Anti-aliasing lines when using CATransform3DRotate

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I copied the code from the answer to How do I apply a perspective transform to a UIView? in order to achieve the following image. Note, however, the jagged lines separating the levels. Is there away to perform this kind of perspective transformation in a way that anti-aliases these jagged lines?

enter image description here

Edit I've tried the solutions offered by DarkDust in the comments on this post. None seem to work. Here's my code:

    levelView = [[UIControl alloc] initWithFrame:CGRectMake(100, 60, 160, 230)];
    [self addSubview:levelView];
    levelView.transform = CGAffineTransformScale(self.transform, .8, .8);

    CALayer *layer = levelView.layer;

    //DarkDust's 2nd comment
    layer.borderWidth = 1;
    layer.borderColor = [[UIColor clearColor] CGColor];

    CATransform3D rotationAndPerspectiveTransform = CATransform3DIdentity;
    rotationAndPerspectiveTransform.m34 = 1.0 / 1000;
    rotationAndPerspectiveTransform = CATransform3DRotate(rotationAndPerspectiveTransform, 45.0f * M_PI / 180.0f, 0.0f, 1.0f, 0.0f);
    layer.transform = rotationAndPerspectiveTransform;

    for (NSString *ID in [NSArray arrayWithObjects:@"Level 1", @"Level 2", @"Level 3", @"Level 4", @"Level 5", nil]) {

        //Note the offset of 5 from the superview in both X and Y directions. This addresses DarkDusk's first comment
        UIControl *ctrl = [[UIControl alloc] initWithFrame:CGRectMake(5, y + 5, levelView.frame.size.width, 220/5)];
        [levelView addSubview:ctrl];
        [ctrl addTarget:self action:@selector(languageSelected:) forControlEvents:UIControlEventTouchDown];
        [self styleEnabled:ctrl];            

        UILabel *label = [[UILabel alloc] initWithFrame:CGRectMake(13, 0, ctrl.frame.size.width - 13, ctrl.frame.size.height)];
        [ctrl addSubview:label];
        label.text = ID;
        label.font = [UIFont systemFontOfSize:20];
        label.textColor = [UIColor colorWithRed:1 green:1 blue:1 alpha:.8];
        label.backgroundColor = [UIColor clearColor];

        y += ctrl.frame.size.height;
    }

The level buttons you see in the image are all instances of UIControl, and they are the direct subviews of UIControl *levelView. levelView is the one that is getting transformed.

The other two of DarkDusk's comments refer specifically to UIImages, which I am not using. If they are still applicable and someone can explain how to implement them, I'll certainly give them a shot.

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iOS 7 brings a new property allowsEdgeAntialiasingto CALayer, as you can see here. When set to YES, your layer will transform with anti-aliased edges.

/* When true this layer is allowed to antialias its edges, as requested
 * by the value of the edgeAntialiasingMask property.
 *
 * The default value is read from the boolean UIViewEdgeAntialiasing
 * property in the main bundle's Info.plist. If no value is found in
 * the Info.plist the default value is NO. */

@property BOOL allowsEdgeAntialiasing;