<div dir="ltr">Spatially variant erosions and dilations are easy enough to visualise but, as you've discovered, openings and closings are less obvious. Some issues are discussed in the following papers.<div><br></div><div><span class="" id="LC155" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(63,110,116)">@Article</span>{cuisenaire2006locally,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC156" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> title = <span class="" style="color:rgb(196,26,22)">"Locally Adaptable Mathematical Morphology Using</span></span><span class="" style="color:rgb(196,26,22);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC157" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(196,26,22)"> Distance Transformations"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC158" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> author = <span class="" style="color:rgb(196,26,22)">"O. Cuisenaire"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC159" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> journal = <span class="" style="color:rgb(196,26,22)">"Pattern Recognition"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC160" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> volume = <span class="" style="color:rgb(196,26,22)">"39"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC161" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> number = <span class="" style="color:rgb(196,26,22)">"3"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC162" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> pages = <span class="" style="color:rgb(196,26,22)">"405--416"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC163" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> year = <span class="" style="color:rgb(196,26,22)">"2006"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC164" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> publisher = <span class="" style="color:rgb(196,26,22)">"Elsevier"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC165" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">}</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC166" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"></span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC167" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(63,110,116)">@Article</span>{bouaynaya2007theoretical2,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC168" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> title = <span class="" style="color:rgb(196,26,22)">"Theoretical Foundations of Spatially-variant</span></span><span class="" style="color:rgb(196,26,22);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC169" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(196,26,22)"> Mathematical Morphology Part {II}: Gray-level Images"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC170" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> author = <span class="" style="color:rgb(196,26,22)">"N. Bouaynaya and D. Schonfeld"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC171" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> journal = <span class="" style="color:rgb(196,26,22)">"IEEE Transactions on Pattern Analysis and Machine</span></span><span class="" style="color:rgb(196,26,22);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC172" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(196,26,22)"> Intelligence"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC173" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> volume = <span class="" style="color:rgb(196,26,22)">"30"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC174" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> number = <span class="" style="color:rgb(196,26,22)">"5"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC175" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> pages = <span class="" style="color:rgb(196,26,22)">"837--850"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC176" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> year = <span class="" style="color:rgb(196,26,22)">"2008"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC177" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> publisher = <span class="" style="color:rgb(196,26,22)">"Published by the IEEE Computer Society"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC178" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">}</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC179" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"></span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC180" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(63,110,116)">@Article</span>{bouaynaya2007theoretical1,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC181" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> title = <span class="" style="color:rgb(196,26,22)">"Theoretical Foundations of Spatially-variant</span></span><span class="" style="color:rgb(196,26,22);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC182" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(196,26,22)"> Mathematical Morphology Part {I}: Binary Images"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC183" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> author = <span class="" style="color:rgb(196,26,22)">"N. Bouaynaya and D. Schonfeld"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC184" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> journal = <span class="" style="color:rgb(196,26,22)">"IEEE Transactions on Pattern Analysis and Machine</span></span><span class="" style="color:rgb(196,26,22);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC185" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><span class="" style="color:rgb(196,26,22)"> Intelligence"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC186" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> volume = <span class="" style="color:rgb(196,26,22)">"30"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC187" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> number = <span class="" style="color:rgb(196,26,22)">"5"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC188" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> pages = <span class="" style="color:rgb(196,26,22)">"823--836"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC189" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> year = <span class="" style="color:rgb(196,26,22)">"2008"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC190" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"> publisher = <span class="" style="color:rgb(196,26,22)">"Published by the IEEE Computer Society"</span>,</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span><span class="" id="LC191" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">}</span><span style="color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">
</span></div><div><span class="" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><br></span></div><div><span class="" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre">I've also done some work on the efficient implementations for circular/spherical structuring elements on binary images:</span></div><div><span class="" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><br></span></div><div><span class="" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco">@InProceedings{Beare2011j,</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco"> author = {Beare, R. and Jackway, P.},</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco"> title = {Parallel Algorithms Via Scaled Paraboloid Structuring Function</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco">s for <span style="color:rgb(255,255,255);background-color:rgb(0,0,0)">Spati</span>ally-Variant and Label-Set Dilations and Erosions},</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco"> booktitle = {International Conference on Digital Image Computing: Techniques and Applications (DICTA) },</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco"> year = 2011,</p><p style="margin:0px;font-size:10px;line-height:normal;font-family:Monaco"> month = {December}}</p>
</span></div><div><span class="" style="display:inline;color:rgb(0,0,0);font-family:Menlo,Monaco,Consolas,'Courier New',monospace;font-size:12px;line-height:16px;white-space:pre"><br></span></div></div><div class="gmail_extra"><br><div class="gmail_quote">On Wed, Sep 30, 2015 at 11:05 PM, Daoued23 <span dir="ltr"><<a href="mailto:mlt.khawla@gmail.com" target="_blank">mlt.khawla@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Hi everyone,<br>
<br>
I want to implement a spatially variant morphological closing, which its<br>
structuring element is estimated for every pixel. I work on binary images.<br>
<br>
<br>
So to apply the aspatially variant morphological closing, we estimate the<br>
structuring element for every pixel surrounded by its neighborhood, apply<br>
the dilation on every neighbouring window with the corresponding estimated<br>
structring element. Then we get the dilated image as the union of all the<br>
dilated windows. Now, we apply the erosion on every neighbouring window of<br>
the dilated image with the transposed of the structuring element already<br>
estimated. The adaptively closed image will be the union of the eroded<br>
windows of the dilated image.<br>
<br>
Now the problem is that, we find background windows (all pixels are black)<br>
in the image, so we cannot estimate a structuring element. when it's the<br>
phase of the dilation we don't apply anything, but some pixels can appear in<br>
this windows as the result of the union of the surrounding windows that have<br>
been dilated. When applying the erosion on these new pixels which<br>
structuring element should we consider knowing that we can't estimate one<br>
based on the original image (background window)???<br>
<br>
Thank you<br>
<br>
Best regards<br>
<br>
<br>
<br>
--<br>
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