330 J-M Deniel : Radioprotection 2024 , 59 ( 4 ), 327 – 337
Fig . 4 . ( a ) Form factor analogies used in the hemicube - picture from ( Cohen and Greenberg 1985 ) - and ( b ) the hemiplane ( Sillion and Puech 1989 ) methods .
Fig . 5 . ( a ) Geometry of the form factor ff p associated with a pixel p in a picture . ( b ) Carre ( Carré 1998 ) method ( picture from this thesis ) to compute ff p .
edges in trigonometric order . The projections of A and B on the unitary sphere are denoted A ’ and B ’. Then A ’ and B ’ are projected on the unitary disc as A ” and B ” respectively . O , A ” and B ” form a portion of the disc . Its S AB area ƒ can ! be positive or negative , as given in equations ( 2.2 ) where N AB is the normal to sensor and N AB is the normal to the ( O , A ’, B ’) plane .
ƒ ! S AB ¼ arccosOA 0 ƒ ! : OB
0 ƒ ! ƒƒ ! N AB: N Rec ƒ ! ƒ ! ƒ ! OA ’ OB ð2:2Þ
N AB ¼ ƒ ! ƒ ! jj OA ’ OB jj
Finally , pixel p form factor ff p is the sum of the successive
S AB
2p
. For example , in Figure 5 , the form factor associated with polygon ABC is S
2p
, while successive edges projected areas are denoted S1 , S2 and �S3 , such that S = S1 þ S2 – S3 .
3 Results
We analyzed pictures of our blackbody and metal furnace , to estimate E IR irradiance . In the same geometrical conditions , at Z distance from the source opening , we measured spectral irradiance .
Z
As explained before , it was scaled by 2 to account for the Z
ðZ 0 þZÞ 2 0 distance offset from Z to the camera optical center .
In the first case , the emissivity database was reduced to m ( l )= 1 . In the second case , it contained all the emissivity data shown in ( Deniel , 2024 ).
3.1 Estimated E IR vs measured irradiance and theoretical blackbody irradiance
In front of our blackbody , we measured E m irradiance in the range [ 1000 ; 2500 nm ] with the AvaSpec NIR256-2.5TEC and took pictures of the blackbody opening , at three Z distances from it . E m was corrected to account for the Z o distance offset .
In addition , blackbody radiance is assumed to be homogeneous and diffuse . In this case , as explained in ( DiLaura et al ., 2011b , 10.3.2 ), and indicated before , the far field approximation allows the simple calculation of E th irradiance , when the “ five times rule ” is respected . In this case , the light source can be considered as a point when irradiance distance exceeds 5 times the size of the source . Then , irradiance at distance d varies as inverse squared d , multiplied by the black body intensity that is pr 2 times its radiance . This explains the p r2 factor in equation ( 3.1 ) below . d
For consistency 2 , E th is limited to the E m wavelength range , as in equation ( 3.1 ), while a theoretical E IR , th value is calculated by expanding this equation over [ 780 ; 3,000 nm ].