What is the first moment of the image irradiance?

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Recently I took on the task of surface reconstruction. And now I try to take a grasp on how to find the illumination direction given gray-scale image under Lambertian assumption. The formula I stuck on is the first foment of the image irradiance:



$$ mu_1= E( E(alpha,beta))=int_0^2pi int_0^pi/2 E(alpha,beta)f(alpha,beta) dbeta dalpha $$



where:



$E(.)$ - expectation operator



$E(alpha,beta)$ - image irradiance



$f(alpha,beta)$ - distribution of the normal vectors



So, my questions are the following:



  1. How to understand what the first moment of the image irradiance is and how to depict it on spherical coordinate system?


  2. Regarding $f(alpha,beta)$ author states that the number of normal vectors with slant equal to $beta$ will be proportional to cos $beta$, hence can be assumed as $f(alpha,beta) = fraccosbeta2pi.$ But I can't interpret it geometrically even though I understand what Gaussian distribution is.


Thanks in advance.




surface-integrals image-processing




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  • $cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
    – Yves Daoust
    Jul 18 at 16:00















up vote
0
down vote

favorite












Recently I took on the task of surface reconstruction. And now I try to take a grasp on how to find the illumination direction given gray-scale image under Lambertian assumption. The formula I stuck on is the first foment of the image irradiance:



$$ mu_1= E( E(alpha,beta))=int_0^2pi int_0^pi/2 E(alpha,beta)f(alpha,beta) dbeta dalpha $$



where:



$E(.)$ - expectation operator



$E(alpha,beta)$ - image irradiance



$f(alpha,beta)$ - distribution of the normal vectors



So, my questions are the following:



  1. How to understand what the first moment of the image irradiance is and how to depict it on spherical coordinate system?


  2. Regarding $f(alpha,beta)$ author states that the number of normal vectors with slant equal to $beta$ will be proportional to cos $beta$, hence can be assumed as $f(alpha,beta) = fraccosbeta2pi.$ But I can't interpret it geometrically even though I understand what Gaussian distribution is.


Thanks in advance.




surface-integrals image-processing




share|cite|improve this question





















  • $cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
    – Yves Daoust
    Jul 18 at 16:00













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Recently I took on the task of surface reconstruction. And now I try to take a grasp on how to find the illumination direction given gray-scale image under Lambertian assumption. The formula I stuck on is the first foment of the image irradiance:



$$ mu_1= E( E(alpha,beta))=int_0^2pi int_0^pi/2 E(alpha,beta)f(alpha,beta) dbeta dalpha $$



where:



$E(.)$ - expectation operator



$E(alpha,beta)$ - image irradiance



$f(alpha,beta)$ - distribution of the normal vectors



So, my questions are the following:



  1. How to understand what the first moment of the image irradiance is and how to depict it on spherical coordinate system?


  2. Regarding $f(alpha,beta)$ author states that the number of normal vectors with slant equal to $beta$ will be proportional to cos $beta$, hence can be assumed as $f(alpha,beta) = fraccosbeta2pi.$ But I can't interpret it geometrically even though I understand what Gaussian distribution is.


Thanks in advance.




surface-integrals image-processing




share|cite|improve this question













Recently I took on the task of surface reconstruction. And now I try to take a grasp on how to find the illumination direction given gray-scale image under Lambertian assumption. The formula I stuck on is the first foment of the image irradiance:



$$ mu_1= E( E(alpha,beta))=int_0^2pi int_0^pi/2 E(alpha,beta)f(alpha,beta) dbeta dalpha $$



where:



$E(.)$ - expectation operator



$E(alpha,beta)$ - image irradiance



$f(alpha,beta)$ - distribution of the normal vectors



So, my questions are the following:



  1. How to understand what the first moment of the image irradiance is and how to depict it on spherical coordinate system?


  2. Regarding $f(alpha,beta)$ author states that the number of normal vectors with slant equal to $beta$ will be proportional to cos $beta$, hence can be assumed as $f(alpha,beta) = fraccosbeta2pi.$ But I can't interpret it geometrically even though I understand what Gaussian distribution is.


Thanks in advance.





surface-integrals image-processing





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 18 at 15:52
























asked Jul 16 at 17:46









Taras Mykhalchuk

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  • $cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
    – Yves Daoust
    Jul 18 at 16:00

















  • $cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
    – Yves Daoust
    Jul 18 at 16:00
















$cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
– Yves Daoust
Jul 18 at 16:00





$cosbeta$ is a geometric factor which probably accounts for the fact that a surface viewed under an angle has a smaller apparent area.
– Yves Daoust
Jul 18 at 16:00
















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