Voxels
 Volume Number: 9 Issue Number: 3 Column Tag: C Workshop

# What are Voxels?

## A hobbyist level overview of voxels, voxel space and Phong shading

By Geoffrey Clements, Chelmsford, Massachusetts

Note: Source code files accompanying article are located on MacTech CD-ROM or source code disks.

Geoffrey Clements can be reached on internet at clementsg@gw1.hanscom.af.mil.

## Introduction

The medical industry is studying ways of viewing medical imagery on the computer. Doctors are using computer graphics to plan operations before going into the operating room. In fact, entire operations are dry run on the computer, before the doctor lifts a knife.

To create the data for a medical image, a radiologist will take a series of computer aided tomography (CAT) scan images at different depths. The images are lined up in order of depth to form a three dimensional representation of the object being scanned. Each data point in this three dimensional cube is called a voxel. The cube of voxels are called a voxel space. Think of a voxel as a measure of the density of matter at that point. You can also think of a voxel as a “volume pixel”.

In this article we’ll draw a sphere that appears three dimensional using the techniques used to draw medical imagery. I chose a sphere to make the problem simpler to understand. But first we’ll check out how a medical image is generated.

A viewing window defines the orientation of the voxel space, and what part of the voxel space is in view. This viewing window corresponds to the computer screen. The image is created by following a ray from the viewing window through the voxels and summing the contributions of the voxels the ray passes through. The ray is perpendicular to the viewing window. Figure 1 shows two voxels spaces. The first has the viewing window on the front face. The second has the viewing window on the top left front corner. The arrow is the ray followed through the voxel space.

Figure 1 Voxel Space and Viewing Windows

The contribution of each voxel is calculated from a transparency and a color. The transparency, sometimes called opacity, is calculated from the intensities associated with each voxel. Usually the intensity is just scaled to yield an opacity. In our example, inside the sphere is opaque, and outside the sphere is transparent. Figure 2 shows the voxel space we will be using and the sphere opacity.

Figure 2 Voxel Space

To characterize the viewing environment we’ll define several vectors. First a normalized vector pointing toward the viewing window. Next we define normal vectors at each voxel. Finally we need a normal vector pointing toward each light source.

The color of a particular pixel is calculated using the intensities of the voxels and these vectors defining the viewing environment. We’ll be using the Phong shading model to calculate the color.

The Phong shading model is used in order to give objects in an image their three dimensional quality. We’ll go through the Phong shading equation by describing each piece. I use the words “shade” and “color” interchangeably. They mean the same thing. We start with:

Shade is the color of the voxel we are calculating. Ip is the intensity of a light source. ka is the ambient light reflection coefficient. The ambient light reflection coefficient, as you might expect, just sets an overall light level for the image. A gray disk is drawn if we use this equation to calculate the color.

Next, we add some depth cueing, which will start to give the image a 3D look.

kd1 and kd2 are depth cueing coefficients. We set kd2 = 1 in the example program. k is the distance from the viewing plane along the ray. This adds a small 3D effect. A gray disk is drawn that gets slightly lighter as we move from the edge of the sphere into the center.

Now we add the effect of diffuse and specular reflected light. (This step is a big jump, but these effects are what makes Phong shading so good.)

Figure 3 shows the difference between diffuse and specularly reflected light. Diffuse reflection is light reflected in all directions. This is caused by the roughness of the reflecting surface. Specular reflection, on the other hand, is light that is reflected in only one direction. Specular reflection is caused by the smoothness of the reflecting surface.

Figure 3 Reflected light

Here is the Phong shading equation with the effect of diffuse and specularly reflected light added.

d is the diffuse reflection coefficient, and s is the specular reflection coefficient. N is a normal vector to the voxel whose shade we are calculating, V is a vector pointing toward the viewer, L is a vector pointing at the light source and H is the normalized sum of V and L. N•L is the dot product of N and L.

We’ve come to the point where we can set our viewing window and define the vector space. Figure 4 shows the vector space. There are two L vectors shown in Figure 4. We’ll use two light sources to make the image a little more interesting. To add another light source we only need to add the effect of diffuse and specular reflection to the shade calculated above.

Figure 4 Vector space

The axes are slightly unusual, but the x and y axes correspond to the Macintosh coordinate system. N, H, L1, L2, and V are drawn to show their relative directions. This space corresponds to a viewer looking into the front face, (through the viewing window). The two light sources are pointing down from the top-left front corner of the cube, and into the right side. These vectors must be set before we start.

We can pre-calculate most of the shade equation before going into the loop that calculates the pixel color. This speeds up drawing the image.

Here are the definitions for V, L1, L2, H, and N.

r is the radius of the sphere, i, j, k are a point on the surface of the sphere. In our example, the center of the sphere is at (cv, cv, cv).

We can generate equations for N•L and N•H.

At this point you may be asking yourself, “How do you sum the contributions from the voxels?” Here’s how the example program does it. The indexes i and j step through the pixels on the viewing window. The index k corresponds to the ray going back into the voxels. As k varies, we check to see if point (i, j, k) is inside or outside the sphere. If it is outside the sphere, return a color of zero. If k gets to the back of the voxel space without hitting the sphere, draw a black pixel at (i, j) on the screen. If we hit the sphere, calculate a color for the sphere surface and draw that color at (i, j) on the screen, then go on to the next pixel. There is no need to process any farther inside the sphere, because there is no way the light can get there. Normally the effects of all voxels the ray passes through are used to calculate a color, but I have chosen to make the problem (and program) simpler to understand.

## The Example Program

The code was developed under Think C. The MacTraps library is the only library that needs to be included. Turn on the Native floating-point format and Generate 68881 instruction switches in the Compiler settings screen if you have a floating point coprocessor. If not, just turn on the Native floating-point format switch. The program takes about a minute to generate a 128x128 pixel image on a Mac IIci with 68881 instructions on, and three minutes with it off.

The example program uses a standard Macintosh event loop shell and main routine. All of the interesting processing is done by DoColor() called by the Init() routine. Init() starts by initializing the the Macintosh managers and setting up the menus. If Color Quickdraw is not available, the program quietly exits. If Color Quickdraw is available, a window is opened and sized to our voxel data. The size of the volume and sphere are set with the defines:

/* size of the voxel data */
#define volSize  64
/* half the size of voxel data */
#define halfVolSize32
/* the radius of the sphere */
#define sphere_r 30
/* sphere_r*sphere_r */
#define volumeMag900.0
/* sqrt(3.0)*sphere_r */
#define sqrt3r   51.9615
/* sphere_r*sqrt(6+2*sqrt(3)) */
#define rsqrt6   92.2913

Notice that some of the constants for the N•L and N•H equation are defined. This speeds up processing. Because this program is calculation intensive, start with a small volume and increase it later when you have the effect you like.

A grayscale palette is loaded and attached to the window using SetPalette(). The palette is a 128 shade grayscale 'pltt' resource created in ResEdit. An offscreen drawing port is used because we only want to calculate the pixel colors once. The offscreen port is set up using the GWorld calls defined in Volume VI of “Inside Macintosh”.

This brings us to the drawing section. The indexes i and j cycle through all the pixels of the viewing window. The k index is the ray moving back through the voxels. All of the work of deciding the color of a pixel is done in the DoColor() routine. Once a non-zero color is returned we move on to the next pixel in the viewing plane. Inside DoColor() the CalcVolumeData() routine calculates whether or not i, j, k is inside the sphere or not. If it is, we calculate a shade. If not, return a RGBColor of zero.

Once the drawing to the offscreen port is done, we set the current port to the onscreen window and exit Init(). At this point the screen is still blank. When WaitNextEvent() receives an update event we use CopyBits() to copy the offscreen bit map onto the screen.

## Stuff To Try

Use a small volume to start off. Start with 64x64x64 voxel set. The following are a couple of other sets of defines to try to get various sized spheres. Remember the bigger the voxel space the more time you have to get coffee while the program runs.

/* 1 */

#define volSize  128
#define halfVolSize64
#define sphere_r 60
#define volumeMag3600.0
#define sqrt3r   103.9230
#define rsqrt6   184.5827
and

#define volSize  256
#define halfVolSize128
#define sphere_r 120
#define volumeMag14400.0
#define sqrt3r   207.8461
#define rsqrt6   369.1654
The defines:

/* these constants define the Phong shading */
/* ambient reflection coefficient */
#define ambientReflCoef 0.1
/* depth cueing coefficient */
#define depthCueCoef 1.0
/* diffuse reflection coefficient */
#define diffReflCoef 2.0
/* specular reflection coefficient */
#define specReflCoef 3.0
/* first light source intensity */
#define light  0.6
/* second light source intensity */
#define light2   1.2
/* coefficient to approx highlight */
#define highlightCoef11

set the constants for the Phong shading. You can play around with these to change the shading effects in the displayed image. But be careful. The value of shade should fall between 0.0 and 1.0. If shade is greater than one, the color will roll over from white to black, and the image will appear with black blotches in the middle of an area that should be white.

A major performance improvement can be made by replacing the floating-point math with suitable integer arithmetic. Some improvement could be made by calling SetCPixel() from a pointer rather than leaving it to the trap dispatcher. Or, the code for the functions could be inserted into Init() to eliminate the overhead of the function calls.

Drawing in 3D is not hard; it just takes some math know-how and a good computer.

## References

Computer Graphics: Principles and Practice, 2nd ed., by J. D. Foley, A. Van Dam, S. K. Feiner, and J. F. Hughes (Addison-Wessley, 1990)

Marc Levoy, “Display of Surfaces from Volume Data,” IEEE Computer Graphics and Applications, May 1988 pp. 29-37

Code Listing
#include <Palettes.h>
#include <SANE.h>
#include <QDOffscreen.h>

/* size of the voxel data */
#define volSize  128
/* half the size of voxel data */
#define halfVolSize64
/* the radius of the sphere */
#define sphere_r 60
/* sphere_r*sphere_r */
#define volumeMag3600.0
/* sqrt(3.0)*sphere_r */
#define sqrt3r   103.9230
/* sphere_r*sqrt(6+2*sqrt(3)) */
#define rsqrt6   184.5827

/* resource numbers for the window, palette and menus */
#define windowRscr 128
#define paletteRscr 128

#define appleID 128
#define appleM 1

#define fileID 129
#define fileM 2
#define fileQuit 1

#define editID 130
#define editM 3
#define editUndo 1
#define editCut 3
#define editCopy 4
#define editPaste 5
#define editClear 6

#define sleepTicks 30

/* these constants define the Phong shading */
/* ambient reflection coefficient */
#define ambientReflCoef 0.1
/* depth cueing coefficient */
#define depthCueCoef 1.0
/* diffuse reflection coefficient */
#define diffReflCoef 5.0
/* specular reflection coefficient */
#define specReflCoef 5.0
/* first light source intensity */
#define light  1.0
/* coefficient to approx highlight */
#define highlightCoef30

char aChar;
WindowPtr currentWindow;
Rect dragRect, growRect;
long newSize;
Boolean doneFlag;
EventRecord event;
WindowPtr whichWindow;
RGBColor pixColor;
short i, j, k;
PaletteHandle palH;
DialogPtr dPtr;
short doneDlg;
OSErr err;
SysEnvRec envRec;

Rect copyRect;
GWorldPtr wallyWorld;
GDHandle savedDevice;
CGrafPtr savedPort;

double PowerOfN (double x, short r) {
double ans;

ans = 1.0;
while (r- > 0) ans *= x;
return ans;
}

double fx, fy, fz;

short CalcVolumeData (short i, short j, short k) {
long x, y, z;

fx = -(double)(i - halfVolSize);
fy = -(double)(j - halfVolSize);
fz = -(double)(k - halfVolSize);
if ((fx * fx + fy * fy + fz * fz) <= volumeMag)
return 1;
else return 0;
}

void DoColor (short i, short j, short k,
RGBColor *RGBVal) {
double n_dot_h, n_dot_l;
unsigned short color;

if (CalcVolumeData (i, j, k)) {
n_dot_l = (fx + fy + fz)/sqrt3r;
n_dot_h = (fx + fy + 2.7321*fz)/rsqrt6;
(light/((double)(k)+depthCueCoef)
*(diffReflCoef*n_dot_l+specReflCoef
*PowerOfN (n_dot_h, highlightCoef)));

/* second light source */
n_dot_l2 = -fx/sphere_r;
n_dot_h2 = (-fx + fz)/(1.4142*sphere_r);
*(diffReflCoef*n_dot_l2+specReflCoef
*PowerOfN (n_dot_h2, highlightCoef));

color = (unsigned short)(shade * 65534.0);

RGBVal->red = color;
RGBVal->green = color;
RGBVal->blue = color;
}
else {
RGBVal->red = 0;
RGBVal->green = 0;
RGBVal->blue = 0;
}
}

void OpenWindow (void) {
currentWindow =  (WindowPtr)GetNewCWindow(
windowRscr, NULL, (Ptr)-1);
SetPort(currentWindow);
SizeWindow(currentWindow, volSize + 25,
volSize + 25, 1);
SetWTitle(currentWindow, &”\pVol3D”);
ShowWindow(currentWindow);
}

void Init (void) {
short i, j, k;

InitGraf(&thePort);
InitFonts ();
FlushEvents (everyEvent, 0);
InitWindows ();
TEInit ();
InitDialogs (NULL);

for (i=appleM;i<=editM;i++)

SetRect(&dragRect, 30, 20,
screenBits.bounds.right - 10,
screenBits.bounds.bottom - 30);
SetRect(&growRect, 50, 50,
screenBits.bounds.right - 20,
screenBits.bounds.bottom - 50);

doneFlag = 0;
err = SysEnvirons(1, &envRec);
if (!envRec.hasColorQD) doneFlag = 1;
else {
OpenWindow ();
palH = GetNewPalette (paletteRscr);
if (palH == NULL) {
doneFlag = 1;
}
else {
SetPalette (currentWindow, palH, 1);
}

/* set up the offscreen drawing port */
GetGWorld (&savedPort, &savedDevice);
SetRect (&copyRect, 0, 0, volSize-1,
volSize-1);
LocalToGlobal (&copyRect.top);
LocalToGlobal (&copyRect.bottom);
err = NewGWorld (&wallyWorld, 0, &copyRect,
NULL, NULL, 0);
GlobalToLocal (&copyRect.top);
GlobalToLocal (&copyRect.bottom);

if (err != noErr)
doneFlag = 1;
else {
SetGWorld (wallyWorld, NULL);
if (LockPixels (wallyWorld->portPixMap)) {
/* draw off screen here */
for(i=0;i<volSize;i++)
for (j=0;j<volSize;j++) {
k = 0;
do {
DoColor(i, j, k, &pixColor);
k++;
} while ((pixColor.red == 0)
& (k < volSize));
SetCPixel (i, j, &pixColor);
}
UnlockPixels (wallyWorld->portPixMap);
}
else doneFlag = 1;

/* the drawing is done set the current port back to the display window
*/
}
SetGWorld (savedPort, savedDevice);
}
}

(Ptr)-1);
do
ModalDialog(NULL, &doneDlg);
while (!doneDlg);
DisposDialog(dPtr);
}

void CleanUp (void) {

HideWindow (currentWindow);
DisposeGWorld (wallyWorld);
DisposePalette (palH);
DisposeWindow (currentWindow);
doneFlag = 1;
}

Str255 daName;
short daErr;

case appleID:
else {
daErr = OpenDeskAcc(daName);
if (currentWindow)
SetPort (currentWindow);
}
break;
case fileID:
case fileQuit:
CleanUp ();
break;
}
break;
}
}

void DoEvent (void) {

switch (event.what) {
case mouseDown:
switch (FindWindow(event.where,
&whichWindow)) {
break;
case inSysWindow:
SystemClick(&event, whichWindow);
break;
case inDrag:
DragWindow(whichWindow, event.where,
&dragRect);
break;
case inGrow:
newSize = GrowWindow(whichWindow,
event.where, &growRect);
SizeWindow(whichWindow, LoWord(newSize),
HiWord(newSize), 1);
InvalRect(&currentWindow->portRect);
break;
case inGoAway:
if (TrackGoAway(whichWindow,
event.where)) CleanUp ();
break;
} /* case findwindow (...) */
break;
case keyDown:
case autoKey:
aChar = (char)(BitAnd (event.message,
if (BitAnd (event.modifiers, cmdKey))
break;
case activateEvt:
if (BitAnd(event.modifiers, activeFlag))
break;

case updateEvt:
BeginUpdate(currentWindow);
EraseRect(&currentWindow->portRect);
DrawGrowIcon(currentWindow);
InsetRect (&currentWindow->portRect, 8, 8);
OffsetRect (&currentWindow->portRect,
-8, -8);

if (LockPixels (wallyWorld->portPixMap)) {
CopyBits(&wallyWorld->portPixMap,
&currentWindow->portBits, &copyRect,
&currentWindow->portRect, srcCopy, NULL);
UnlockPixels (wallyWorld->portPixMap);
}

OffsetRect (&currentWindow->portRect, 8, 8);
InsetRect (&currentWindow->portRect, -8, -8);
EndUpdate(currentWindow);
break;
}
}

void main (void) {
currentWindow = NULL;
Init ();
InitCursor ();

do {
if (WaitNextEvent (everyEvent, &event,
sleepTicks, NULL)) DoEvent ();
} while (!doneFlag);
}

Community Search:
MacTech Search:

Parallels Desktop 10.2.0 - Run Windows a...
Parallels Desktop is simply the world's bestselling, top-rated, and most trusted solution for running Windows applications on your Mac. With Parallels Desktop for Mac, you can seamlessly run both... Read more
LaunchBar 6.2 - Powerful file/URL/email...
LaunchBar is an award-winning productivity utility that offers an amazingly intuitive and efficient way to search and access any kind of information stored on your computer or on the Web. It provides... Read more
Firefox 37.0 - Fast, safe Web browser. (...
Firefox offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals and casual... Read more
Arq 4.11 - Online backup to Google Drive...
Arq is super-easy online backup for the Mac. Back up to your own Google Drive storage (15GB free storage), your own Amazon Glacier (\$.01/GB per month storage) or S3, or any SFTP server. Arq backs up... Read more
MacFamilyTree 7.3.4 - Create and explore...
MacFamilyTree gives genealogy a facelift: it's modern, interactive, incredibly fast, and easy to use. We're convinced that generations of chroniclers would have loved to trade in their genealogy... Read more
Yummy FTP 1.10.2 - FTP/SFTP/FTPS client...
Yummy FTP is an FTP + SFTP + FTPS file transfer client which focuses on speed, reliability and productivity. Whether you need to transfer a few files or a few thousand, schedule automatic backups, or... Read more
VueScan 9.5.08 - Scanner software with a...
VueScan is a scanning program that works with most high-quality flatbed and film scanners to produce scans that have excellent color fidelity and color balance. VueScan is easy to use, and has... Read more
Iridient Developer 3.0.1 - Powerful imag...
Iridient Developer (was RAW Developer) is a powerful image conversion application designed specifically for OS X. Iridient Developer gives advanced photographers total control over every aspect of... Read more
Monodraw 0.8.4.1 - Powerful ASCII art ed...
Monodraw allows you to easily create text-based art (like diagrams, layouts, flow charts) and visually represent algorithms, data structures, binary formats and more. Because it's all just text, it... Read more
Air Video Server HD 2.1.0 - Stream video...
Air Video Server HD streams videos instantly from your computer on your iPhone, iPad, iPod touch or Apple TV. No need to worry about converting or transferring files. We took everything that was... Read more

## Latest Forum Discussions

Marvel Mighty Heroes, the Ultimate Marve...
DeNA and Marvel Entertainment have brought us a new action-packed brawler: Marvel Mighty Heroes. You can play with up to four of your friends as you favorite marvel heroes like Iron Man, Thor, Spider-Man, Captain America, Star-Lord, Hulk, and... | Read more »
Take Note - The Jot Script 2 Evernote Ed...
Adonit's Jot Script 2 sylus has an all-new edition that's meant to be Evernote's BFF. The sylus has been redesigned to work better with iPads and give you faster stroke tracking, smoother line rendering, and better tip-to-line accuracy. | Read more »
INFINIT Lets Mobile Users Send Files of...
Infinit  is a file sharing app that ignores file size limits. It maintains the original quality of your photos and videos so you can be sure they look awesome after sending. You can move entire albums of photos or full HD films from your iPhone to... | Read more »
Serious Sam Double D Developer is Making...
There's nothing like the thrill of watching a horse race, especially when the horses are drunk.  | Read more »
2K Announces WWE 2K, Mobile's First...
It seems like this month has been pretty big for wrestling. First Wrestlemania, then 2K has announces that they're releasing  WWE 2K for iOS. It's a simulation-based WWE game where you'll get to play with several WWE superstars such as John Cena, ... | Read more »
How the Apple Watch Could Change the Fac...
The Apple Watch is still a ways out, but my previous musings on the wearable’s various features got me thinking: what might it be like a year after launch? Two years? Five years? What if it becomes a symbiotic part of the iOS framework to the point... | Read more »
You Can Start Challenging Your Friends t...
Last year we reported that Sebastian Gosztyla's new game, Dual, was theorized to be released sometime during the summer of 2014. Sadly that did not become a reality, but the good news it there is now an official release date. | Read more »
Forgotten Memories : Alternate Realities...
Forgotten Memories : Alternate Realities, from Psychoz Interactive, is planned for release on April 23. The Resident Evil, Silent Hill, and Alone in the Dark inspired horror game puts you in the shoes of Rose Hawkins as she searches for a missing... | Read more »
Pie In The Sky: A Pizza Odyssey (Games)
Pie In The Sky: A Pizza Odyssey 1.0 Device: iOS Universal Category: Games Price: \$2.99, Version: 1.0 (iTunes) Description: A game about delivering pizza. In space. | Read more »
Chosen Gives Hopeful Singers, Songwriter...
If YouTube videos and reality TV shows like The Voice have taught us one thing, it’s that there are a lot of people out there who are anxious to show the world their talents. And if they’ve taught us a second thing, it’s that there’s an almost... | Read more »

## Price Scanner via MacPrices.net

Will Microsoft’s Surface 3 Give The 12-inch M...
The more I ruminate over the new 12-inch MacBook, the more it occurs that it’s probably going to be more of a cannibalization threat to high-end iPads (including a new 12-inch ‘Pad when/if that... Read more
13-inch 2.4GHz Retina MacBook Pro available f...
MacMall has the 2013 13″ 2.4GHz/128GB Retina MacBook Pro available for \$949.99 for a limited time. Shipping is free. Their price is \$350 off original MSRP, and it’s the only sub-\$1000 new Retina... Read more
Adobe today announced the availability of Adobe Comp CC, a free iPad app that enables rapid creation of layout concepts for mobile, Web and print projects. With Comp CC, designers can rough out and... Read more
13-inch 2.6GHz/256GB Retina MacBook Pro avail...
Best Buy has clearance 2014 13″ 2.6GHz/256GB Retina MacBook Pros available for \$1199.99 including free shipping. Their price is \$300 off original MSRP, and it’s the lowest price for this model.... Read more
Updated Mac Price Trackers
We’ve updated our Mac Price Trackers with the latest information on prices, bundles, and availability on systems from Apple’s authorized internet/catalog resellers: - 15″ MacBook Pros - 13″ MacBook... Read more
21-inch 1.4GHz iMac on sale for \$999, save \$1...
Best Buy has the 21″ 1.4GHz iMac on sale for \$999.99 on their online store. Choose free shipping or free local store pick up. Price is for online orders only, in-store prices may vary. Their price is... Read more
2.6GHz Mac mini on sale for \$649, save \$50
Amazon has the 2.6GHz Mac mini on sale for \$649.99 including free shipping. Their price is \$50 off MSRP, and it’s the lowest price available for this model. Read more
Textkraft Professional 3.2 Powerful iPad Text...
Finally it’s springtime, at least theoretically in my neck of the woods, where we’re still navigating canyons between towering snowbanks with temperatures well below freezing in winter weather that... Read more
Apple offering refurbished 27-inch 5K iMacs f...
The Apple Store is offering Apple Certified Refurbished 27″ 3.5GHz 5K iMacs for \$2119 including free shipping. Their price is \$380 off the price of new models, and it’s the lowest price available for... Read more
16GB iPad mini on sale for \$199, save \$50
Walmart has 16GB iPad minis (1st generation) available for \$199.99 on their online store, including free shipping. Their price is \$50 off MSRP. Online orders only. Read more

## Jobs Board

DevOps Software Engineer - *Apple* Pay, iOS...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more
*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
Sr. Technical Services Consultant, *Apple*...
**Job Summary** Apple Professional Services (APS) has an opening for a senior technical position that contributes to Apple 's efforts for strategic and transactional Read more
Lead *Apple* Solutions Consultant - Retail...
**Job Summary** Job Summary The Lead ASC is an Apple employee who serves as the Apple business manager and influencer in a hyper-business critical Reseller's store Read more
*Apple* Pay - Site Reliability Engineer - Ap...
**Job Summary** Imagine what you could do here. At Apple , great ideas have a way of becoming great products, services, and customer experiences very quickly. Bring Read more