超對焦距（hyperfocus）-- 大光圈也可以有無窮長景深 Large aperture can make depth of field infinite

 攝影想要有長景深，一般人直接會想到小光圈（大f數），或許加上短焦鏡。然而在一個特定光圈和焦距的設定下，我們都可以找到一個對焦距離（物距），使得後景深無窮長，而前景深等於二分之一的這個物距。換句話說就是，將相機對焦在這個「超對焦距（hyperfocus)」時，從二分之一的超對焦距的位置到無窮遠的物體都能清晰的成像。

In photography  most people would think of a small aperture (large f-number) to have a long depth of field (DoF), and perhaps with a short focal length.  However, at a specific aperture and focal-length setting, we can find a focusing distance (object distance), which is called the "hyperfocus", so that the rear DoF becomes infinite, and the front DoF is equal to one-half of this hyperfocus.  In other words, when the camera is focused on this hyperfocus, the image of object that locates from one-half of the hyperfocus to infinity should be all clear.

f/5.6, 1/6, 9.3mm, ISO2500, 5968x3352

f/5.6, 1/6, 9.3mm, ISO1600, 5968x3352

      為了能在有限空間（家裡）做實驗，我選用f/5.6和F＝9.3mm來展示此現象。三個（黑、紅和銀色）水瓶分別相距約一公尺，而在最遠處放置一包衛生紙。左邊兩張照片都是對焦在紅色水瓶上。第一張照片的物距約為10公分，顯示只有在像平面的部分紅色水瓶是清晰的，一公尺之外的銀色水瓶已相當模糊，更無法看清衛生紙上的字。在第二張照片中紅色水瓶與相機距離（物距）設定約2公尺（超對焦距）。這張照片顯示包括在紅色水瓶（主體）前方1公尺（二分之一的超對焦距）的黑色水瓶到衛生紙（離主體約1.5公尺遠）上的字都是相當清晰。
In order to be able to experiment in a limited space (home), I chose f/5.6 and focal length F=9.3mm to demonstrate this phenomenon.  Three (black, red, and silver) water bottles are about one meter apart, and a pack of toilet paper is placed at the farthest place.  The two photos on the left are both focused on the red water bottle with different focusing distance.  The fusing distance of the first photo is about 10 cm, showing that only part of the red water bottle in the image plane is clear, and the silver water bottle at one meter away is quite blurred and even making it impossible to see the words on the toilet paper.  In the second photo, the distance between the red water bottle and the camera (object distance) is set to about 2 meters (approximately equals to hyperfocus).  This photo shows that the black water bottle, which is 1 meter in front of the red water bottle (the subject) at one-half of the hyperfocus, is quite clear even for the toilet paper (about 1.5 meters away from the subject).
      在光學上我們可以推導出，在固定光圈和鏡頭焦距之下，超對焦距＝F＾2/(f*像素尺寸）。將使用參數代入得：
超對焦距＝(9.3mm)^2/(5.6*7微米）～2公尺。所以，前述紅色水瓶與相機距離（物距）設定約2公尺。如果使用iPhone 7+手機，f/1.8和F3.99mm，計算得超對焦距＝6.7m，所以將手機對焦在6.7m處，我們可以得到從3.3m到無窮遠的物體都能清晰成像的結果。
In Optics, we can deduce that under the fixed aperture and focal length, the hyperfocus follows the expression:
hyperfocus = F ^ 2 / (f * C) = (9.3mm) ^ 2 / (5.6 * 7 microns) ~ 2 meters, where C = pixel size  of camera, ~7 microns.  Therefore, I set the distance between the aforementioned red water bottle and the camera (object distance) to about 2 meters.  If one uses the iPhone 7+ mobile phone, f/1.8 and F3.99mm, the calculated hyperfocus = 6.7m, so one can get the result of clear imaging from 3.3 m to infinity as he or she focuses the phone at 6.7m, since the hyperfocus = F ^ 2 / (f * pixel size).

943x437

     為了證實第二張照片有超長景深，我們將第二張照片裁切接近第一張照片的大小。如果兩張照片有相同景深，裁切成一樣大小（同樣影像大小），則背景模糊層度也會相同。左圖是由5968x3352個像素裁切到943x437像素的這張照片。影像不但不會因為裁切後像素變少而模糊掉，銀色水瓶還很清楚，連衛生紙袋上印的文字都相當清晰。所以景深超過2.5公尺。
To confirm that the second photo indeed has a very long DoF, we cut the second photo closer to about the size of the first photo.  If the two photos have the same DoF and are cropped to the same size (same image size), the background blur will be the same.  The picture above is cropped from 5968x3352 pixels to 943x437 pixels. The image will not be blurred even if the total pixel numbers after cutting are reduced.  The silver water bottle is also very clear, and the printed text on the toilet paper bag is quite clear.  Therefore, the DoF should be more than 2.5 meters.


結論：
1) 雖然通常景深隨光圈f數成正比與影像縮小倍率平方成正比（由於繞射，在f > 11，景深正比於f的平方），在固定光圈和鏡頭焦距之下，對焦在超對焦距時，可以得到從二分之一的超對焦距到無窮遠的物體都能清晰的成像的超（無窮）長景深。
2) 計算超對焦距公式：對f < 11，超對焦距＝F＾2/(f*像素尺寸）；而對f > 11，由於已達繞射極限，則使用: 超對焦距 =1.37* (F/f)^2。
3) 不知道或忘掉如何計算，只要記得：將光圈設定在f>=11，鏡頭對焦調在「無窮大標誌（或平躺的8字）」的腰即可。f越大，稍微往無窮大標誌短的方向調一點即可；其實不調也無仿。

Conclusion：
1) Although the DoF is usually proportional to the aperture f-number and proportional to the square of the image reduction coefficient (notice that the DoF is proportional to the square of  f-number due to the diffraction for f > 11).  Under the fixed aperture and focal length, one can find a specific focusing distance called the hyperfocus, at which  the DoF is infinite, i.e., subjects which are located from one-half of the hyperfocus to the infinity will all be clear.
2) Formula to calculate the hyperfocus is: for f < 11, hyperfocus = F^2/(f*pixel size) ; whereas, for f > 11, since it has reached the diffraction limit so
hyperfocus =1.37*(F/f)^2.
3) It is not big deal to forget the formula to make a long DoF shot.   The trick is just remember to set the aperture at f >= 11, and focus the lens on the waist of the "infinity" symbol (the lie-down figure 8).  For the larger the f, one could slightly adjust focus to the near-side of the "infinity" symbol; even keeping it fixed is also fine.