相對近視過度.. https://imgur.com/6XvHvfd.jpeg

在MPMVA狀態下注視無窮遠處.. 受檢者仍舊會被量測到調節超前.. 此時調節超前的量值為[DoF(lead) - auto calibration] https://imgur.com/cJlqMvD.jpeg

由幽靈稜鏡吞吐篇章..我們可以知道當人類清晰注視近物時.. 若瞬時深度的落點在焦深外..人眼可以靠深度適應將落點拉入焦深.. 此時呈現模糊落後的焦深位置剛好就落在視網膜上.. 也就是自動驗光儀讀值會呈現調節落後。 https://imgur.com/x9Frlob.jpeg

在MPMVA狀態下注視極近處.. 受檢者會被量測到調節落後.. 此時調節超前的量值為[DoF(lag) + auto calibration] https://imgur.com/lgymndC.jpeg

---------------------------------------------------------------------- 深度感知引理數學詮釋張力性調節(tonic accommodation) https://imgur.com/ITkcgoO.jpeg

交叉鏈結生而協調深度感知 R*depth acc sti = k1*depth ver ..........(1) 深度感知在統計上是等強的 R*(ΔBlur - depth acc res) + k2*(ΔDisparity - depth ver)= 0 ..........(2) 調節邊界條件 acc res = acc sti*(1-0.018*age) ..........(3) 起始條件 ΔBlur = 0 , ΔDisparity = anatomical position ..........(4) 解上(1)(2)(3)(4)聯立方程組可得 depth acc res = (17/4.3)*k1*k2*(1-0.018*age)/[k1*(1-0.018*age)+k2] = (17/4.3)*(1-0.018*age)/(2-0.018*age) ..........overall statistically 深度引理預測張力性調節為 tonic acc = depth acc res + [DoF(lead) - auto calibration] = (17/4.3)*k1*k2*(1-0.018*age)/[k1*(1-0.018*age)+k2] + [DoF(lead) - auto calibration] = (17/4.3)*(1-0.018*age)/(2-0.018*age) +0.25 ..........overall statistically https://imgur.com/gfjhUs8.jpeg

------------------------------------------------------------------- 深度感知引理數學詮釋新生兒屈光偏差(infant refractive error) https://imgur.com/kYX0wEp.jpeg

嬰童時期各個深度感知的指向統計上應該是準確的 refractive error (postnatal) = depth acc res |age = 0 = (17/4.3)*(1-0.018*age)/(2-0.018*age) |age = 0 = 1.98 in diopter 由此我們可以知道出生時的嬰童屈光偏差應該要是在遠視+200上下節錄自Emmetropization, refraction and refractive errors: control of postnatal eye growth, current and developing treatments https://imgur.com/aJdPQrH.jpeg

------------------------------------------------------------------- 深度感知引理數學詮釋張力性輻輳(tonic vergence) https://imgur.com/YodA4es.jpeg

嬰童時期各個深度感知的指向統計上應該是準確的 depth ver = (R/k1)*depth acc res/(1-0.018*age) = (17/6)*k2/[k1*(1-0.018*age)+k2] tonic ver (intrinsic) = anatomical position - depth ver |age = 0 = (17/6)*[k1/(k1+k2)] = (17/12) in meter angle ..........overall statistically = (17/2) in prism/diopter ..........overall statistically https://imgur.com/Odjx8Vz.jpeg

交叉鏈結生而協調深度感知 R*depth acc sti = k1*depth ver ..........(1) 深度感知在統計上是等強的 R*(ΔBlur - depth acc res) + k2*(ΔDisparity - depth ver)= 0 ..........(2) 調節邊界條件 acc res = acc sti*(1-0.018*age) ..........(3) 起始條件 ΔBlur = 0 , ΔDisparity = anatomical position tonic ver (intrinsic) = (17/6)*[k1/(k1+k2)] ..........(4) 解上(1)(2)(3)(4)聯立方程組可得張力性輻輳 tonic ver (age) = depth ver (age) + tonic ver (intrinsic) - anatomical position = (17/6)* k2/[k1*(1-0.018*age)+k2] + (17/6)*[k1/(k1+k2)] - (17/6) = (17/6)*{ k2/[k1*(1-0.018*age)+k2] + [1-k1/(k1+k2)] - 1 } = (17/6)*[1/(2-0.018*age) - (1/2)] in meter angle ....overall statistically = 17*[1/(2-0.018*age) - (1/2)] in prism/diopter .....overall statistically = 9.86*[1/(2-0.018*age) - (1/2)] in degree ..........overall statistically 深度引理預測張力性輻輳為 tonic ver (age) = (17/6)*{ k2/[k1*(1-0.018*age)+k2] + [1-k1/(k1+k2)] - 1 } - (4.3/6)*k2*[DoF(lead) - auto calibration] = (17/6)*[1/(2-0.018*age) - (1/2)] - (4.3/6)*0.25 in meter angle .......... overall statistically = 17*[1/(2-0.018*age) - (1/2)] - 4.3*0.25 in prism .......... overall statistically = 0.58*17*[1/(2-0.018*age) - (1/2)] - 0.58*4.3*0.25 in degree ..........overall statistically 節錄自tonic vergence, age and clinical presbyopia https://imgur.com/hgmfN37.jpeg

深度感知引理預測..吻合人類歷史臨床實驗結果.. https://imgur.com/tifICFX.jpeg

#至於數學上為何必須先執行離焦模糊和輻輳偏差兩感知的協合.. 再進行張力性輻輳(本質上)的加減運算.. 必須由後續的深度感知三位一體來闡明.. --------------------------------------------------------------------- 深度感知引理數學詮釋遠方斜位(distant heterophoria) https://imgur.com/ZOXcaxR.jpeg

交叉鏈結生而協調深度感知 R*depth acc sti = k1*depth ver ..........(1) 深度感知在統計上是等強的 R*(ΔBlur-depth acc res) + k2*(ΔDisparity-depth ver) = 0 ..........(2) 調節邊界條件 acc res = acc sti*(1-0.018*age) ..........(3) 起始條件 ΔBlur = - tonic acc , ΔDisparity = - tonic ver , tonic acc = (17/4.3)*(1-0.018*age)/(2-0.018*age) + auto-calibration tonic ver = (17/6)*{ k2/[k1*(1-0.018*age)+k2] + [1-k1/(k1+k2)] - 1 } ....(4) 解上(1)(2)(3)(4)聯立方程組可得 depth ver = - [ (4.3/6)*tonic acc + k2*tonic ver ]/[(k1*(1-0.018*age)+k2] = - [ (4.3/6)*tonic acc + tonic ver ]/(2-0.018*age) ....overall statistically 輻輳適應前的遠方斜位為 heterophoria(distance) #before adaptation = tonic ver + depth ver 由幽靈稜鏡吞吐篇章，我們可以得到 heterophoria(distance) #after adaptation = 0.5*(tonic ver + depth ver) 深度引理預測遠方斜位為 heterophoria(distance) = 0.5*(tonic ver + depth ver) among them , tonic acc = (17/4.3)*(1-0.018*age)/(2-0.018*age) + 0.25 tonic ver = (17/6)*[1/(2-0.018*age) - (1/2)] -(4.3/6)*0.25 depth ver = - [ (4.3/6)*tonic acc + tonic ver ]/(2-0.018*age) https://imgur.com/FqMFqKo.jpeg

------------------------------------------------------------------------ 深度感知引理數學詮釋近方斜位(near heterophoria) https://imgur.com/9GFPjej.jpeg

交叉鏈結生而協調深度感知 R*depth acc sti = k1*depth ver ..........(1) 深度感知在統計上是等強的 R*(ΔBlur-depth acc res) + k2*(ΔDisparity-depth ver) = 0 ..........(2) 調節邊界條件 acc res = acc sti*(1-0.018*age) ..........(3) 起始條件 ΔBlur = (1/WD) - tonic acc, ΔDisparity = (1/WD) - tonic ver tonic acc = (17/4.3)*(1-0.018*age)/(2-0.018*age) + auto-calibration tonic ver = (17/6)*{ k2/[k1*(1-0.018*age)+k2] + [1-k1/(k1+k2)] - 1 } .....(4) 解上(1)(2)(3)(4)聯立方程組可得 depth ver = {(4.3/6)*[(1/WD)-tonic acc]+k2*[(1/WD)-tonic ver]}/[(k1*(1-0.018*age)+k2] = {(4.3/6)*[(1/WD)-tonic acc]+[(1/WD)-tonic ver]}/(2-0.018*age) ..........overall statistically 輻輳適應前的近方斜位為 heterophoria(near) #before adaptation = tonic ver + depth ver 由幽靈稜鏡吞吐篇章，我們可以得到 heterophoria(near) #after adaptation = 0.5*[tonic ver + depth ver - 6*(1/WD)] 深度引理預測近方斜位為 heterophoria(near) = 0.5*[tonic ver + depth ver - 6*(1/WD)] among them , tonic acc = (17/4.3)*(1-0.018*age)/(2-0.018*age) + 0.25 tonic ver = (17/6)*[1/(2-0.018*age) - (1/2)] - (4.3/6)*0.25 depth ver = {(4.3/6)*[(1/WD)-tonic acc]+[(1/WD)-tonic ver]}/(2-0.018*age) https://imgur.com/uTVlKJG.jpeg

-------------------------------------------------------------------------- 深度感知引理數學詮釋近方加入度#交叉圓柱鏡 (near addition # Fused Cross Cylinder) https://imgur.com/IlrrMI5.jpeg

交叉鏈結生而協調深度感知j R*depth acc sti = k1*depth ver ..........(1) 深度感知在統計上是等強的 R*(ΔBlur - depth acc res) + k2*(ΔDisparity - depth ver)= 0 ..........(2) 調節邊界條件 acc res = acc sti*(1-0.018*age) ..........(3) 起始條件 ΔBlur = (1/WD) - tonic acc - FCC, ΔDisparity = (1/WD) - tonic ver tonic acc = (17/4.3)*(1-0.018*age)/(2-0.018*age) + auto-calibration tonic ver = (17/6)*{ k2/[k1*(1-0.018*age)+k2] + [1-k1/(k1+k2)] - 1 } .....(4) 解上(1)(2)(3)(4)聯立方程組可得 depth acc res = {k1*[(1/WD)-tonic acc -FCC ]+k1*k2*(6/4.3)*[(1/WD)-tonic ver]} *(1-0.018*age)/[(k1*(1-0.018*age)+k2] = {[(1/WD)-tonic acc - FCC]+(6/4.3)*[(1/WD)-tonic ver]} *(1-0.018*age)/(2-0.018*age) ...........overall statistically 若近方加入度(交叉圓柱鏡)可以使調節誤差為零值..則下等式成立.. (1/WD) + [DoF(lead) - auto calibration ] =FCC+ tonic acc + depth acc res = FCC+ tonic acc + {[(1/WD)-tonic acc - FCC]+(6/4.3)*[(1/WD)-tonic ver]} *(1-0.018*age)/(2-0.018*age) 不考慮遠方與近方的球差變化.. 解上方程式可得近方加入度(交叉圓柱鏡)..如下所示.. https://imgur.com/nGASdra.jpeg

由後述的深度感知三位一體.. 我們將會知道深度的最終位置為偏差感知所支配.. https://imgur.com/t5uu8nD.jpeg

瞳孔的穩態收縮和輻輳偏差成正相關..和調節幅度無關.. 吻合2017年莫里茲·菲爾及芭芭拉·莫澤臨床實驗結果.. https://imgur.com/rcdy684.jpeg

而根據2018年胡安·薩帕塔-迪亞斯及赫瑪·拉達克里希南所建立的焦深模型.. 我們知道個體所呈現的人眼正球差與瞳孔半徑成正相關.. 連帶與注視距離成負相關.. https://imgur.com/FlvQFi3.jpeg

因此我們知道若利用主觀去感知最佳視力及模糊.. 與客觀交叉圓柱鏡去量測最小模糊圓.. 若初始位置為注視遠方而結束位置為近方.. 將不可避免的面臨負度數偏移..也就是加入度需求降低.. https://imgur.com/6qEntS4.jpeg

又根據2005年桑吉夫·卡斯圖裡蘭甘及阿德里安·格拉瑟的臨床實驗.. 注視近物時瞳孔直徑縮減的速度會隨著年齡的增加而下滑.. https://imgur.com/ploXTYV.jpeg

這也說明當注視遠方切換到近方時.. 雖然因為球差降低的影響致使加入度需求比理想狀態來得要少.. 但隨著年齡的增加..瞳孔直徑縮減的速度下滑.. 連帶致使球差變化影響減少..因此在老年時這種效應將會消退.. https://imgur.com/BsLLmAk.jpeg

這個效應會出現在所有的近用加入度的年齡函數.. 我們可以用下線性方程來修正使其較貼近經驗法則.. near addition correction = 0.7 - 0.01*age https://imgur.com/116zxtJ.jpeg

經驗法則(rule of thumb) https://imgur.com/RMc7gga.jpeg

---------------------------------------------------------------------------- 深度感知引理數學詮釋近方加入度#正負相對性調節(near addition # NRA/PRA) 邊界條件 0 ≦ φBlur ≦ Amplitude of accommodation 負相對性調節運動的讀值..即為邊界條件的下界.. φBlur = (1/WD) - NRA 0 It can be solved NRA = (1/WD) = 2.5 我們知道個體所呈現的人眼正球差與瞳孔半徑成正相關.. 連帶與注視距離成負相關.. 若初始位置為注視遠方而結束位置為近方.. 將不可避免的面臨負度數偏移..也就是加入度需求降低.. 深度引理預測負相對性調節運動為 NRA = 2.5 - near addition correction = 2.5 - (0.7 - 0.01*age) = 1.8 + 0.01*age https://imgur.com/pSJjx8A.jpeg

正相對性調節運動的讀值即為邊界條件的上界 φBlur = (1/WD) + PRA Amplitude of accommodation that is 2.5 - PRA = 9*(1-0.018*age) It can be solved PRA = -6.5 + 9*0.018*age) 我們知道個體所呈現的人眼正球差與瞳孔半徑成正相關.. 連帶與注視距離成負相關.. 若初始位置為注視遠方而結束位置為近方.. 將不可避免的面臨負度數偏移..也就是加入度需求降低.. 深度引理預測正相對性調節運動為 PRA = (-6.5 + 9*0.018*age) - near addition correction = (-6.5 + 9*0.018*age) - (0.7 - 0.01*age) = -7.2 + 0.172*age https://imgur.com/6jvzxQl.jpeg

確保老花處方是否適合被檢者.. 則工作距離應該在清晰視力範圍的中間.. near addition (NRA/PRA) = (NRA + PRA)/2 深度引理預測近方加入度(NRA/PRA)為 near addition (NRA/PRA) = [(1.8 + 0.01*age) + (-7.2 + 0.172*age)]/2 = -2.7 + 0.0865*age https://imgur.com/wCgAx6c.jpeg

我們將近方加入度(NRA/PRA)與近方加入度(交叉圓柱鏡)兩相比較.. 可以發現兩者讀值很接近.. https://imgur.com/3TB6VhN.jpeg

-------------------------------------------------------------------- 通過這些函數.. 我們不僅為老花處方的經驗推估提供了科學支撐.. 也揭示了年齡對調節與輻輳系統的影響機制.. 為開發革新的視覺矯正策略..奠定了堅實的理論基礎.. -- "The science I see delivers to me a feeling of great beauty, but few others see it. This makes me sad." —Feyman's Letters: The Beat of a Different Drum, October 1967 -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 111.249.161.94 (臺灣) ※ 文章網址: https://www.ptt.cc/bbs/optical/M.1743063667.A.2CA.html

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