{"id":1412,"date":"2025-06-03T12:39:14","date_gmt":"2025-06-03T10:39:14","guid":{"rendered":"https:\/\/cammonte.com\/?page_id=1412"},"modified":"2025-06-03T13:42:17","modified_gmt":"2025-06-03T11:42:17","slug":"volume-rendering-draft","status":"publish","type":"page","link":"https:\/\/cammonte.com\/index.php\/volume-rendering-draft\/","title":{"rendered":"Volume Rendering Draft"},"content":{"rendered":"\n

When traveling through a volume, light interacts with it, resulting in a change in radiance<\/mark><\/strong><\/p>\n\n\n\n

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Change in radiance [math]dL[\/math] along [math]ds[\/math]:<\/mark><\/strong><\/p>\n\n\n\n

[math]dL(x, \\omega) = L_o(x, \\omega) – L_i(x, \\omega) = \\text{emission} + \\text{in-scattering} – \\text{out-scattering} – \\text{absorption}[\/math]<\/mark><\/strong><\/p>\n\n\n\n

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  • Radiance [math]L[\/math]<\/strong>: intensity of the light beam after it has traveled through the volume\n
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    • Incoming radiance [math]L_i[\/math]:<\/strong> intensity of light shone on the cylinder<\/li>\n\n\n\n
    • Outgoing radiance [math]L_o[\/math]<\/strong>: how light much leaves the volume on the other end<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • View direction [math]\\omega[\/math]<\/strong>: camera ray direction<\/li>\n<\/ul>\n\n\n\n

      Light\/Medium Interactions<\/h1>\n<\/div>\n\n\n\n
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      \"\"<\/figure>\n<\/div>\n<\/div>\n\n\n\n
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      • Radiance decreasing interactions<\/strong>\n
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        • Absorption<\/strong>: some of the light is absorbed<\/li>\n\n\n\n
        • Out-scattering<\/strong>: photons making up the light beam traveling towards the eye are scattered in another direction<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Radiance increasing interactions<\/strong>\n
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          • In-scattering<\/strong>: some of the light is redirected towards the eye<\/li>\n\n\n\n
          • Emission<\/strong>: medium emits light, electrons gain energy released in the form of photons. Directions taken by these photons are random but some of them will travel towards the eye<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n\n\n\n
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            \"\"<\/figure>\n<\/div>\n<\/div>\n\n\n\n

            Absorption (Beer-Lambert Law)<\/h2>\n\n\n\n

            Internal transmittance: amount of light absorbed by the volume as light travels a certain distance through it<\/mark><\/strong><\/p>\n\n\n\n

            [math]T = e^{- \\text{distance} \\cdot \\sigma_a}[\/math]<\/mark><\/strong><\/p>\n\n\n\n

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            • [math]T=0[\/math]: volume blocks all light; [math]T=1[\/math]: all light transmitted<\/li>\n\n\n\n
            • Absorption coeffcient [math]\\sigma_a[\/math]:<\/strong> the denser the volume, the higher the absorption coefficient<\/li>\n\n\n\n
            • Colour that we see when looking at a surface with colour [math]c_s[\/math] through a volume with colour [math]c_v[\/math] is given by:<\/li>\n<\/ul>\n\n\n\n

              [math]C = T \\cdot c_s + (1 – T) \\cdot c_v[\/math]<\/p>\n\n\n\n

              Out-scattering<\/h2>\n\n\n\n

              Light initially traveling towards the eye through a volume interacts with particles in the volume, changing its direction<\/mark><\/strong><\/p>\n\n\n\n

              In-scattering<\/h2>\n\n\n\n

              Light passing through a volume is redirected toward the eye due to a scattering event<\/mark><\/strong><\/p>\n\n\n\n

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              • In-scattering can happen anywhere between [math]t_0[\/math] and [math]t_1[\/math]<\/li>\n\n\n\n
              • Amount scattered toward the eye’s direction [math]\\omega[\/math] where [math]x[\/math] is a point on the segment [math]t_0[\/math] to [math]t_1[\/math]:<\/li>\n<\/ul>\n\n\n\n

                [math]\\int_{x=t_0}^{t_1} L_i(x, \\omega) dx[\/math]<\/p>\n<\/div>\n\n\n\n

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                \"\"<\/figure>\n<\/div>\n<\/div>\n\n\n\n
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                No analytical solution -> approximate through Riemann’s sum<\/strong>:<\/p>\n\n\n\n

                For each segment [math]dx[\/math]:<\/p>\n\n\n\n

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                • Shoot a ray from [math]x[\/math] (middle of subsegment) towards light source to get the distance light has traveled through the volume between light source and [math]x[\/math]<\/li>\n\n\n\n
                • Beer’s law gives amount of light [math]L_i(x, \\omega)[\/math] arriving at [math]x[\/math]<\/li>\n\n\n\n
                • Multiply [math]L_i(x, \\omega)[\/math] by [math]dx[\/math] and accumulate the contribution<\/li>\n<\/ul>\n<\/div>\n\n\n\n
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                  \"\"<\/figure>\n<\/div>\n<\/div>\n\n\n\n

                  Two things missing:<\/p>\n\n\n\n

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                  • We dont know what fraction of light is actually being in-scattered<\/li>\n\n\n\n
                  • The in-scattered light will also be absorbed on it’s way back to the eye<\/li>\n<\/ul>\n\n\n\n

                    Coefficients<\/h2>\n\n\n\n
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                    Absorption coefficient [math]\\sigma_a[\/math]<\/strong><\/p>\n\n\n\n

                    Absorption increases whether you double either the absorption coefficient or the distance traveled by the light through the volume<\/p>\n<\/div>\n\n\n\n

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                    Scattering coefficient [math]\\sigma_s[\/math]<\/strong><\/p>\n\n\n\n

                    In-scattering and out-scattering are part of the same scattering phenomenon -> proba that a photon is being scattered is the same and defined by single coefficient<\/p>\n<\/div>\n\n\n\n

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                    Extinction coefficient [math]\\sigma_t[\/math]<\/strong><\/p>\n\n\n\n

                    [math]\\sigma_t = \\sigma_a + \\sigma_s[\/math]<\/strong><\/p>\n\n\n\n

                    Out-scattering and absorption both result in loss in radiance -> combine them when calculating this loss -> extinction or attenuation coefficient<\/p>\n<\/div>\n<\/div>\n\n\n\n

                      \n
                    • They all represent probability densities<\/strong>: likelihood that a random event (ex: absorption event) occurs at a given point<\/li>\n\n\n\n
                    • Unit<\/strong>: reciprocal distance, inverse of a distance\n
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                      • Mean free path<\/strong>: average distance a photon will travel through a volume before the photon and the medium interact with one another, inverse of the coefficient<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

                        When traveling through a volume, light interacts with it, resulting in a change in radiance Change in radiance [math]dL[\/math] along [math]ds[\/math]: [math]dL(x, \\omega) = L_o(x, \\omega) – L_i(x, \\omega) = \\text{emission} + \\text{in-scattering} – \\text{out-scattering} – \\text{absorption}[\/math] Light\/Medium Interactions Absorption (Beer-Lambert Law) Internal transmittance: amount of light absorbed by the volume as light travels a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ub_ctt_via":"","site-container-style":"default","site-container-layout":"default","site-sidebar-layout":"default","disable-article-header":"default","disable-site-header":"default","disable-site-footer":"default","disable-content-area-spacing":"default","footnotes":""},"class_list":["post-1412","page","type-page","status-publish","hentry"],"featured_image_src":null,"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/pages\/1412","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/comments?post=1412"}],"version-history":[{"count":13,"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/pages\/1412\/revisions"}],"predecessor-version":[{"id":1428,"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/pages\/1412\/revisions\/1428"}],"wp:attachment":[{"href":"https:\/\/cammonte.com\/index.php\/wp-json\/wp\/v2\/media?parent=1412"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}